Lars Koopman
Impact of Modeling Assumptions
on the Holistic Balance Sheet
MSc Thesis 2013-035
Impact of Modeling Assumptions on the
Holistic Balance Sheet
by
Lars Koopman [212850]
[BSc. Tilburg University 2012]
A thesis submitted in partial fulfillment of the requirements for the
degree of Master of Science in Quantitative Finance & Actuarial
Science
Tilburg School of Economics and Management
Tilburg University
Supervisors:
Prof. Dr. J.M. Schumacher
Drs. E.A.J.P. Fransen AAG
October 22, 2013
Abstract
EIOPA is forced to provide more guidance concerning the modeling assumptions
for the holistic balance sheet, before implementation of the new supervisory
framework for pension funds within the European Union, IORP II, is possible.
When varying the underlying short rate models, this results in significantly
different holistic balance sheets and solvency capital requirements. The impacts
on the SCR when changing between the Hull-white one- and two-factor are in
the order of magnitude of -0.8% to 1.2% of the current value of liabilties for our
data.
The impact on the holistic balance sheet is less sensitive to the variation in
short rate parameters per model, where the variation in parameters comes from
altering the weights in a short rate calibration over a set of liquid swaptions.
We see that the Hull-White one- and two-factor are suitable for modeling purposes, however the Cox-Ingersoll-Ross++ model is not suitable due to its large
probability of generating complex values.
We also see that the correlation structure does have an impact on the holistic
balance sheet and solvency capital requirement. The exact size of the impact
is yet to be investigated, however the implementation of correlations in the
economic scenario generator resulted in a decrease in the capital requirement.
Due to the assumption of a static portfolio for our fictitious fund, we have that
the impact on the SCR may be positive for a different correlation structure as
well.
Besides modeling assumptions for the generating of risk-neutral scenarios,
this thesis also discussed the impact on the holistic balance sheet and solvency
capital requirement when varying policies. The more steering mechanisms available to the fund, the higher the loss absorbing capacity will be. This results in
a significant decrease of the solvency capital requirement.
Finally the impact on the solvency capital requirement is shown when the
evaluation horizon of the options of the holistic balance sheet decreases from
15 to 10 years. This results in a decrease of the loss absorbing capacity, after
which the resulting solvency capital requirement decreases as well. Although
it may be difficult for EIOPA to argument some of their decisions concerning
modeling assumptions for the holistic balance sheet, such as the length of the
evaluation horizon, one does have to keep in mind the main purpose of the
solvency capital requirement. This is to prevent any large shocks on the pensions
of the particants of a pension fund over time.
Contents
1 Introduction
1.1 About this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Structure Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 From Solvency II to IORP II using Steering Mechanisms
2.1 FTK & IORP II . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Steering Mechanisms . . . . . . . . . . . . . . . . . . . . . .
2.3 The Holistic Balance Sheet . . . . . . . . . . . . . . . . . .
2.4 Calculating the SCR . . . . . . . . . . . . . . . . . . . . . .
2.5 Main Strategy throughout this Thesis . . . . . . . . . . . .
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3 Calibration of Short Rate Models
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3.1 Data & Optimization Problem . . . . . . . . . . . . . . . . . . . 16
3.2 Algorithms for Optimizing Non-Linear Problems . . . . . . . . . 17
3.2.1 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . 17
3.2.2 Levenberg-Marquardt-Fletcher Algorithm . . . . . . . . . 18
3.2.3 Tricks when Calibrating One-Factor Models . . . . . . . . 19
3.3 Calibrating the Hull-White One-Factor Model . . . . . . . . . . . 21
3.3.1 Finishing the SMM Approximation for the HW1F Model
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3.3.2 Finishing the Jamshidian’s Decomposition for the HW1F
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.3 How to Perform Calibration . . . . . . . . . . . . . . . . . 26
3.3.4 Results Calibration Hull-White one-factor Model . . . . . 28
3.4 Calibrating the Hull-White Two-Factor Model . . . . . . . . . . 30
3.4.1 Two-Additive-Factor Gaussian Model . . . . . . . . . . . 30
3.4.2 From G2++ to HW2F . . . . . . . . . . . . . . . . . . . . 33
3.4.3 Results Calibration Hull-White Two-Factor Model . . . . 34
3.5 Calibrating the Cox-Ingersoll-Ross ++ Model . . . . . . . . . . . 35
3.5.1 The Cox-Ingersoll-Ross Model . . . . . . . . . . . . . . . 35
3.5.2 Swaption Premium under CIR++ Model . . . . . . . . . 36
3.5.3 Calibration Results CIR++ Model . . . . . . . . . . . . . 37
4 The Economic Scenario Generator
4.1 Defining other State Variables . . . . . . . .
4.1.1 Stocks . . . . . . . . . . . . . . . . .
4.1.2 Inflation . . . . . . . . . . . . . . . .
4.1.3 Correlations . . . . . . . . . . . . . .
4.2 Valuing Embedded Options using the ESG
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4.2.1
4.2.2
4.2.3
Sponsor Support . . . . . . . . . . . . . . . . . . . . . . .
Adjustment Mechanism . . . . . . . . . . . . . . . . . . .
Residue Option . . . . . . . . . . . . . . . . . . . . . . . .
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5 Specifications of the Fund
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5.1 Pension Plan of the (Dutch) Pension Fund Sector . . . . . . . . . 48
5.2 Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 The Impact on the Holistic Balance Sheet
6.1 Projection Assumptions of the Holistic Balance Sheet
6.2 Varying ESG Assumptions . . . . . . . . . . . . . . . .
6.2.1 Varying Short Rate Assumptions . . . . . . . .
6.2.2 Adding Correlations . . . . . . . . . . . . . . .
6.3 Decreasing the Evaluation Horizon . . . . . . . . . . .
6.4 Varying Policies . . . . . . . . . . . . . . . . . . . . . .
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7 Conclusions & Recommendations
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Bibliography
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A Technical Background: Risk-Neutral Valuation
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A.1 Arrow-Debreu Securities . . . . . . . . . . . . . . . . . . . . . . . 66
A.2 Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.3 First Fundamental Theorem of Asset Pricing . . . . . . . . . . . 68
B Technical Background: Term Structure Modeling
B.1 Term Structure Derivatives . . . . . . . . . . . . . . . . .
B.2 Black’s Model . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 From Lognormal Implied Volatility to Swaption Premium
B.4 Black’s Model Fit . . . . . . . . . . . . . . . . . . . . . . .
B.5 Lognormal and Normal Implied Volatilities . . . . . . . .
B.6 Nelson-Siegel function . . . . . . . . . . . . . . . . . . . .
B.7 Other Short Rate Models . . . . . . . . . . . . . . . . . .
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C Proofs
C.1 Jamshidian’s Decomposition . . . . . . . . . . . . .
C.2 Derivation theta function in HW1F Model . . . . .
C.3 Dynamics Short Rate Integral under HW1F Model
C.4 Deriving Short Rate Solution HW2F Model . . . .
C.5 Dynamics short rate integral G2++ Model . . . .
C.6 Exact Swaption Formula under G2++ Model . . .
C.7 Equivalence G2++ and HW2F Model . . . . . . .
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D Calibration Results
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D.1 Hull-White one-factor Model . . . . . . . . . . . . . . . . . . . . 93
D.2 Hull-White two-factor Model . . . . . . . . . . . . . . . . . . . . 99
D.3 Cox-Ingersoll-Ross++ Model . . . . . . . . . . . . . . . . . . . . 101
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E Results on Simulated Holistic Balance
E.1 Varying Short Rate Assumptions . . .
E.2 Including Correlations . . . . . . . . .
E.3 Smaller Evaluation Horizon . . . . . .
E.4 Varying Policies . . . . . . . . . . . . .
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Sheets
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Chapter 1
Introduction
Currently the Institution for Occupational Retirement Provision (IORP) Directive provides a regulatory framework for occupational pension funds within
member states of the European Union. This Directive prescribes quantitative
and qualitative requirements that a pension fund within the European Union
should meet. The IORP framework however is very outdated. The European
Commission wanted to create more harmonization among pension funds within
the EU. The EC therefore requested the European Insurance and Occupational
Pensions Authority (EIOPA) to revise the old IORP (I) Directive. This revision
will result in the IORP II Directive. This framework aims to allow for comparison between pension funds within the EU. The implementation date for this
framework is unknown when writing this thesis.
By implementing IORP II the European Commission aims to harmonize
quantitative requirements for all pension funds within member states of the
EU. Currently the quantitative requirements by EIOPA are based on the quantitative requirements for the new Solvency II framework. Solvency II aims to
harmonize supervision for all insurers within the EU. EIOPA also recognizes the
difference between pension funds and insurers, as pension funds are able to use
risk-mitigating instruments in order to alter its financial position. These riskmitigating instruments, also called steering mechanisms, include for example
sponsor support, indexation and benefit cuts. Insurers are not able to use these
mechanisms in the measurement within the risk-based supervision framework.
EIOPA advises to incorporate these steering mechanisms available to pension
funds in the new IORP II Directive. EIOPA therefore proposes the implementation of a holistic balance sheet. The holistic balance sheet values each steering
mechanism separately by valuing it as an embedded option. Eventually all embedded option values of each steering mechanism will be added to the traditional
balance sheet in order to get the holistic balance sheet. The general idea behind
the holistic balance sheet is that it is able to give a better representation of the
financial position of the fund. In the end EIOPA wants to base quantitative
and/or qualitative requirements on this holistic balance sheet. Another aim of
EIOPA is, by implementing the holistic balance sheet, to create a level playing
field for pension funds within the EU.
The valuation of embedded options in this thesis is done using risk-neutral
valuation. The big advantage of risk-neutral valuation is that no risk-premium
needs to be simulated in order to value an option. Based on some model as1
sumptions, one is able to generate a risk-neutral scenario set using an economic
scenario generator (ESG). Different model assumptions obviously provide different scenario sets. In this thesis we want to measure the impact of different
models and different models assumptions on the holistic balance sheet. In other
words, we want to see to what extent the holistic balance sheet is consistent
when using different short rate modeling assumptions. These short rate models
will be calibrated to swaption premiums, as a swaption is considered to be very
market representative due to its high liquidity. We will also measure the impact
on the HBS using varying policies, as pensions fund in other EU-countries are
restricted to different steering mechanisms compared to pension funds in the
Netherlands. Finally we will measure the impact on the HBS when decreasing
the evaluation horizon of the steering mechanisms. Obviously this will decrease
all option values, the question remains how this affects the solvency capital
requirement of the fund.
Throughout this thesis we will see that there is a significant impact on the
HBS when varying between short rate models, which are used for generating the
economic scenarios. Multiple HBSs were also derived using different parameters
choices per short model, which did not lead to very different HBSs. This variety
in parameter-choices was based on the calibration of the corresponding short
rate models using different weights in the optimization problem. We will also
see that the CIR++ model is not usable in the ESG when valuing the HBS.
In this thesis we also wanted to show the impact on the HBS of assumptions
that do not concern the ESG. We will see that a smaller evaluation horizon
results in significantly different solvency capital requirement, and how the implementation of multiple steering mechanism results in a higher loss absorbing
capacity of the fund.
1.1
About this Thesis
This thesis was written during an internship at PGGM. PGGM is a large pension administrator located at Zeist, the Netherlands. It manages pension assets
worth around e140 billion of more than 2.5 million Dutch participants. My
internship was located at the department Actuarial Advice & ALM (Asset Liability Management), under the supervision of Drs. E.A.J.P. Fransen AAG.
This thesis was written in order to complete the master ’Quantitative Finance & Actuarial Science’, which I followed at Tilburg University. My supervisor from Tilburg University for writing this thesis was Prof. Dr. J.M.
Schumacher.
1.2
Structure Thesis
In chapter 2 we start off by looking more closely into the current situation
of the Dutch pension industry. We discuss some of the upcoming changes for
Dutch pension funds when IORP II will be implemented. This chapter continues
by discussing steering mechanisms more extensively, after which we create and
discuss a holistic balance sheet. We end chapter 2 by presenting the main
strategy throughout this thesis.
In chapter 3 we calibrate short rate models, which will be used to create
2
our economic scenario generator. In this chapter we calibrate 3 different models
under different assumptions in order to have a variety of calibrated short rate
models. Eventually we will these results to see what the impact will be on the
holistic balance sheet.
Chapter 4 will be used to complete our ESG by adding other models to our
calibrated short rate models from chapter 3. We do this by defining stochastic
processes for stocks, price inflation and wage growth. Using these four processes
we can simulate the pension fund for future years, so that we can define the
holistic balance sheet as well.
Chapter 5 will be used in order to define our fictive pension fund. We will
not only discuss the pension plan of this fund, but also the different policies
the fund can act. As each policy may represent pension funds from different
countries within the EU, we eventually want to see how the holistic balance
sheet holds up when comparing pension funds using different policies.
Chapter 6 is where everything comes together. After discussing projection assumptions for the holistic balance sheet briefly, we will discuss the results found when assuming different model assumptions, varying policies and
a smaller evaluation horizon of the option values. Any impacts will be shown
using the resulting holstic balance sheet and corresponding solvency capital requirements. In the end we will sum up conclusions and recommendations in
chapter 7.
3
Chapter 2
From Solvency II to IORP
II using Steering
Mechanisms
In this chapter we will introduce the current situation of the Dutch pension
fund regulation, and discuss how the implementation of IORP II may change
this. We will do this by introducing the current Dutch regulatory framework
called Financiële Toetsingskader (FTK). As IORP II will probably match the
quantitative measures of Solvency II, we will discuss the quantitative measures
stated in the latest Quantitative Impact Study [5]. This is a study, published by
EIOPA, concerning the development of IORP II, which is also based on Solvency
II. Next we will discuss steering mechanisms in more detail, and explain in
outlines how a holistic balance sheet is created using embedded option values.
Subsequently we will discuss how to compute a SCR based on several confidence
levels as stated in the latest QIS. Finally we end the chapter by discussing the
main strategy throughout this thesis.
2.1
FTK & IORP II
The pension system in the Netherlands consists of three pillars. The first pillar,
provided by the government, consists of a basic collective pension for almost
every Dutch citizen. The third pillar concerns private saving, often provided
by insurers and banks. This pillar however is not very popular among society in the Netherlands, which is mainly due to the revision of the tax law in
2001. After this revision the fiscal margin for Dutch citizens decreased, to which
private saving became less attractive. This thesis mainly concerns the second
pilar, which consists of occupational pension schemes, financed by employer and
employees. Often participation for this layer is forced by employers so that the
participation rate for employees is quite high.
In the Netherlands we have several occupational pension schemes. The majority of the participants of these occupational pension schemes, around 87%,
participates in a career-average Defined Benefit (DB) scheme. In such a scheme,
pension payments are determined by averaging the salary received of the par-
4
ticipant during service time. This payment can be adjusted for wage growth or
price inflation, typically on a conditional basis. Besides the DB-scheme other
schemes known in the Netherlands may define pension payments based on the
last-earned salary, or via an individual Defined Contribution (DC) scheme.
A funding ratio is computed in order to determine whether a fund is solvent.
This ratio equals the value of the assets over the value of the liabilities. A fund
is considered to be solvent, whenever it can (in theory) meet all of its obligations
in the future with certainty1 . In other words, pension funds are considered to
be solvent whenever their funding ratio is above 100%. As this funding ratio is
calculated using assets and liabilities, it is important to value these consistently.
Therefore one typically values assets and liabilities based on market-consistent
valuation. The idea behind this is that one should value financial contracts
consistently to other alternatives available on the current market.
In order to absorb shocks on the funding ratio, a pension fund holds a buffer.
This buffer has to meet certain requirements according to the supervisory frameworks applicable to that particular fund. In the Netherlands the local supervisory framework for pension funds is called FTK. The required buffer we call the
Solvency Capital Requirement (SCR). Implemented in the current FTK there is
also a supervisory framework for all pension funds within member states of the
European Union. This framework is called Institutions for Occupational Retirement Provision (IORP) I Directive. In the Netherlands the IORP I Directive is
embedded in the Dutch Pension Law.
IORP I originates from 2003. In order to have a prudent framework, IORP I
turns out to be outdated. Since it is outdated, the European Commission asked
the European Insurance and Occupational Pensions Authority (EIOPA) to revise the old framework. This new framework is called IORP II. Although not
implemented yet, IORP II will have different quantitative measures compared to
FTK. These different measures will also probably require Dutch pension funds
to alter their buffers. Although the impact on these buffers may be small for
Dutch pension funds, it may be larger for other pension funds within the EU.
As mentioned before, the current development of quantitative measures of
IORP II are discussed in the latest QIS. This QIS report was based on the
development of Solvency II, which is scheduled to be implemented January
20142 . Solvency II will be the new regulatory framework for insurers within the
EU.
Solvency II uses three pillars. The first pillar describes quantitative measures
that a fund will be restricted to. Under this pillar for instance the Solvency
Capital Requirement (SCR) is defined, indicating to what extent insurers can
take risks. Our focus is also on this pillar, as it will likely be used for IORP
II. In order to be complete, the second pillar is defined by mostly qualitative
measures. Requirements for governance, risk management and supervision are
stated under this pillar. If a fund is willing to implement its own model to assess
its risks, this pillar also states requirements to carry out an ORSA (Own Risk
and Solvency Assessment). Finally, the third pillar focuses on requirements
towards disclosure and transparency.
IORP II will have the same pillars as Solvency II. Therefore in outlines IORP
1 Assuming the fund has to meet nominal obligations only, and that all risks the fund is
exposed to are fully hedged.
2 Currently the implementation date of Solvency II at January 2014 is unlikely. Indications
have been made that the implementation of Solvency II will be postponed until January 2016.
5
II will resemble Solvency II. One of the differences will probably be the level
of the quantitative requirements. This is because insurers and pension funds
are two completely different institutions. Pension funds operate on a longer
term, while insurers operate on a shorter term. Besides that, pension funds are
able to use steering mechanisms which can alter the financial position of the
fund. Steering mechanisms include for example recovery premiums or providing
indexation to pension payments. That is why in the end the confidence level
when calculating the SCR may be smaller for IORP II compared to Solvency
II.
These fundamental differences between pension funds and insurers result
into EIOPA implementing the holistic balance sheet. In this balance sheet
all steering mechanisms are valued in order to have a better representation
of the financial position of the fund. In the next subsection we will discuss
the various steering mechanisms a Dutch pension fund has access to, before
discussing the holistic balance sheet itself. Subsequently we will explain how
the holistic balance sheet may result in a different SCR. We end by explaining
our main strategy throughout this thesis.
2.2
Steering Mechanisms
Sponsor support can be identified by recovery payments that can occur whenever
the funding ratio falls below some certain threshold, but also by additional
funding provided by employers. Sponsor support can also be identified by new
accrual, however according to the latest QIS this may not be incorporated when
valuing the holistic balance sheet. Nevertheless, the extra funding increases the
value of the assets. The value of sponsor support is calculated by computing
the value of the corresponding embedded option. This value is dependent upon
several factors. The amount of recovery premiums in the future for instance is
dependent on the current funding ratio. Typically we have that the higher the
current funding ratio, the lower the value of the sponsor support.
Other factors may influence the value of sponsor support as well. The threshold value, for which pension funds require recovery payments if its funding ratio
is below this value, is also an important factor. If the threshold value increases,
recovery payments are made more often, and so will the value of the sponsor
support increase. The threshold value also depends upon the local regulation
and policy of the fund. For example, say that a fund is forced to increase its
funding ratio to 105% within a recovery period whenever it falls below this
value. If local regulation only allows for a recovery period of 5 years, the fund
can set a different recovery premium compared to a fund in the same situation
with a recovery period of 3 years. The latter fund probably will set a higher
premium as it is forced to recover faster and therefore results in an increase in
the value of sponsor support.
A second kind of steering mechanism we define that is available to pension
funds is indexation. The value of this steering mechanism is identified by the
embedded option to give indexation in the future. This value is again influenced by the threshold value for which indexation is given if the funding ratio
is above this value. In the Netherlands pension funds typically give indexation
conditional on the funding ratio, which is called conditional indexation. Nevertheless, the value of the indexation option becomes larger as the threshold value
6
decreases, since then the pension fund is more likely to give indexation in the
future. The current funding ratio also influences the indexation option. The
larger the current funding ratio, the larger the value of the indexation option.
Also note that the value for indexation is subject to restrictions stated in local
regulatory frameworks.
Benefit cuts are a measure of last resort in case the funding ratio is drastically
low. By cutting benefits, pension obligations decrease. The value of the option
to cut benefits is subject to a number of factors. First, in order to determine
the value of the option of benefit cuts, one has to determine the threshold for
which benefits will be cut if the funding ratio falls below this threshold. This
is again subject to the fund’s own policy and local regulation, for instance the
length of the recovery period. Second, one has to determine by how much the
benefits will be cut. This will again be subject to policy restrictions by local
regulation.
Now that we discussed valuing all three steering mechanisms, one should
keep in mind two things. First note that the values for these three embedded
options are subject to the horizon of evaluation. After all, the values of an
embedded option over 15 years will be larger compared to an embedded option
over 5 years. Therefore, the most reasonable length of the evaluation period
also is an important value yet to be determined.
Besides that, note that the option values of steering mechanisms are correlated. For example, a fund that announces recovery premiums more frequently
does not have to announce benefit cuts so often. This correlation between options is an important topic, which we will see in chapter 6 when discussing the
holistic balance sheet for several policies. Also note that the current funding
ratio has impact on the option values, as for example the value of an indexation
option becomes higher when the current funding ratio increases. We will discuss
these topics later on in this thesis.
2.3
The Holistic Balance Sheet
Currently EIOPA is considering whether to implement the holistic balance sheet
in IORP II. Insurers typically do not have the oppurtunity to use the steering
mechanisms explained in the previous chapter, therefore this approach will be
only available for pension funds. Whether the HBS actually will be implemented
depends upon the consistency of the HBS. As discussed before, evaluation horizon and correlation of the steering mechanisms are very important in determining their values. More insight on these problems is given later on in this thesis.
Here we will discuss the main concept of the holistic balance sheet, after which
we will discuss the calculation of the SCR based on the holistic balance sheet.
We will start with a traditional balance sheet presented in table 2.1. Here
premiums paid by the participant will be added to the assets. At the same time,
new pension accrual when premiums are paid will be added to the liabilities.
The residue between assets and liabilities is given by R0 . If this is a surplus, R0
is positive. If the residue is negative, so that it is a deficit, R0 becomes negative
or may be written positive on the left hand side of the table.
The holistic balance sheet allows for valuation of the steering mechanisms as
discussed in the previous section. The current value of the assets and liabilities,
A0 and L0 , remain the same as in the traditional balance sheet. However note
7
Table 2.1: Traditional Balance Sheet
Assets A0
Liabilities
L0
Residue
R0
A0
L0 + R0
that EIOPA intends to model the holstic balance sheet based on a closed fund.
In other words, there are no incoming premiums or new accrual to be added
to the current value of assets and liabilities. A reason for this approach is that
the holistic balance sheet becomes less sensitive for differences in subjective
interpretations. Residue R0 however changes into an option V0RO , as the value
of the residue is dependent upon the policy-making of the fund. The holistic
balance sheet is presented in table (2.2) and in figure (2.1).
Table 2.2: Holistic Balance Sheet
Assets
A0
Liabilities
Sponsor Support V0SPS
Adjustment Mechanism
Residue Option
AHBS
L0
V0AM
V0RO
LHBS
Figure 2.1: A Holistic Balance Sheet
Note that the scaling in figure (2.1) is not representative, and that assigned
values to the steering mechanisms are fictive. Also note that the adjusment
mechanism (AM) in figure (2.1) is split up into two parts, a positive and a
negative part. This is because the adjument mechanism consists of several
options, which can be positive and negative. Therefore these options each have
a different impact on the liabilities under the holistic balance sheet. Next we
will define these various options more extensively.
8
In table (2.2), the adjustment mechanism option V0AM is the aggregated
option value of the steering mechanisms indexation and benefit cuts. According
to De Haan et al. [9], one can split up sponsor support and the adjustment
mechaism in table (2.2) up into several types of options. These options are
displayed in table (2.3), and will be discussed briefly.
Assets
Table 2.3: Holistic Balance Sheet (extended)
A0
Liabilities
Sponsor Support
Employer contribution option
Employer guarantee option
Premium cut option
Premium damping option
V0SPS
AHBS
L0
Adjustment Mechanism
Indexation option
Surplus sharing option
Benefit cut option
Catch up indexation option
V0AM
Residue Option
Surplus option
Deficit option
V0RO
LHBS
• Employer contribution option: The option for a fund to ask for recovery
premiums when its funding ratio is too low.
• Employer guarantee option: The promise of an employer to provide additional funding when the funding ratio is very low.
• Premium cut option: The option for a fund to ask for lower premiums
whenever the funding ratio of that fund is sufficiently high. Note that the
premium cut option in table (2.3) is negative.
• Premium damping option: In order to prevent fluctuations in the premium
to be asked by the fund, the fund typically aks a little higher premium than
the cost covering premium.3 This option typically applies to countries
where the determination of the premium is based on the term structure
of interest rates, so that the cost covering premiums fluctuate a lot.
• Indexation option: The option to give indexation to accrued rights whenever the funding ratio is sufficiently high.
• Surplus sharing option: The option for the fund to give extra indexation
above full indexation, whenever the catch up indexation from the past has
already been redeemed and the funding ratio is still high.
• Benefit cut option: When the funding ratio is drastically low, a pension
fund can reduce accrual rights in order for the funding ratio to reach some
kind of minimum. This can be seen as negative indexation, which can
be offset using catch up indexation. Note that there is only a reduction
3 The premium damping does not always suggest a premium higher than the cost covering
premium, it may also suggest a lower premium than the cost covering premium.
9
of accrued rights whenever the funding ratio falls behind on a recovery
plan. This recovery plan is being made whenever the funding ratio falls
below some critical level. Note that the benefit cut option in table (2.3)
is negative.
• Catch up indexation option: The option for the fund to give extra indexation in order to catch up missed indexaton from the past. The funding
ratio has to be high in order to give this type of indexation
• Surplus option: The ’option’ value of the fund to hold a surplus at the
end of the evaluation horizon.
• Deficit option: The ’option’ value of the fund to hold a deficit at the end
of the evaluation horizon. Note that the deficit option in table (2.3) is
negative.
Note that we define employer contribution, employer guarantee, premium
cut and premium damping to be sponsor support. Indexation, surplus sharing,
benefit cut and catch up indexation we define to be the options of the adjustment
mechanism. The residue option consists of a surplus and deficit option. In
chapter 5 we will opt in and opt out particular options in order to define various
policies for which we will model the holistic balance sheet.
Using the holistic balance we also have a different behavior of the SCR. Under the traditional balance sheet, the SCR changes with the risks taken. For
instance, the more invested in risky stocks, the higher the SCR. Under the
holistic balance sheet this is not so straightforward. The level of the SCR will
depend on the values of the steering mechanisms. Assuming all steering mechanisms are available to a pension fund, the values of these steering mechanisms
will decrease the impact on the free capital of the fund. We call this the loss
absorbing capacity. We will explain this using an example.
As an example we use the fund discussed by Fransen et al. [7]. In the base
case this fund has a current funding ratio of 105%. In table (2.4) we present the
values for the assets and liabilities under the traditional- and holistic balance
sheet. These values are given under base case, as well as after an equity, interest
rate or longevity shock. As one can see, the change in surplus when moving from
base case to any of the cases after a shock is lower under the holstic balance
sheet compared to the traditional sheet. This we call the loss absorbing capacity
(LAC) of the fund.
The buffer required by the regulatory frameworks FTK and IORP II are
computed using the stressed balance sheets of several risks. Both SCRs are
computed by aggregating the changes in (option) surplusses after pre-defined
risks using a pre-defined correlation matrix. Under FTK the SCR becomes 21,
where under IORP II the SCR becomes 7. This decrease in SCR is a direct
result of the loss absorbing capacity. In the next section we will discuss in
outlines how the SCR under IORP II is computed.
2.4
Calculating the SCR
In this section we will show in outlines how to calculate the solvency capital
requirements (SCRs) for pension funds under the IORP II framework. In this
10
Table 2.4: Loss Absorbing Capacity
Base
Shock in:
case Equity Interest rates
Longevity
Assets
Sponsor support
Total assets
105
13
118
92
16
108
116
18
134
105
14
119
Liabilities
Adjustment mechanism
Total liabilities
100
12
112
100
7
107
121
9
130
104
10
114
5
6
−8
1
−5
4
1
5
−13
−5
−10
−2
−4
−1
R0
V0RO
∆R0
∆V0RO
thesis calculation is done using spreadsheets in Excel, which are supplied by
PGGM. These spreadsheets are based on the QIS report on IORP II [5], which
is a working document for testing purposes. This report does not represent or
pre-judge any proposals of the European Commission.
The SCR is calculated by aggregating the change in surplusses on the holistic
balance sheet under different shocks. These pre-determined shocks are stated
in the QIS report. The aggregation of the changes in surplus is done using
various correlation matrices. We will define the risk factors and their correlation
matrices below.
The calculation of the SCR is done using the following formula:
SCR = BSCR + SCROp + Adj1 + Adj2
where SCROp stands for operational risk. BSCR stands for basic solvency capital requirement, which basically is the SCR before any adjustments. This value
combines the capital requirement of five major risk categories: market risk,
counterparty default risk, pension liability risk, health risk and intangible assets risk. Aggregation of the buffer requirements of these risk factors is done
using the following correlation matrix:
Table 2.5: Correlation matrix four major risk factors
Market Default Pension liability Health
Market
1
0.25
0.25
0.25
Default
1
0.25
0.25
1
0.25
Pension liability
Health
1
Counterparty default risk stands for the risk that the fund loses money
whenever the counterparty defaults. Health risk is typically equal to zero, as
11
a pension fund does not run that risk. Here one can notice the relation with
Solvency II, as health risk often is non-zero for insurers. The last two major
risk factors are pension liability risk and market risk. We will discuss these two
in more detail.
Market risk is defined by the risk that the surplus of the HBS decreases
because of sudden changes in the market. Calculating the capital requirement
of market risk is again done using a correlation matrix, where market risk is
split up into several smaller risk factors. These risk factors include: interest rate
risk, equity risk, property risk, spread risk, currency risk, concentration risk and
counter-cyclical premium risk. The calculation of the capital requirement for
every risk factor is based on the change in surplus of the holistic balance sheet
when a corresponding shock occurs. The aggregation of the smaller risk factors
is done using the following correlation matrix displayed in table (2.6).
Table 2.6: Correlation matrix market risk
Interest Equity Prop. Spread
Interest
1
0.5
0.5
0.5
Equity
1
0.75
0.75
Property
1
0.5
Spread
1
Currency
Concentration
Counter-cyclical
factors
Curr. Conc.
0.25
0
0.25
0
0.25
0
0.25
0
1
0
1
C-C
0
0
0
0
0
0
1
Note that market risk, and thereby all risk factor in table (2.6), can affect
both the value of assets as well as liabilities. The remaining large risk factor not
mentioned before from table (2.5), pension liability risk, only affects the value
of the liabilities.
Pension liability risk denotes the risk of a sudden increase of liabilities. In
the QIS report this risk factor is decomposed into seven smaller risk factors:
mortality risk, longevity risk, disability risk, benefit risk, expense risk, revision
risk and catastrophe risk. These risks are aggregated using the correlation
matrix displayed in table (2.7).
Table 2.7: Correlation matrix pension liability risk factors
Mort. Long. Dis. Benefit Exp. Rev. CAT
Mortality
1
−0.25 −0.25 −0.25
0.25 0
0.25
Longevity
1
0
0.25
0.25 0.25 0
Disability
1
0
0.5
0
0.25
1
0.5
0
0.25
Benefit
Expenses
1
0.5
0.25
Revision
1
0
CAT
1
For more detail on calculation of the SCR is referred to the latest QIS report
[5].
12
2.5
Main Strategy throughout this Thesis
This section we will show how a steering mechanism can be valued using riskneutral valuation. We will show how this type of valuation depends on a riskneutral scenario set, which is dependent on several assumptions one has to
make. These assumptions can vary a lot between scenario sets. The question
here is how will these different scenario sets influence the value of the steering
mechanisms, and thereby the holistic balance sheet as well. We will deal with
this impact later on in the thesis.
The advantage of risk-neutral valuation is that one does not need to simulate
a risk-premium when valuing. This simplifies the assumption one has to make.
If one wants to value the embedded option of a steering mechanism C, this can
be done using the numeraire dependent pricing formula (NDPF):
C0
QN CT
=E
(2.5.1)
N0
NT
where N is some numeraire. The expectation here is under the QN -measure.
This pricing formula is called the numeraire dependent pricing formula (NDPF)
[16]. The NDPF is based on the first fundamental theorem of asset pricing,
which states that every relative price process is a martingale under some riskneutral measure. For more elaboration on the NDPF, see appendix A.
When valuing steering mechanisms we typically cannot solve (2.5.1) analytically. Therefore we will use a scenario set in order to simulate the payoff and
corresponding numeraire. Based on these simulations we can compute the expectation in order to derive the current value of the steering mechanisms. The
simulations we call Monte-Carlo simulations, and the scenario generator we call
the economic scenario generator (ESG).
In this thesis our ESG only models short rate, stocks, price inflation and
wage growth. Our emphasis however will be on the modeling of the short rate.
A short rate process is able to describe an entire term structure over time. This
relation can be shown by valuing a zero-coupon bond using the NDPF and the
money market account as numeraire. Then the relation between term structure
and short rates becomes as follows:
"
!#
Z
T
P (t, T ) = EtQM exp −
rs ds
t
"
!#!
Z T
1
QM
R(t, T ) = − log Et
exp −
rs ds
T
t
where P (t, T ) and R(t, T ) denote the discount and interest rate respectively.
For derivations see appendix B.
In the remainder of this thesis we will among other things discuss the effect
of the choice between different short rate models as well as the input parameters
for these models. The parameters one chooses are typically found by doing a
short rate calibration to market data. This calibration can be done to several
market data, for example caplets or swaptions. In this thesis we choose to
calibrate short rate models to swaptions, as swaptions are considered to be very
liquid. The high liquidity of these contracts ensures that the swaption premiums
13
are very market representative. In the remainder of this section we will discuss
a swaption contract and how it is typically valued in practice.
A swaption is a contract that gives the holder the right, but not the obligation, to enter a swap contract at a specified date with a specified strike rate. For
instance define a swaption with strike rK expiry date T0 , tenor dates as before,
and current time t, t < T0 . The corresponding payoff of this swaption at T0 can
be defined as the maximum of the value of the swap at T0 and zero:
!
N
X
max 0, 1 − P (T0 , TN ) − rK
P (T0 , Ti )
i=1
If a swaption is set to be at-the-money, strike rate rK will at initiation of the
contract be set such that the value of the underlying swap equals zero. Denote
ATM
this strike rate by rK
. In order for a payer swaption to be in-the-money at
ATM
initiation, the strike rate should be smaller than rK
. This payer swaption is
out-of-the-money when, at the initiation date of the contract, the strike rate is
ATM
larger than rK
. For a receiver swaption this is vice versa.
Using the NDPF one can price a swaption. Typically in practice swaptions
are priced using the famous Black’s Formula [1].4 Black’s Formula prices swaptions using the NDPF, while defining the annuity factor as numeraire. Black
also made the assumption that the forward swap rate is lognormally distributed.
After derivation the price of a swaption can be expressed analytically as follows:
πtBlack
=L
N
X
P (t, Ti ) [Ft (T0 , TN )Φ(d1 ) − rK Φ(d2 )]
i=1
where:
d1
=
d2
=
ln
Ft (T0 ,TN )
rK
√
σB
2
+ 12 σB
T0
T0
√
d1 − σB T0
For derivations, see appendix B. Note that using this way of pricing, one can
state a swaptions price by giving the premium or the lognormal (Black’s) implied
volatility. The volatility leads to the premium using Black’s Formula, and the
premium leads to the volatility.5
In the next chapter we will calibrate short rate models to swaptions. We
do this by providing an analytical expression for the swaption under the corresponding short rate model. This allows us to ’match’ these expressions to
the observed swaption premiums by minimizing the difference between the two.
This describes the calibration. The calibration can be done in a lot of ways,
therefore we will devote a large part of this thesis to this topic.
In the end we want to see how different models and parameters choices
influence the holistic balance sheet. We do this by performing some sensitivity
analysis to the HBS. This analysis will also be done using several policies. For
4 Actually nowadays traders value swaptions based on the assumption of a normal implied
volatility, however this implied the same premium as Black’s Formula. For more information
on valuing swaptions based on a normal implied volatility, see appendix B.
5 Using, for example, Newton’s scheme. Newton’s scheme is an iterative method which
is able to find the roots of a particular equation. The objective function must be twice
differentiable when using this method.
14
now we start defining the economic scenario generator by calibrating short rate
models.
15
Chapter 3
Calibration of Short Rate
Models
The holistic balance sheet requires valuation of all steering mechanism options
discussed in the previous chapter. This valuation is done using a risk-neutral
scenario set, which we will generate using an economic scenario generator (ESG).
This ESG will, amongst other processes, consist of a short rate process. In
this chapter we will calibrate different short rate models to observed market
premiums of at-the-money swaptions. Again note that we use swaptions as
they are considered to be the most liquid contracts available, and thereby the
most representative for the market.
3.1
Data & Optimization Problem
We use swaptions in order to calibrate the short rate models, as these are very
representative for the market. For this thesis data of several characteristics of
swaptions traded at ultimo 2003, 2005, 2008, 2011 and 2012 were available.
Besides swaption premiums, we could also use normal and lognormal implied
volatilities. For each year, each data set consisted of swaptions of which expiry
and maturity dates ranged from 1 month to 30 years and 1 year to 30 years
respectively.1 However in order to guarantee the high liquidity and market
representative quotes only swaptions with an expiry date from 1 year to 10 years
were used. The same holds for the corresponding maturity date. Finally, it is
important to note that all swaptions used are at-the-money (ATM), as these are
considered to be most liquid as well. For more information on ATM-swaptions
or swaptions as a whole, see appendix B.1.
In this thesis we will calibrate short rate models (SRMs) to market representative swaption premiums by minimizing least squared errors:
X
2
min
wi ViObserved − ViSRM (θ)
(3.1.1)
θ
where V can be the swaption’s premium or the swaption’s volatility, as the premium can be defined by the volatility and vice versa. The vector θ is the vector
1 Note
that by the expiry date we mean the expiry date of the option, and by maturity
date we mean the maturity date of the underlying swap.
16
of parameters to be optimized over, which corresponds to the input parameters
of the corresponding short rate model. Finally weights can be applied by defining wi , so that a particular subset of swaptions can have more weight. These
weights can also be defined such that one calibrates absolute errors (wi = 1) or
relative errors (wi = (V Observed )−2 ). Later on we will see if this has any effect
on results as well.
The calibration’s fit is determined universally among all weights. We do this
by computing the root mean squared error (RMSE), which is defined as follows:
v
u n
u1 X
2
RMSE = t
ViObserved − ViSRM (θ)
n i=1
where calibration is performed over n swaptions.
The problem stated in equation (3.1.1) is a non-linear optimization problem. In order to solve this we use two algorithms; simulated annealing and the
Levenberg-Marquardt-Fletcher algorithm. Both algorithms are used in order
to be certain a global minimum has been reached. In the next section we will
discuss these two algorithms in more detail.
3.2
Algorithms for Optimizing Non-Linear Problems
3.2.1
Simulated Annealing
Simulated annealing is an algorithm that is able to find a global optimum when
local optima are present as well. This algorithm is used in order to provide a
numeric solution to equation (3.1.1). In this section some more insight is given
on the algorithm. The general idea of this algorithm is that the next point in
finding the optimal solution is drawn randomly from a normal distribution. One
can perform simulated annealing by following these steps:
1. Choose an initial point x0 and an initial standard deviation σ0 . This σ0
is the standard deviation of our initial random draws from the normal
distribution. Also define the running variable xr = x0 . Eventually this
running variable will be the solution of our optimization problem:
min ||r||2 = min rT r
x
where:
r=
√
x
w V Observed − V SRM (x)
as in our original optimization problem, equation (3.1.1).
2. Next draw a next sample x from a normal distribution with mean xr and
standard deviation σi where:
i
σi = σ0 e−γ N −1
for some σ0 chosen in step 1. Note here that i denotes the i-th iteration
of the algorithm.
17
3. If ||r(x)||2 < ||r(xr )||2 , then define x as our new running variable: xr = x.
If ||r(x)||2 > ||r(xr )||2 , then forget x.
4. If i 6= N , then go back to step 2. If i = N , the optimal solution is found.
Note that N , γ and σ0 are variables for input. N stands for the number
of iterations one would like to run in order to find the optimum, and γ stands
for the speed of convergence of the σi ’s. Both variables should be chosen such
that it allows the algorithm to find an optimum. Too few iterations result in
the algorithm stopping too fast. A slow convergence of the σi ’s could result in
a unprecise solution, however a too fast convergence of the σi ’s could result in
the algorithm stopping too fast as well. Therefore, the choice of N , γ and σ0
will always be a choice between computational effort and precision.
3.2.2
Levenberg-Marquardt-Fletcher Algorithm
In this section some information is given towards the Levenberg-MarquardtFletcher algorithm. This algorithm was used in order to optimize equation
(3.1.1). The algorithm finds a global optimum, even if the optimization function
is non-linear and local minima exist.
Suppose we want to minimize:
min ||r||2 = min rT r
c
where:
r=
√
c
w V Observed − V SRM (c)
as in equation (3.1.1). Suppose we want to optimize for some unknown coefficients c. By definition, we are looking for the optimal solution c∗ such that
vector v equals the zero vector in the following equation:
δ||r||2
δrT
=2
r = 2J T r = 2v
δc
δc
Here J is called the Jacobian matrix. Elements of J are defined according to
δri
Ji,j = δc
. The optimal solution is found by doing iterations, where every new
j
draw follows from the old draw. For example, after the k-th iteration we have:
c(k+1) = c(k) + ∆c(k)
Let r(c) to be smooth functions, so that:
r(k+1) = r(k) +
By rewriting (3.2.1) using Ji,j =
J
(k)
δri
δcj ,
δr(k) (k)
∆c + . . .
δc(k)
(3.2.1)
and multiplying the whole equation times
, we get:
T
A(k) ∆c(k) − J (k) r(k+1) = −v (k)
T
(3.2.2)
where we have that A(k) = J (k) J k . However, this equation is impossible to
solve, as r(k+1) is unknown before iterating. Therefore Levenberg substituted
T
J (k) r(k+1) by λ∆c(k) in equation (3.2.2), which results in:
18
A(k) − λI ∆c(k) = −v (k)
(3.2.3)
For λ = 0, this method turns into a Newton method, where for λ → ∞ the
algorithm turns into the stable steepest descent method.
Later on Marquardt changed the scale parameters λ, by making it variable
per iteration. When the iterations show slow convergence, the algorithm will be
sped up by setting:
λ(k+1) = λ(k) /v
(3.2.4)
For some value of v, which usually is set between 2 and 10. In case the iterations
show divergence, the new value for λ will be set according to:
λ(k+1) = λ(k) v
(3.2.5)
Fletcher made some final modifications to the algorithm, which resulted in a
better adaptation of λ per iteration. This causes an even faster running time of
the algorithm. The modifications by Fletcher consisted of replacing the identity
matrix in equation (3.2.3) by a diagonal matrix of scales. Besides that, he also
made the parameter v in equation (3.2.4) and (3.2.5) variable per iteration.
3.2.3
Tricks when Calibrating One-Factor Models
In this section we show two different tricks when calibrating one-factor models. This method called the Swap Market Model (SMM) approximation [8].
Using this trick we can derive the swaption’s volatility from the volatility of
the bond ratio using only limited computational effort. The second trick is
called Jamshidian’s Decomposition [12]. This method rewrites the swaption’s
premium into a combination of zero-coupon bond put-options in an analytical
way. Again note that both methods can only be used for one-factor models. In
the remainder of this section we will discuss both tricks more extensively. Any
effect on calibrations will be discussed in the subsequent sections.
SMM Approximation
The swap market model approximation derives the swaption’s volatility in terms
of the volatility of the bond ratio. The advantage here is that computational
speed can be reduced when calibrating. The approximation however may have
significant effects on the calibration results. Besides that, this method only
works with a one-factor short-rate model. Any results on calibration will be
discussed later on. We start by stating the forward swap rate:
P (t, T0 ) − P (t, TN )
Ft (T0 , TN ) = PN
i=1 δ(Ti−1 , Ti )P (t, Ti )
One can approximate this rate by stating that the discount rate does not change
from time 0 to time T0 , so that:
P (0, TN )
P (t, T0 )
F̃t (T0 , TN ) = PN
−1
(3.2.6)
P (t, TN
i=1 δ(Ti−1 , Ti )P (0, Ti )
19
where we take the discount curve at time 0 as known. From (3.2.6) follows that:
dF̃t (T0 , TN )
1
P (0, TN )
=
d
PN
F̃t (T0 , TN )
F̃t (T0 , TN ) i=1 δ(Ti−1 , Ti )P (0, Ti )
P (t, T0 )
P (t, TN )
(3.2.7)
(t,T0 )
represents the stochastic differential equation for the bond
Here d PP(t,T
N)
ratio. This has to be specified inorder to compute the volatility of the forward
(t,T0 )
swap rate. The expression for d PP(t,T
remains yet unknown as it is modelN)
specific, however we do know the drift term of this expression should be zero in
order to have absence of arbitrage. In the next section we will finish expression
(3.2.7) for the Hull-White one-factor model.
In order to calibrate using the SMM approximation, we will optimize equation (3.1.1) using:
VtObserved = Vol FtObs (T0 , TN )
VtSMM = Vol FtSMM (T0 , TN )
In other words, we will calibrate the volatility of the observed forward swap
rate to the volatility of the forward swap rate. The observed volatility of the
forward swap rate can be computed using the normal- or the lognormal implied
volatility. In the remainder of this section we will derive the observed volatility
of the forward swap rate using a normal implied volatility. For a lognormal
volatility the volatility of the forward swap rate can be derived easily using
Itô’s lemma.
Say we have a normal implied volatility σN , for which the process of the
forward swap rate for some swaption Ft = Ft (T0 , TN ) is defined via the following
stochastic differential equation:
dFt = σN dWt
(3.2.8)
By integrating (3.2.8), one can easily see that the volatility of the forward
swap rate equals σN (which is correct by definition), so that the volatility of the
forward swap rate over the difference between expiry and settling date equals:
p
Vol FtObs (T0 , TN ) = σN T0 − t
(3.2.9)
In section 3.4.1 we will finish the optimization problem using the SMM approximation by deriving Vol(FtSMM (T0 , TN )), which varies per short rate model. We
will do this by completing expression (3.2.7).
Jamshidian’s Decomposition
The second method is not an approximation, unlike the previous method. Using
Jamshidian’s Decomposition we can calibrate swaption premiums using exact
expressions. This method is based on a paper by Jamshidian [12], which states
that one can write the value of a swaption as a combination of put options on
zero coupon bonds. Jamshidian’s Decomposition does have some restrictions.
The underlying short rate model must be a one-factor model. Besides that, this
method requires that the derivative of the expression for the zero coupon bond
with respect to the short rate must be decreasing. In formula:
20
δΠ(t, s, r)
<0
for all0 < t < s
δr
Here we define Π(t, s, r) as the (unknown) expression for the zero coupon
bond price for some one-factor short rate model. Next we will write down the
expression that belongs to a payer swaption valued at time t. This swaption has
expiry and maturity date T0 and TN respectively. The strike of the swaption is
defined by K and the notional amount is defined as N . The expression using
Jamshidian’s Decomposition becomes:
PS(t, T0 , TN , L, K) = L
N
X
ci ZBP (t, T0 , Ti , Π(T0 , Ti , r∗ ))
(3.2.10)
i=1
where ZBP (t, T0 , Ti , Π(T0 , Ti , r∗ )) denotes a put option on a zero coupon bond,
valued at time t. This option has expiry date T0 , and the underlying zero
coupon bond expires at time Ti . The expression Π(T0 , Ti , r∗ ) denotes the strike
rate of the put option, which is the price of a zero coupon bond at time T0 with
maturity Ti . The short rate r∗ can be derived by solving the following equation:
N
X
ci Π(T0 , Ti , r∗ ) = 1
i=1
where the weights ci are defined such that:
ci
cN
= K
= 1+K
for each i = {1, . . . , N − 1}
For proof see appendix C.1.
3.3
Calibrating the Hull-White One-Factor Model
In this section we will discuss the calibration results of the Hull-White onefactor model, after we have derived an analytical expression for the swaption
premium using Jamshidian’s Decomposition. We will also finish the expression
for the volatility of the forward swap rate using the SMM approximation.
The short rate according to a Hull-White One-Factor (HW1F) model under a
risk-neutral measure is defined according to the following stochastic differential
equation:
drt = (θ(t) − art ) dt + σdWtQ
(3.3.1)
In case θ is a constant, we have the Vasicek model. However the advantage
of a time-dependent function θ(t) is that it allows for a perfect fit of the initial
term structure. In order to have an even better fit, one can additionally make
parameters a and/or σ dependent on time. In case we only make the parameter a
dependent on time, we have the extended Vasicek model. In this thesis however
we only calibrate the Hull-White one-factor model (3.3.1), which uses a constant
a and σ only.
In order to derive the expression for the zero coupon bond price, we first have
to derive dynamics of the short rate under the HW1F model. The function θ(t)
is chosen so that the model will fit the term structure of interest rates perfectly.
21
We define f M (0, t) and P M (0, t) to be the market instantaneous forward rate
and market discount factor respectively, which are observed at time 0. The
relation between the two, per definition, is defined as follows:
δlnP (0, T )
δT
which is also shown in appendix B.1. Using this, one can write the function θ(t)
as follows:
f M (0, T ) = −
δf M (0, t)
σ2
+ af M (0, t) +
(1 − e−2at )
(3.3.2)
δt
2a
For proof, see appendix C.2. Now we can proceed defining the exact solution for
the short rate. Using Itô’s lemma, we can first derive the stochastic differential
equation for eat rt :
d eat r(t) = eat dr(t) + aeat r(t)dt = θ(t)eat dt + eat σdW (t)
θ(t) =
After integrating this equation, one can divide by eat in order to get the analytical expression for the short rate:
Z t
Z t
−a(t−s)
−a(t−u)
r(t) = r(s)e
+
θ(u)e
du + σ
e−a(t−u) dW (u)
s
s
which can be rewritten using (3.3.2) as:
r(t) = r(s)e
−a(t−s)
+ α(t) − α(s)e
−a(t−s)
Z
+σ
t
e−a(t−u) dW (u)
s
where:
σ2
(1 − e−at )2
2a2
One can see that the short rate follows a normal distribution. The mean and
variance of this distribution are defined as follows:
α(t) = f M (0, t) +
E[r(t)]
Var[r(t)]
= r(s)e−a(t−s) + α(t) − α(s)e−a(t−s)
=
σ2
2a (1
− e−at )
2
So that the variance of the short rate in the long run is defined by σ2a . Later on
we will use this variance in order to compare calibrations from different dates.
A drawback of the HW1F model is that short rate can become negative. As
the distribution of the short rate is known to us, we can compute the probability
of the short rate becoming negative:


α(t)

Q(r(t) < 0) = Φ − q
σ2
−2at ]
[1
−
e
2a
Next, we want to define the structure of the zero-coupon bond P (t, T ). We
do this as eventually we need this structure in order to price swaption premiums.
Recall the relation between the short rate and P (t, T ), which is also derived in
appendix B.1:
22
"
P (t, T ) =
EtQ
Z
exp −
!#
T
r(u) du
(3.3.3)
t
Since the process of the short rate is normally distributed, we can compute
the expectation of the equation above by computing the mean and variance of
RT
r(u)du conditional on all information known at time t, Ft . When doing so,
t
we get the following normal distribution:
RT
t
M
(0,t)
∼ N (B(t, T )[r(t) − α(t)] + ln PPM (0,T
)
r(u) du|Ft
+ 12 [V (0, T ) − V (0, t)], V (t, T ))
where:
B(t, T ) =
1
[1 − e−a(T −t) ]
a
(3.3.4)
2 −a(T −t)
1 −2a(T −t)
3
σ2
− e
−
V (t, T ) = 2 T − t + e
a
a
2a
2a
For proof, see appendix C.3. As the expectation of the exponential taken of
random variable equals the exponential of the mean plus half of the variance of
this random variable, we can remove the expectation in equation (3.3.3).
P (t, T )
M
(0,t)
exp(−B(t, T )[r(t) − α(t)] − ln PPM (0,T
)
=
− 12 [V (0, T ) − V (0, t)] + 21 V (t, T ))
which can be written into:
P (t, T ) = A(t, T )e−B(t,T )r(t)
where B(t, T ) and A(t, T ) are defined respectively according to (3.3.4) and:
σ2
P M (0, T )
exp B(t, T )f M (0, t) −
(1 − e−2at )B(t, T )2
A(t, T ) = M
(3.3.5)
P (0, t)
4a
3.3.1
Finishing the SMM Approximation for the HW1F
Model
Now that we have all analytical expressions for the zero-coupon bond structure,
we can finish our approximation of the swaption’s volatility in section 3.3.1. We
do this by defining the stochastic differential equation describing process of the
zero-coupon bond. We start by defining the stochastic differential equation for
the zero coupon bond structure, which can be deriving using Itô’s lemma:
dP (t, T ) = r(t)P (t, T )dt − σB(t, T )P (t, T )dW Q
Next, again using Itô’s lemma, we can define the stochastic differential equation
of the bond ratio with fixing and paying times TF and TP :
P (t, TF )
P (t, TF )
=
σ (B(t, TP ) − B(t, TF )) dW QTP
(3.3.6)
d
P (t, TP )
P (t, TP
23
Where dW QTP represents a Brownian under the TP -forward measure. Using this
measure we can for instance price financial contracts using the zero-coupon bond
maturing at time TP as numeraire, when applying the numeraire dependent
pricing formula.
Note that (3.3.6) may also introduce a drift. However as we want absence
of arbitrage, we assume that this drift equals zero. According to the first fundamental theorem of asset pricing, every relative price process under the riskneutral measure must be a martingale. For more information on the NDPF
and the FTAP, see appendix A.3. Using (3.3.6), we can rewrite equation (3.2.7)
into:
dF̃t (T0 ,TN )
F̃t (T0 ,TN )
F0 (T0 ,TN )P (0,TN )P (t,T0 )
F̃t (T0 ,TN )[P (0,T0 )−P (0,TN )]P (t,TN )
=
· σ[B(t, TN ) − B(t, T0 )]dW QTN
In determining the swaption’s volatility, we use a second approximation to
(t,T0 )
by their initial valthe equation above. By replacing F̃t (T0 , TN ) and PP(t,T
N)
ues F̃0 (T0 , TN ) and
measure, we get:
P (0,T0 )
P (0,TN )
respectively, and changing measure to the annuity
P (0, T0 )
dF̃t (T0 , TN )
≈ drift + σ
[B(t, TN ) − B(t, T0 )]dW QA
P (0, T0 ) − P (0, TN )
F̃t (T0 , TN )
As this may introduce a drift, we again assume it to be equal to zero in order to
have absence of arbitrage. Using the equation above, one can find the volatility
of the swap rate by multiplying the volatility of the approximated forward swap
rate times the difference between the initial and expiry date of the option. We
get the following expression:
Vol FtSMM (T0 , TN ) = σ
p
P (0, T0 )
B(T0 , TN ) T0 − t
P (0, T0 ) − P (0, TN )
(3.3.7)
Note that σ represents the
volatility of the short rate process defined in (3.3.1),
and Vol FtSMM (T0 , TN ) represents the volatility of the approximated forward
swap rate under the corresponding short rate model until expiry, where t, T0 and
TN represent the swaption’s settle, expiry and the underlying swap’s maturity
date respectively.
Using Vol(FtSMM ) and Vol(FtObs ) from equations (3.3.7) and (3.2.9) respectively, the Hull-White one-factor model can be calibrated to market data using
the SMM approximation. As we have discussed already, results may not be
very market representative as the SMM approximation uses two approximations. This method therefore may only be preferred when computation time is
(very) costly.
3.3.2
Finishing the Jamshidian’s Decomposition for the
HW1F Model
This section we will finish the analytical expression for the swaption premium
under a Hull-White one-factor model using Jamshidian’s Decomposition. In
section 3.3.2 we showed that a swaption can be written as a combination of
24
put options on a zero-coupon bond. The analytical expression for the zerocoupon bond put option (ZBP) is dependent on the underlying short rate model.
Therefore we will derive the expression for a ZBP first, after which we will define
the expression for the swaption.
A zero-coupon bond put option (ZBP) gives the holder the right to buy a
zero coupon bond at the exercise date of the option, for a predetermined strike
price. Say at current time 0, we are the holder of a zero coupon bond with
exercise date T0 . The underlying zero coupon bond pays off e1 at maturity
date TN . The strike of the option equals X. Payoff at exercise equals:
[X − P (T0 , TN )]+
One can compute the initial value of this payoff in several ways. This can be
done using Margrabe’s formula [15] or again by using the numeraire dependent
pricing formula. The value of a zero coupon bond put option under the HW1F
model becomes:
ZBP(T0 , TN , X) = XP (0, T0 )N (d1 ) − P (0, TN )N (d2 )
where:
d1
d2
=
=
ln
√
P (0,T0 )X
P (0,TN
Vp (0,T0 ,TN )
P (0,T )X
ln P (0,T0
N
√
Vp (0,T0 ,TN )
+
1
2
p
Vp (0, T0 , TN )
−
1
2
p
Vp (0, T0 , TN )
Here Vp (0, T0 , TN ) stands for the variance of the bond forward price, which is
defined as follows:
Vp (0, T0 , TN ) = σ 2
1 − e−2a(T0 −t)
B(T0 , TN )2
2a
Jamshidian [12] shows that the option on a swap can be expressed as a
combination of zero coupon bond put options, assuming the short rate process
is defined by a one-factor model. The value for a payer swaption can be expressed
as follows:
N
X
ci ZBP(T0 , Ti , Xi )
πt =
i=1
where:
ci
=
Kδ(Ti−1 , Ti )
cN
=
1 + Kδ(TN −1 , TN )
and where:
Xi = A(T0 , Ti )exp(B(T0 , Ti )r∗ )
Here A(T0 , Ti ) and B(T0 , Ti ) are defined in equations (3.3.5) and (3.3.4) respectively, and we must have that r∗ satisfies the following equation:
N
X
ci A(T0 , Ti )exp(−B(T0 , Ti )r∗ ) = 1
i=1
25
3.3.3
How to Perform Calibration
In this section we will discuss how calibration was performed for the Hull-White
one-factor model. In order to take into account some of the conventions in
the data, multiple calibrations were done in order to see the effect of these
conventions on the results.
In the following table we displayed the different choices we had during this
calibration. Note that not all calibrations are not possible. We will discuss each
choice below:
Table 3.1: Choices during Calibration
Term structure: Nelson-Siegel function / Splines / Calculated
Calibrated to:
Observed premiums / Implied volatilities
SMM approximation / Jamshidian’s Decomposition
Method:
Optimization:
Simulated Annealing / Levenberg-Marquardt-Fletcher
Weights:
Absolute / Absolute-Relative / Relative
Calibrations were based on three different term structures. The first curve
is based on the term structure defined by the Nelson-Siegel function, which is
calibrated to certain data points. This allows for a smooth term structure of
interest rates, however the data points are not fitted perfectly. The Nelson-Siegel
function is defined more extensively in appendix B.6.
The second curve is based on the term structure which is found by interpolating data points using splines. Splines are sufficiently smooth polynomial
functions. Each of those functions are defined between every two subsequent
data points. Using splines all data points fit perfectly, however the disadvantage
here is that the curve between data points might be curved as in reality.
Figure 3.1: Term structures interpolated using Nelson-Siegel function & Splines
In figure (3.1) the discount curves for the calibration dates ultimo 2011
and 2012 are displayed for both Nelson-Siegel’s method and interpolating using
26
splines. All curves are plotted up to a maturity of 20 years only, since we do
not need any longer when calibrating swaptions with a maximal option expiry
and swap’s maturity of 10 years. Differences between curves are small, however
may have a large impact on the results of calibration. This impact on calibrated
parameters will be discussed later on.
Finally, a third discount curve was calculated, since for the data originating
from ultimo 2003, 2005 and 2008 a correct term structure was missing. According to the specifications of the swaptions originating from that dates, the
discount curve used to price these swaptions is based on the EURIBOR-curve.
However when using this EURIBOR-curve, relative pricing errors when using
Black’s formula are as high as 80% of the observed premiums.2 Therefore in
order to use the data from these years, the discount rates were computed by
inverting Black’s formula. As a result, we get the implied discount rates for the
maturities 2 years up to 20 years. In Figure (3.2) these discounts are plotted,
when interpolating using splines. However one should keep in mind here, as
discussed before, that the observed premiums are not the fundamental prices.
Therefore this calculated discount curve is based on several conventions, so that
calibration results should be handled with care.
Figure 3.2: Observed EURIBOR & calculated term structures by inverting
Black’s Formula
As expected, figure (3.2) shows that the calculated curve does not fit the observed curve very well. The calculated curve is not very smooth either. However
according to the pricing of the swaptions, which is based on Black’s Formula,
this should be the curve we are looking for. Therefore we will use this curve
as a third option when calibrating. Note that the figure is only plotted for the
maturities 2 up to 10 years, as only these maturities can only be derived using
the data we have. Extrapolation of these data points is done using linear lines
between data points, connecting one point with the other using a straight line.
Another choice we make when calibrating is whether to calibrate to observed
2 Swaption premiums can be calculated using Black’s formula. For more information on
Black’s formula see appendix B.2. For more information on the fit of the calculated premiums
see appendix B.4.
27
premiums or implied volatilities. Note that observed premiums are actually the
average of the corresponding bid- and ask-price. These premiums also may have
some transaction costs implemented. Although these conventions may introduce
pricing errors, these should be small as the swaption is considered a very liquid
product.
The calibration based on the implied volatility is a calibration based on the
swaption premium calculated by implementing the implied volatility in Black’s
formula in appendix B.2. By definition, the swaption’s price is defined using
its implied volatility and vice versa. As the implied volatility is given by a
single number per swaption, no bid- or ask-volatilities, the premium calculated
using this volatility can also be seen as market representative. Note here that
calibrating the HW1F-model using a calculated term structure and a calculated
premium is not possible, as one is implied by the other and vice versa.
Next we can choose between a calibration using the SMM approximation or
Jamshidian’s Decomposition, where the first method is considered an approximation and the latter is exact. Both methods were introduced in section 3.3
and applied to the Hull-White one-factor model in section 3.4.1 and 3.4.2.
The algorithm finding the optimal solution for our calibration problem can
be simulated annealing or the Levenberg-Marquardt-Fletcher algorithm. Both
methods are used in order to be more certain of our solution, as one could
come up with a local minimization point. As it turns out, the solutions to
all calibrations are the same for both methods. For more elaboration on both
simulated annealing as the Levenberg-Marquardt-Fletcher algorithm see section
3.2.
Finally we define three different weights to implement in our optimization
problem, equation (3.1.1). The first type we discuss are called the ‘absolute
weights’, defined as follows:
wi = 1
So that in this case we calibrate by minimizing the absolute difference between
the observed and calibrated premiums. Next we define ‘relative weights’. Obviously, using these weights we minimize the relative error between observed and
calibrated premiums:
−2
wi = ViObserved
As our third choice in weights, we will define weights which lie in between
absolute and relative weights. This we call the ‘absolute-relative weights’:
wi = ViObserved
−1
Now we have discussed all options when calibrating, we can discuss the results.
3.3.4
Results Calibration Hull-White one-factor Model
The results for the calibration of the Hull-White one-factor model are displayed
in appendix D, tables (D.1) up to (D.6). We will the discuss the results by
discussing the difference in various options defined in the previous section. We
also define a ‘base case’ calibration, which makes use of the following, most
reasonable, options:
Before discussing the various choices, note that the value for a is negative for
quite some calibrations. This is unrealistic as the short rate process would be
28
Table 3.2: Base Case Calibration:
Term structure: Nelson-Siegel function
Observed premiums
Calibrated to:
Method:
Jamshidian’s Decomposition
Levenberg-Marquardt-Fletcher
Optimization:
Weights:
Absolute-Relative
mean-fleeing for these parameters choices. These calibrations therefore become
useless for generating economic scenarios. Also note that the calibration results
when using the SMM approximation or the calculated term structure should be
handled with care. The conventions and approximations used in these calibrations may give misleading results.
The difference in results when calibrating based on Nelson-Siegel functions
or splines is significant. Not only do parameter values change, the RMSE is also
significantly different. Although the difference in values is lower for the base
case calibration, we will have to determine this effect on the HBS.
Next we will discuss the results when calibrating to implied volatilities instead of the observed premiums. Note that we only use the calibrations based
on Jamshidian’s Decomposition here, as there is no comparison for the SMM
approximation. We see that parameters values as well as the RMSE changes
very slightly. Since these small changes will not have such an effect on the HBS
as the different interpolation methods will have, we will omit this choice when
generating the HBS.
The SMM approximation gives us entirely different parameters compared to
a calibration based on Jamshidian’s Decomposition. Note here that one should
interpret the RMSE of a calibration based on SMM approximation in terms of
the volatility instead of a premium. We also see that the short rate variances
implied by the calibrations performed using the SMM approximation are too
small. For most of these calibrations this variance is a factor 10 too small
compared to the base case calibrations.
Finally we will discuss the different weights one can use. The first thing
we notice, is that the parameter values for a and σ become smaller when we
go from absolute weights to absolute-relative weights to relative weights in the
base case calibration. The second thing we notice is that the RMSEs for all
calibrations using absolute weights are smaller compared to the RMSE from
corresponding calibrations using absolute-relative weights. The same holds for
all RMSEs using absolute-relative weights, which are smaller compared to their
corresponding RMSEs using relative weights. This is correct per definition, as
the RMSE is equal to the value computed in equation (3.1.1) using absolute
weights. Therefore we have that the RMSEs using absolute weights are the
smallest overall.
In chapter 4 we will use the calibration results in order to create a riskneutral scenario set. We will do this several times in order to take into account
the variability we see over data sets and choices we make when calibrating. In
chapter 6 we will see what impact this variability has on the holistic balance
sheet. For now, we will discuss the calibration of (two) two-factor models.
29
3.4
Calibrating the Hull-White Two-Factor Model
In this section we will discuss the results for calibrating the Hull-White twofactor model [11]. As the name indicates, this model was based on the HullWhite one-factor model discussed in the previous section. The difference between the two models is that the two-factor model adds a stochastic process
{ut } in the drift of the process of the short rate. This extra stochastic term is
also correlated to the process of the short rate. Whether the extra term also
provides a better fit when calibrating will be shown later in this section.
For the actual calibrating, we will use a different two-factor model. This
model is called the Two-Additive-Factor Gaussian (G2++) model. We will do
this as both models are equivalent to each other, and it is easier to derive an
analytical expression under the G2++ model. The analogy between the G2++
and HW2F model will be discussed as well.
Before we define this analytical formula, we start by giving a little introduction on the Hull-White two-factor model. The Hull-White two-factor model
defines the process of the short rate according to:
(θ(t) + ut − art ) dt + σ1 dW1,t
drt
=
dut
= −but dt + σ2 dW2,t
r(0) = r0
(3.4.1)
where θ(t) is a deterministic function of time, such that the short rate process
fits the observed term structure perfectly. Here W1,t and W2,t are two Brownian
motions under the risk-neutral measure. These Brownian motions are correlated
as well:
ρ dt = dW1,t dW2,t
The exact expression for the short rate follows after integrating and rewriting (3.4.1). We get the following:
Rt
Rt
r(t) = r0 e−at + 0 θ(v)e−a(t−v) dv + σ1 0 e−a(t−v) dW1 (t)
R t −b(t−v)
σ2
[e
− e−a(t−v) ] dW2 (t)
+ a−b
0
The proof for this expression is shown in appendix C.7.
3.4.1
Two-Additive-Factor Gaussian Model
In this section we will discuss the two-additive-factor Gaussian model, in order
to provide an analytical swaption formula for calibration purposes. Later on we
will provide the analogy of the G2++ model with the HW2F model. In order to
keep notation distinctive, we will write parameters of the G2++ model barred.
Under the G2++ model the short rate process is defined as:
rt = xt + yt + ϕ(t)
(3.4.2)
where ϕ(t) is a deterministic function in time, in order for the short rate process
to fit the term structure perfectly. The other terms on the right hand side, xt
and yt , are two stochastic processes which are defined as follows:
dxt
= −āxt dt + σ̄ dW1,t
x(0) = 0
dyt
= −b̄yt dt + η̄ dW2,t
y(0) = 0
30
(3.4.3)
Again we denote W1,t and W2,t to be two Brownian motions under the riskneutral measure Q, which are correlated as follows:
ρ̄ dt = dW1,t dW2,t
In the G2++ model we have that ā, b̄, σ̄ and η̄ are positive constants, and
−1 ≤ ρ̄ ≤ 1. The analytical solution for the short rate (3.4.2) can be derived by
integrating both stochastic differential equations in equation (3.4.3):
Rt
r(t) = x(s)e−ā(t−s) + y(s)e−b̄(t−s) + σ̄ s e−ā(t−u) dW1,t
(3.4.4)
Rt
+ η̄ s e−b̄(t−u) dW2,t + ϕ(t)
We can rewrite (3.4.4) for two independent Brownian motions W̃1,t and W̃2,t ,
by substituting the dependent Brownians motions in for independent Brownian
motions:
dW1,t = dW̃1,t
p
(3.4.5)
dW2,t = ρ̄ dW̃1,t + 1 − ρ̄2 dW̃2,t
So that we can rewrite (3.4.4) using (3.4.5), which gives us the following expression for the short rate:
Rt
r(t) = x(s)e−ā(t−s) + y(s)e−b̄(t−s) + σ̄ s e−ā(t−u) dW̃1,t
p
Rt
Rt
+ η̄ ρ̄ s e−b̄(t−u) dW̃2,t + η̄ 1 − ρ̄2 s e−b̄(t−u) dW̃2,t + ϕ(t)
Now that we have the solution for the short rate, we can define the zerocoupon bond structure under the Gaussian two-factor model. Like in the onefactor model, this structure will be important when finding an analytical expression for pricing swaptions. Again recall the pricing formula using the numeraire
dependent pricing formula when pricing a zero-coupon bond using the money
market account as numeraire:
"
!#
Z
T
P (t, T ) = EtQM exp −
r(u) du
(3.4.6)
t
Note that the expectation of a lognormally distributed variable K is E[exp(mK +
vK )], where where mK and vK denote the mean and variance of the normally
distributed variable log(K). Therefore we want to determine the mean and variRT
ance of normally distributed variable t r(u) du. Define the following integral:
Z
T
I(t, T ) =
[x(u) + y(u)]du
(3.4.7)
t
which is normally distributed with mean M (t, T ) and V (t, T ), defined respectively as:
M (t, T )
=
1−eā(T −t)
x(t)
ā
V (t, T )
=
σ̄ 2
ā2
+
1−eb̄(T −t)
y(t)
b̄
T − t + ā2 e−ā(T −t) −
1 −2ā(T −t)
2ā e
For proof of equation (3.4.8), see appendix C.5.
31
(3.4.8)
−
3
2ā
Using (3.4.8), we can rewrite equation (3.4.6) in order to define the zerocoupon bond structure:
P (t, T )
=
exp[−
RT
t
ϕ(u)du − B(ā, t, T )x(t)
−B(b̄, t, T )y(t) + 21 V (t, T )]
(3.4.9)
where:
1 − e−z(T −t)
(3.4.10)
z
Next we want to define the deterministic function ϕ(·), which ensures that
the G2++ model fits the currently observed term structures of interest perfectly.
The G2++ model fits the observed term structure of interest rates only if we
have the following:
#
" Z
B(z, t, T ) =
T
P M (0, T ) = exp −
0
ϕ(u) du + 12 V (0, T )
which is an application of equation (3.4.9), and where P M (0, T ) denotes the
currently observed discount factor. If we take logs, differentiate with respect to
T and take the negative, we get:
− δP
M
(0,T )
δT
= ϕ(T ) −
σ̄ 2
2ā2 (1
− e−āT )2 −
η̄ 2
(1
2b̄2
− e−b̄T )2
− ρ̄ σ̄āη̄b̄ (1 − e−āT )(1 − e−b̄T )
which must be equal to the instantaneous forward rate f M (0, T ), discussed in
appendix B.1, equation (B.1.6). Therefore we can rewrite in order to define
ϕ(T ) as:
ϕ(T )
= f M (0, T ) +
σ̄2
2ā2 (1
− e−āT )2 +
η̄2
(1
2b̄2
− e−b̄T )2
+ρ̄ σ̄āη̄b̄ (1 − e−āT )(1 − e−b̄T )
(3.4.11)
Using (3.4.11), we can also rewrite the zero-coupon bond structure, equation
(3.4.9), into:
P (t, T ) = A(t, T )exp −B(ā, t, T )x(t) − B(b̄, t, T )y(t)
where B(z, t, T ) as in equation (3.4.10), and A(t, T ) such that:
P M (0, T )
1
A(t, T ) = M
exp
(V (t, T ) − V (0, T ) + V (0, t))
P (0, t)
2
(3.4.12)
(3.4.13)
Now we completely defined our dynamics for the G2++ model and its underlying zero-coupon bond structure, we can price some swaptions analytically.
Since Jamshidian’s Decomposition from section 4.3.2 is only applicable to onefactor models, we cannot use this method. Therefore we give the analytical
premium of a swaption using integrals below.
Say we want to price a European swaption at current time t = 0. The option
expires at time T0 and the last tenor date of the underlying swap is defined at
time TN . We have that 0 < T0 < TN . The strike of the swaption is defined by
32
K, the notional amount by L. The arbitrage-free premium at time 0 under the
G2++ model becomes:
π0G2++
=
R∞
e
−1
2
2
x
( x−µ
σx )
LP (0, T0 ) −∞ σ 2π [Φ (−h1 (x))
x
PN
− i=1 λi (x)eκi (x) Φ (−h2 (x))]dx
√
where the following functions are used in the equation above:
ŷ−µy
√
−
2
ρxy (x−µx )
√
σx 1−ρ2xy
h1 (x)
=
h2 (x)
=
h1 (x) + B(b̄, T0 , Ti )σy
λi (x)
=
κi (x)
=
ci A(T0 , Ti )e−B(ā,T0 ,Ti )x
h
−B(b̄, T0 , Ti ) µy − 12 (1 − ρ2xy )σy2 B(b̄, T0 , Ti )
i
x
+ρxy σy x−µ
σx
σy
1−ρxy
q
1 − ρ2xy
(3.4.14)
and where ŷ = ŷ(x) is the unique solution of the following:
N
X
ci A(T0 , Ti )e−B(ā,T0 ,Ti )x−B(b̄,T0 ,Ti )ŷ = 1
i=1
In the equation above, the weights ci resemble those used in Jamshidian’s Decomposition for the one-factor model:
ci
cN
= Xτi
=
for i = {1, . . . , N − 1}
1 + XτN
where τi denotes the year fraction between time Ti−1 and Ti .
Finally we define the functions used in equations (3.4.14), which were not
yet defined:
µx = −MxT0 (0, T0 )
µy
σx
σy
ρxy
= −MyT0 (0, T0 )
q
−2āT
= σ̄ 1−e2ā 0
q
−2b̄T
= η̄ 1−e2b̄ 0
i
h
η̄
= (ā+ρ̄σ̄
1 − e−(ā+b̄)T0
b̄)σx σy
For proof, see appendix C.6.
3.4.2
From G2++ to HW2F
Now that we have the analytical expression for a swaption premium under the
G2++ model, we can calibrate the parameters for this model. After calibration,
we can convert these parameters to parameters for the Hull-White two-factor
model using the analogy between the two models. This conversion can be proven
by equalizing the exact expressions for the short rate under both models. Below
we will give the equivalence between each parameters. Note that the barred
33
parameters correspond to the parameters from the G2++ model (3.4.3). The
‘plain’ parameters are from the Hull-White two-factor model, corresponding to
(3.4.1).
The conversion from G2++ to Hull-White two-factor model parameters is
given as follows:
a = ā
b
σ1
= b̄
p
σ̄ 2 + η̄ 2 + 2ρ̄σ̄ η̄
=
σ2
= η̄(ā − b̄)
ρ
=
σ̄ ρ̄+η̄
σ1
θ(t)
=
dφ(t)
dt
+ aφ(t)
In order to be complete, we also give the conversion from Hull-White two-factor
to G2++ model parameters:
ā
= a
b̄
σ̄
= b
q
=
σ12 +
η̄
=
ρ̄
=
φ(t)
σ22
(a−b)2
1 σ2
+ 2ρ σb−a
σ2
a−b
σ1 ρ−η
σ̄
= r0 e−at +
Rt
0
θ(v)e−a(t−v) dv
For proof, see appendix C.7.
3.4.3
Results Calibration Hull-White Two-Factor Model
In this section we will interpret the results when calibrating the Hull-White
two-factor model, displayed in tables (D.7), (D.8) and (D.9) in appendix D.
Note that we did not perform every calibration possible, as the calibration
of the Hull-White one-factor showed us some calibrations are not very useful.
As discussed in section 3.3.4 for the calibration of the one-factor model, we
see that the difference in results between calibrating exactly using observed
premiums or observed implied volatilities is very small when using Jamshidian’s
Decomposition. Therefore we chose only to calibrate the two-factor model to
the observed premiums. We also did not perform any calibration based on
approximations, as the SMM approximation is applicable to one-factor models
only. What we will do is calibrate the two-factor model using different weights,
which we defined in section 3.3.3. This way we can see how the results differ
when giving other weights to the same set of swaptions.
The results for the calibration of the G2++ and the HW2F-model are displayed in tables (D.7), (D.8) and (D.9) in appendix D. On first sight the calibration of the two-factor in general seems more consistent over different weights
and over different years. Also note that the variance of the short seems a bit
lower compared to the one-factor base case calibration. This could result in a
34
significantly different holistic balance sheet, when switching from one-factor to
two-factor model.
Using these results, we will see what the impact will be on the holistic balance
sheet and the SCR in chapter 6. We will not only consider using different short
rate models, but also using different parameters. This variety in parameterchoices will come from the calibrations using different weights. For now we will
discuss our third short rate model, and see if this model is useful for scenario
generating purposes.
3.5
Calibrating the Cox-Ingersoll-Ross ++ Model
In this section we will discuss the third calibrated short rate model, the CoxIngersoll-Ross (CIR)++ in this thesis. The CIR++ model is a modification of
the CIR model, which allows for a perfect fit of the current term structure. The
CIR model is not able to provide a perfect fit. The most important difference
of the CIR++ model compared to the Hull-White models is that the CIR++
model implements a volatility term conditional on the short rate itself.
In order to derive an exact swaption formula for the CIR++ model, we will
discuss the CIR model first. The difference between the CIR model and its
successor is that the CIR++ model adds a value determined by a deterministic
function ϕ(·) to the stochastic process in order to provide a perfect fit of the
initial term structure.
3.5.1
The Cox-Ingersoll-Ross Model
The Cox-Ingersoll-Ross model [4] was introduced in 1985 as an extension of
the Vasicek model. The Vasicek model uses a constant mean-reversion level,
constant speed of mean-reversion and constant volatility. The CIR model makes
this volatility term variable on the current short rate. The CIR model also does
not imply a perfect fit of the initial term structure. In formula, we have that
the CIR model is stated as follows:
√
drt = k (θ − rt ) dt + σ rt dWt
where the mean and variance of the short rate implied by the CIR model are
given as follows:
E[r(t)|Fs ]
=
Var[r(t)|Fs ]
=
r(s)e−k(t−s) + θ(1 − ek(t−s) )
2
r(s) σk e−k(t−s) − e−2k(t−s)
2
2
+θ σ2k 1 − e−k(t−s)
(3.5.1)
Like in the Hull-White one-factor model, we will use Jamshidian’s Decomposition in order to derive an exact swaption formula for the CIR model. Before
we define zero coupon bond put option prices, we will define the zero-coupon
bond structure as:
P (t, T ) = A(t, T )e−B(t,T )r(t)
35
where
A(t, T )
=
B(t, T )
=
h =
h
2hexp{(k+h)(T −t)/2}
2h+(k+h)(exp{(T −t)h}−1)
i2kθ/σ2
(3.5.2)
2(exp{(T −t)h}−1)
2h+(k+h)(exp{(T −t)h}−1)
√
k 2 + 2σ 2
Using this, we can define the price of a zero coupon bond call option. Assume
current time t, option maturity T0 , strike price K and underlying maturity of
the zero coupon bond to be TN .
ZBC(t, T0 , TN , K) =
2ρ2 r(t)exp{h(T −t)}
P (t, TN )χ2 2r̄[ρ + ψ + B(T0 , TN )]; 4kθ
,
σ2
ρ+ψ+B(T0 ,TN )
2
4kθ 2ρ r(t)exp{h(T −t)}
2
−KP (t, T0 )χ 2r̄[ρ + ψ]; σ2 ,
ρ+ψ
where
ρ =
ρ(T0 − t)
ψ
=
k+h
σ2
r̄
= r̄(TN − T0 )
=
2h
σ 2 (exp[h(T0 −t)]−1)
=
ln(A(T0 ,TN )/K)
B(T0 ,TN )
(3.5.3)
Using the put-call parity, we can define the zero coupon bond put option:
ZBP(t, T0 , TN , K) = ZBC(t, T0 , TN , K) − P (t, TN ) + K P (t, T0 )
(3.5.4)
So that by using Jamshidian’s Decomposition, equation (3.2.10), we can derive
the swaption premium under the CIR model.
3.5.2
Swaption Premium under CIR++ Model
In this section we will modify the formulas from the previous section for the
CIR++ model. As discussed before, the difference between the CIR and CIR++
model is that the CIR++ model is able to provide a perfect fit of the initial
term structure. The CIR++ model is defined as follows:
√
dxt = k(θ − xt ) dt + σ xt dWt ,
x(0) = x0
r(t)
= x(t) + ϕ(t)
ϕ(t)
= f M (0, t) − f CIR++ (t)
where:
f CIR++ (t)
=
2kθ(exp{th}−1)
2h+(k+h)(exp{th}−1)
2
4h exp{th}
+ x0 [2h+(k+h)(exp{th}−1)]
2
The mean and variance of the process {x} remains equal to (3.5.1). The
zero coupon bond structure becomes equal to:
P (t, T ) = Ā(t, T )e−B(t,T )r(t)
where
Ā(t, T ) =
P M (0, T )A(0, t)exp{−B(0, t)x0 }
A(t, T )eB(t,T )ϕ(t)
P M (0, t)A(0, T )exp{−B(0, T )x0 }
36
and where A(t, T ) and B(t, T ) are defined in (3.5.2). Therefore we only have to
modify the zero coupon bond call option for the new zero coupon bond structure.
This modification can be defined as follows:
M
(0,TN )A(0,t)exp{−B(0,t)x0 }
ZBC(t, T0 , TN , K) = PPM (0,t)A(0,T
N )exp{−B(0,TN )x0 }
P M (0,T0 )A(0,TN )exp{−B(0,TN )x0 }
CIR
·Ψ
t, T0 , TN , K P M (0,TN )A(0,T0 )exp{−B(0,T0 )x0 } , r(t) − ϕ(t)
where ΨCIR is the CIR zero coupon bond call option price defined in (3.5.3),
with r(t) = x(t). By rewriting the equation above, we get:
ZBC(t, T0 , TN , K) =
2ρ2 [r(t)−ϕ(t)]exp{h(T −t)}
P (t, TN )χ2 2r̄[ρ + ψ + B(T0 , TN )]; 4kθ
,
2
σ
ρ+ψ+B(T0 ,TN )
2
4kθ 2ρ [r(t)−ϕ(t)]exp{h(T −t)}
2
−KP (t, T0 )χ 2r̄[ρ + ψ]; σ2 ,
ρ+ψ
(3.5.5)
where
r̂ =
1
B(T0 ,TN )
h
i
M
0 ,TN )
0 )A(0,TN )exp{−B(0,TN )x0 }
ln A(TK
− ln PP M(0,T
(0,TN )A(0,T0 )exp{−B(0,T0 )x0 }
So that when calculating the payer swaption premium under the CIR++ model,
one can use Jamshidian’s Decomposition in combination with the analytical
expression for the zero coupon bond put option. Note that this expression can
be derived using the zero coupon bond call option, equation (3.5.5), and the
put-call parity, equation (3.5.4).
3.5.3
Calibration Results CIR++ Model
In this section we will discuss the calibration results for the Cox-IngersollRoss++ model. Like for the Hull-White one-factor and two-factor model, we
calibrate the exact swaption expression for the corresponding short rate model
to the observed swaption premiums. This is again done using three different weights: absolute, absolute-relative and relative weights. For definitions of
these weights see section 3.3.3. The calibrations were also done using the observed EONIA-curve, interpolated using the Nelson-Siegel function. The results
are displayed in tables (D.10), (D.11) and (D.12). Note that the results were
found using both the simulated annealing as the Levenberg-Marquardt-Fletcher
algorithm, discussed in section 3.2.
The first thing we notice is that the calibration results per year differ a lot.
Although for both year the speed of mean-reversion k is very small, the meanreversion level of 2011 is very small compared to those of 2012. The parameter
σ seems quite constant, around 0.06.
The RMSEs of the calibrations in 2011 are smaller compared to those of
2012. As a matter of fact, these RMSEs are smaller compared those values
corresponding to the Hull-White one-factor and Hull-White two-factor. This
indicates the CIR++ is able to provide a better fit over the observed swaptions.
Note that this is only the case for 2011. For the 2012 calibrations we see better
results under the HW2F model.
As for the economic scenario generator, used to value the holistic balance
sheet, one could argue to use the short rate model providing the best fit, or
37
smallest RMSE. There is however a big downside to the CIR++ model. When
generating 2500 scenarios under the CIR++ using the parameters defined in
tables (D.10), (D.11) and (D.12), we see that the process {x} becomes negative.
For 2011 this happens after 5 years, for 2012 after 10 years. When {x} becomes
negative, the next short rate value in the process will be a complex value due
to the square root in the volatility term of the process. Therefore the CIR++
model becomes no good for scenario generating purposes, and we have to omit
this model out of our sensitivity analysis in chapter 6.
38
Chapter 4
The Economic Scenario
Generator
The short rate models, discussed in the previous chapter, are only a part of the
economic scenario generator used to value the holistic balance sheet. The ESG
actually exists of multiple processes, describing several state variables. In this
chapter we will finish the ESG by defining the behavior of other state variables
such as price inflation, wage growth and stocks. In section 4.2 we will show
how one can use the ESG in order to value steering mechanisms of our fictitious
fund.
4.1
4.1.1
Defining other State Variables
Stocks
In this section we will define the stock behavior in our ESG. We will define only
one stock process in which our fictitious fund can invest in. The behavior of this
stock will be derived from an average of stocks traded in the current market.
But first, we want to define the process under the risk-neutral measure.
In order to define our stock process, we will use the famous process defined
by Black and Scholes [2]:
dSt = µS St dt + σS St dWtP
(4.1.1)
This equation states that the stock process follows a Geometric Brownian
motion. For clarity, the Brownian motion dWtP denotes a Brownian motion
under the objective probability measure P. We can use the change of numeraire
in order to rewrite (4.1.1) under the risk-neutral measure. The change is measure
is explained in appendix A.3. We get the following equation:
dSt = rt St dt + σS St dWtQ
(4.1.2)
so that under the risk-neutral measure Q the stock behavior is influenced by
the process of the short rate. In order to be complete, we will also compute the
correlation between the short rate and stock process in section 4.1.3. For now
we will discuss the data used to derive the volatility term in equation (4.1.2).
39
As one can see, the volatility term for the stock process under Black-Scholes
remains unchanged when changing measure from objective to risk-neutral measure. Therefore we only have to define the parameter σS under the Q-measure,
as the short rate will be defined by another process and the initial value S0 is
given. The parameter σS will be defined per year of calibration, like we did with
the calibration of the short rate models. Therefore we will define σS by limiting
ourselves to stock data available until ultimo 2003, 2005, 2008, 2011 and 2012.
The stock data consists of average returns per region. The distinction here is
made between four regions: North-America, Europe, Asia and other emerging
markets. The following weights per region were used in order to derive the
volatility:
Table 4.1: Regionweights per market
Regionweights:
Market:
North-America
44%
24%
Europe
Asia
15%
Emerging markets
17%
Total
100%
Using these weights we can define σS in the stock process (4.1.2), which are
stated in table (4.2). Note that σS is extremely high in 2008, which is due to
the financial crisis.
Table 4.2: Stock volatility parameter σS per calibration year
Year:
σS :
2003
18.78%
2005
25.45%
35.92%
2008
2011
15.04%
2012
17.24%
4.1.2
Inflation
In this section we will define the process for price inflation, as well as the process
for wage growth. These processes will be heavily correlated. Both are needed
in order to derive the holistic balance sheet for our fictive fund. We define both
processes in the same way as we defined our stock process according to BlackScholes. Therefore we assume that price inflation and wage inflation each follow
a Geometric Brownian motion:
dPtP
=
Q
P
P P
µP
P Pt dt + σP Pt dW1,t
dPtW
=
Q
W
W W
µW
P Pt dt + σP Pt dW2,t
Where we assume some correlation between the Brownian motions W1,t and
W2,t .
40
4.1.3
Correlations
The correlations used in this thesis are derived using historical returns on stocks,
as well as historical indices of price inflation. In the ESG we will use the
Cholesky decomposition in order to create a correlation matrix. However, as we
will have some difficulty identifying the correlations with respect to both processes of the HW2F (G2++) model, as well as the problem of so-called ’broken’
correlation matrices, we must define these correlations with care. Therefore we
will model the holistic balance sheet with and without any correlations between
the defined processes, in order to see the impact on the various options as well
as the solvency capital requirement.
Using historical returns and indices, we get the following correlations between the short rate, price inflation and stocks for ultimo 2003, 2005, 2008,
2011 and 2012:
Table 4.3: Correlations between
2003
2005
ρP S -0.44102 0.12648
ρP r
0.91617 0.49042
-0.37165 0.11204
ρSr
short rate,
2008
-0.49921
0.26968
0.06089
price inflation and stocks
2011
2012
-0.09333 -0.05038
0.76112 -0.36302
-0.13355 -0.18910
As one can see, there is no clear relation between short rate, price inflation
and stocks. We can therefore question whether the correlations stated above
are indeed significantly different from zero. We will not test this throughout
this thesis, but we will investigate what the impact on the HBS will be when
correlations are included or not.
The second problem that arises is the phenomenon of so-called ’broken’
matrices. Using table (4.3) we can create a 3-by-3 correlation matrix. By definition, in order for the correlation matrix to be mathematically correct, this
matrix must be positive definite. A matrix is positive definite whenever the
corresponding eigenvalues of that matrix are all positive. In our case the correlation matrices are not positive definite, so the requirement is not met. These
matrices we call ’broken’. Joubert and Langdell [13] sum up some methods in
order to ’fix’ these broken matrices, however none of these methods are mathematically grounded. The phenomenon of broken matrices is still a widely known
problem of which its solution is not known (yet).
Whenever correlations are used in the ESG, we use one of the methods
described by Joubert and Langdell to fix the correlation matrix. Althrough
mathematically ungrounded, correcting the correlation matrix by setting negative eigenvalues to be positive does allow us to create a correlation matrix which
is based on the original. After all, it is still interesting for us to see what the
correlation does to the simulated HBS and SCR.
4.2
Valuing Embedded Options using the ESG
In this section we will discuss how to value each steering mechanism defined in
section 2.4. The valuation methods are similar to those presented in the paper
by De Haan et al. [9]. We start of valuing the options of the sponsor support,
41
after which we will discuss the five options of the adjustment mechanism. We
will end this chapter by discussing the residue option.
4.2.1
Sponsor Support
Sponsor support consists of the aggregated value of four options: employer contribution option, employer guarantee option, premium cut option and premium
damping option. In formula:
V0SPS = V0EC + V0EG + V0PC + V0PD
Note that premium cuts and premium damping may also be considered as negative sponsor support. We will discuss each option below.
Employer Contribution Option
The employer contribution option mainly consists of the recovery premiums that
the fund requires participants to pay whenever the funding ratio is too low. In
other words, the employer contribution option consists of the amount of excess
contribution paid in the future, which is valued today.
Define ccccr,t to be the cost covering contribution rate at time t. This rate
ccccr,t will equal the new accrued benefits divided by the number of participants,
multiplied with the return on equity:1
N ABt · (1 + S)
ccccr,t = P64
x=25 P opx,t
Here N ABt stands for the value of the newly accrued benefits, and P opx,t
equals the number of participants of age x at time t. Note that the pension
fund always receives the cost covering contribution rate from its participants in
order to cover the new accrual. Therefore this rate itself is not considered an
option.
In order to value the option of the employer contribution, or recovery premiums, we want to value the amount of premium that is paid in excess of the
cost covering contribution. For instance, say ct equals the total premium paid
at time t. This means that the aggregated value of recovery premiums equals
cEC
pool,t :
cEC
pool,t =
64
X
(ct − cccr,t )wx,t · P opx,t
x=25
where wx,t equals the average wage level at time t for age x. In order to value
the option of recovery premiums at current time (say t = 0), we will apply the
numeraire dependent pricing formula. As a numeraire we use the money market
account, so that the employer contribution option value becomes:
!
T
−1
X
1
Q
EC
EC
t
V0 = E0
cpool,t · Πk=0
1 + rk
t=0
1 Since
the cost covering contribution rate is defined uniformly among all participants.
42
where rk denotes the short rate at time k. Note that we will not be able to
compute the expectation used above exactly. We therefore use Monte-Carlo
simulations in order to compute the value of the estimation. In other words, by
simulating the fictive fund under the risk-neutral measure for several years we
can value the estimation by simply taking the average. Say we have N scenarios
in order to compute the employer contribution option as follows:
!
n
T −1
1
1X X s
t
EC
c
Π
(4.2.1)
V0 =
n s=1 t=0 pool,t k=0 1 + rks
Here cspool,t and rks are the recovery premium and short rate respectively for
scenario s, modeled under the risk-neutral measure Q.
Employer Guarantee Option
When modeling our holistic balance sheet our employer guarantee consists of
the extra premium the employer contributes to the premium paid by the participant. Say this is extra premium is given in terms of a rate cEG
t , which may be
conditional to the funding ratio of the fund at that time. In our case we assume
employer guarantee is only given if the funding ratio is less than 130%. We can
define the employer guarantee as:
cEG
pool,t
=
64
X
cEG
t wx,t P opx,t
x=25
So that we can value the employer guarantee option as:
V0EG
=
E0Q
T
−1
X
t
cEG
pool,t Πk=0
t=0
1
1 + rk
!
Which can also be stated in terms of a estimated value of simulated scenarios,
as in equation (4.2.1).
Premium Cut Option
A premium cut may be given whenever the funding ratio is sufficiently high.
For our own fund, we define the premium cut to be conditional on the funding
ratio. Therefore we define the cut to be equal to:
cPC
t
=
0
if F Rt ≤ f loor
F Rt −f loor
cap−f loor
if f loor < F Rt < cap
cPC
max
if F Rt ≥ cap
For our fund we will typically use the floor equal to 130% nominal funding
ratio, and cap equal to 100% real funding ratio. Next we can define the absolute
pooled value for the premium cuts at time t to be equal to:
cPC
pool,t =
64
X
cPC
t wx,t P opx,t
x=25
43
Using the same numeraire, we can value the premium cut option just like the
previous two options:
!
T
−1
X
1
Q
PC
PC
t
V0 = E0
cpool,t Πk=0
1 + rk
t=0
Premium Damping Option
A fund is required to determine the cost covering contribution rate, discussed
in section 4.2.2. This rate however fluctuates a lot, as it is based on the fluctuating term structure at that time. Therefore a fund is allowed to dampen
the cost covering contribution rate by determining this using an averaged term
structure. This creates some value differences between the actual cost covering
D
contribution rate. We define this difference to be cP
pool,t , which is the pooled
value for premium damping at time t. Again, using the numeraire dependent
pricing formula we can value this option at time 0:
!
T
−1
X
1
Q
PD
PD
t
V0 = E0
cpool,t Πk=0
1 + rk
t=0
Again, this value can be estimated using Monte-Carlo simulations.
4.2.2
Adjustment Mechanism
The adjustment mechanism consists of the aggregated value of four options:
Indexation option, surplus sharing option, benefit cut option and the catch up
indexation option. In formula:
V0SPS = V0IND + V0SUR + V0BC + V0CAT
Typically each mechanism influences the level of indexation given to the
pension payments. For indexation and catch up indexation this is obvious.
Surplus sharing and benefit cuts may also be seen as some kind of indexation,
working in opposite directions. As all four mechanisms are expressed in the
level of indexation only, we will derive the value of the option in this section. In
the next sections we will see how each mechanism affects the indexation level.
Say we have some average level of obligations per participant, aged x, valued
at time t: Bx,t . Whenever indexation is given, the benefits will be multiplied
with the fraction of total indexation: indt+1 . The benefits become as follows:
ind
Bx,t+1
= indt+1 · Bx,t
(4.2.2)
In order to have seperate options eventually, we can split up indt+1 into four
different values:
indt+1 =
4
X
indi,t+1
i=1
where indi,t+1 for i = 1, . . . , 4 stands for indexation, surplus sharing, benefit
cuts and catch up indexation respetively. Using (4.2.2), we can derive the value
of the pooled benefits of the fund, indexed using one type of mechanism i:
44
indi
Bpool,t+1
=
∞
X
indi
male
female
Bx,t+1
Dx,t
· P opmale
· P opfemale
x,t + Dx,t
x,t
x=25
Now we can use the numeraire dependent pricing formula as in the previous
sections. Using the money market account as numeraire, we can value each
option using the following equation:
!
T
X
1
Q
indi
indi
t
= E0
V0
Bpool,t Πk=0
1 + rk
t=1
which we will again estimate using Monte Carlo estimations:
!
n
T
X
X
1
1
ind
,s
V0indi =
B i Πt
n s=1 t=1 pool,t k=0 1 + rks
(4.2.3)
In the next sections we will define ind for each option in the adjustment
mechanism.
Indexation Option
In general we have three types of indexation: No indexation, full indexation
and conditional indexation. Our fund will only have the choice between full
indexation and conditional indexation. In case the fund gives full indexation,
define ind1 = w. Here w will be the rate benefits are indexed to. For our fund
we will use wage growth.
In case we have conditional indexation, we have that ind1 at time t is equal
to:
if F Rt ≤ f loor
0
ind1,t = wt ·
F Rt −f loor
cap−f loor
if f loor ≤ F Rt ≤ cap
if F Rt ≥ cap
wt
Where we typically use f loor = 105% and cap = 120%. Using ind1,t we can
value the indexation option using (4.2.3).
Surplus Sharing Option
Surplus sharing is given in a form of indexation, whenever the funding ratio is
above some level. The surplus is smoothed out over a number of years. Note
that the surplus sharing is only given if the funding ratio is still above the
pre-determined floor-level. In formula:
if F Rt ≤ f loor
0
ind2,t =
1
γ
F Rt
f loor
−1
if F Rt ≥ f loor
Where γ denotes the pre-determined number of years the surplus is to be shared
over.
45
Benefit Cut Option
Whenever the funding ratio falls below some critical level, a recovery plan has
to be submitted in which is stated how the funding ratio should be behave in
the upcoming years. Whenever the funding ratio falls below on the recovery
plan, benefit cuts may be applied to increase the value of the funding ratio.
Although a maximum has been set on these benefit cuts, the cut itself will be
such that the next year’s funding ratio equals the critical level.
ind3,t =
F Rt
f loor
−1
if F Rt < f loor
if F Rt ≥ f loor
0
Catch Up Indexation Option
Whenever indexation is missed due to a low funding ratio, catch up indexation
may be given to compensate participants whenever the funding ratio is high.
Therefore assume catch up indexation is only given when the funding ratio is
above some cap-level. Define the indexation missed, indmissedt , to be the
missed indexation over all mechanisms discussed in previous section at time t:
1 + wt
−1
1 + it
where it is the actual indexation given at time t:
indmissedt =
it =
4
X
(1 + indk,t ) − 1
k=1
So that we can define the indexation level defined according to catch up indexation:
ind4,t =
indmissedt
Rt
if F Rt > cap, Fcap
> 1 + indmissedt
F Rt
cap
Rt
if F Rt > cap, Fcap
≤ 1 + indmissedt
−1
0
4.2.3
otherwise
Residue Option
Finally we will discuss the residue option, which actually consists of the surplus
and deficit option. The valuation of these options is straightforward. Denote
AHBS
and VtHBS as the value of the assets and liabilities at the end of the
T
evaluation period T , while implementing all steering mechanisms. In order to
value the residue option, again use the numeraire dependent pricing formula:
1
Q
RO
HBS
HBS
T
V0 = E0 f (AT , LT )Πk=0
1 + rk
where f (AHBS
, LHBS
) = AHBS
− LHBS
. In order to value the surplus and deficit
T
T
T
T
option respectively, use:
f Surplus (AHBS
, LHBS
)
T
T
=
(AHBS
− LHBS
)+
T
T
f Deficit (AHBS
, LHBS
)
T
T
=
(LHBS
− AHBS
)+
T
T
46
Note again that we estimate these option values using Monte-Carlo simulations.
In these simulations we simulate the values for AHBS
and LHBS
per scenario in
T
T
order to value by averaging over all scenarios.
47
Chapter 5
Specifications of the Fund
In this chapter we will define all necessary specifications in order to create the
holistic balance of a fictive pension fund in the next chapter.
5.1
Pension Plan of the (Dutch) Pension Fund
Sector
In this section we will define the main characteristics of our own pension fund
sector, which we will use in order to define our pension fund. In order to have
a somewhat realistic pension fund sector, we will use the characteristics known
in the Netherlands.
• The pension plan is an average wage plan. This means that the pension
fund aims to give the participant, when retired, 70% of his/her average
wage during his/her carreer. Note that there are other pension plans in
the Netherlands, however the average wage plan is the most dominant
type.
• There is no age differentiation. This means that the contributions paid and
benefits accrued are uniform accross generations, as well as the indexation
given.
• When indexation is given, this is based on wage growth.
• The accrual rate is set to 2% of the pensionable wage income. Here an
individual is assumed to enter the pension fund at the age of 25, and
accrues right until the age of 64. From age 65 the participants receives
benefits until death, which is at the age of 115 at maximum. The life
expectancies used are a projection of the life expectancies of the Dutch
population.
• The investment strategy of the fund is assumed to be as follows: 40%
of the portfolio will be invested in stocks and the remaining 60% will be
invested in bonds.
Now that we have defined the pension plan, we continue defining the different
policies a fund can act.
48
5.2
Policies
This section we will devote to the various policies a fund can act. We do
this as pension funds within the EU may be restricted to different policies.
This again will results in different holistic balance sheets, as the values for the
various steering mechanisms will differ over these funds. Since EIOPA aims
to implement one regulatory framework, creating a one-level playing field for
pension funds within the EU, we want to see the impact different policies have
on the holistic balance sheet.
In chapter 2 we already divided the value of the steering mechanisms into
two parts: sponsor support and the adjustment mechanism. The sponsor support was added to the value of the assets, and the adjustment mechanism was
added to the value of the liabilities. The residue changed into an option, which
was again added to the value of the assets or liabilities depending upon whether
it concerned a deficit or surplus option respectively. We also divided the sponsor
support and adjustment mechanism into various options. Sponsor support consists of an employer contribution option, employer guarantee option, premium
cut option and a premium damping option. The adjustment mechanism consists
of an indexation option, surplus sharing option, benefit cut option and a catch
up indexation option. For more elaboration on each option we refer to section
2.3.
As in De Haan et al. [9], we will define the policies by including or discarding
the specific action corresponding to the option described above. Note that the
surplus- and deficit options are included automatically.
Policy
1
2
3
4
5
6
7
8
IND
X
X
X
X
X
X
X
Table 5.1: Policies
Steering Mechanism
EC BC+CAT SUR PC
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
EG
PD
X
X
X
Table 5.2: Abbreviations for the steering mechanisms used in table (5.1)
IND Conditional indexation
EC
Employer contribution
BC
Benefit cuts
CAT Catch up indexation
SUR Surplus sharing
EG
Employer guarantee
PC
Premium cuts
PD
Premium damping
When looking at policies 2 to 8, one should keep in mind that the indexation
49
option actually is a conditional indexation option, and that we assume full
indexation in policy 1. Also note that almost every steering mechanism defined
is conditional to the funding ratio of the fund at that point in time. That is
why we need to define for which funding ratio a specific action is performed.
Next we will discuss policy 1 to 9, indicating which steering mechanisms are
added to the subsequent policy, and how this steering mechanism is subject to
the funding ratio.
• In policy 1 we only have full indexation. This means that the obligations
of the pension fund will be compensated for wage inflation every year.
This policy is particularly interesting as it is the only policy which does
not have any loss absorbing capacity.
• Policy 2 assumes conditional indexation instead of full indexation. Here
the indexation given to the obligations will be dependent on the funding
ratio at that time. This is also called a indexation ladder. For our fund
we assume no indexation below a funding ratio of 105%, full indexation
above 130% and linear indexation in between. Extra indexation will be
given if the real funding ratio is above 100%, which typically comes down
to extra indexation above a nominal funding ratio of around 150%.
• In policy 3 we add employer contribution to policy 2. In other words, in
this policy employers will pay recovery premiums whenever the funding
ratio falls behind on its recovery plan. A recovery plan typically is determined when the funding ratio falls below 105%. This plan states a 3-year
recovery path of the funding ratio in order to bring this up to 105% again.
Whenever the recovery path is not met, a recovery premium will be paid
by participants.
• Policy 4 adds catch up indexation and benefit cuts. Catch up indexation ensures that indexation will be given in order to compensate for the
missed indexation in the past. This missed indexation will only be given
if the funding ratio is above 105%. Benefit cuts are applied whenever the
funding ratio at the end of the recovery plan is not back to 105%. For our
fund the maximal percentage of benefit cuts is 10% uniformly. For active
participants we increase this percentage with an extra 5%-points.
• In policy 5 we also account for surplus sharing. Whenever the funding
ratio is (very) high, the pension fund gives even more extra indexation,
which we call surplus sharing. For our fund we only give this type of
indexation when the real funding ratio is above 120(!)%.
• Employer guarantee stands for the additional funding provided by the
employer. For our fund this additional funding level is set at 2%, when
the funding ratio is between 105% and 130%. If the fund enter a recovery
period this increases to 2.5%, and if the funding ratio is above 130% we do
not assume any additional funding at all. In policy 6 we add this employer
guarantee option.
• Policy 7 adds premium cuts. These premium cuts are given according to
a ladder-scheme. When the nominal funding ratio is less than 130%, no
premium cuts are given. If the real funding ratio is higher than 100%, the
50
entire premium will be cut. In between the amount of premium cuts will
be determined by a linear relation with the funding ratio.
• In policy 8 we include premium damping. Premium dampening ensures
that the premium does not fluctuate that much, by taking into account
some small buffer in the premium. Note that policy 9 includes all steering
mechanisms defined.
Table 5.3: Active Options per Funding Ratio
Full indexation, catch up indexation,
premium cuts and surplus sharing.
120% real
Full indexation, catch up indexation
and premium cuts.
100% real
Full indexation, catch up indexation
and conditional premium cuts.
130% nominal
Full indexation, catch up indexation
and employer guarantee.
120%
Conditional indexation and employer
guarantee.
105%
Employer guarantee, employer
contribution and benefit cuts.
51
Chapter 6
The Impact on the Holistic
Balance Sheet
In this chapter we intend to answer to the main question of this thesis: ”What
happens to the holistic balance sheet if we change some modeling assumptions?”
In the previous chapters we defined the ESG and our fictitious pension fund, so
that we can focus on the modeling of the holistic balance sheet in this chapter.
Before we discuss actual results, we will define the projection assumptions of the
HBS. In the sections thereafter, holistic balance sheets and their corresponding
SCRs will be discussed extensively under various modeling assumptions. These
assumptions vary in short rate model, short rate parameters, correlations, evaluation horizon and policies. Note here that the first three assumptions directly
affect the ESG, and thereby indirectly affect the holistic balance sheet. The
latter two assumptions on evaluation horizon and policies directly influence the
holistic balance sheet.
6.1
Projection Assumptions of the Holistic Balance Sheet
For modeling the holistic balance sheet, the following projection assumptions
were used. Note that we define these assumptions as part of the base case
holistic balance sheet, as we will vary some assumptions in order to see what
its effect is on the results.
• The pension fund will be considered a closed fund. This means that the
fund does not have any new accrual nor any new premiums coming in.
• The initial nominal funding ratio will be set at 105%.
• The initial evaluation horizon for the option values we set at 15 years. In
section 6.4 we will decrease this value to 10 years.
• The number of scenarios generated using the ESG equals 2500. The horizon of each scenario spans 140 years, where we simulate 10 times per year.
This results in a total of 1400 generated timesteps per scenario.
52
6.2
Varying ESG Assumptions
In this section we will discuss the impact on the holistic balance sheet and
the corresponding SCR when altering short rate models and/or short rate parameters. The alterations of these assumptions will be based on our calibration
results from chapter 3, where we calibrated three short rate models under different assumptions. The other three processes of the economic scenario generator,
wage growth, price inflation and stocks, we keep the same when changing short
rate assumptions. No correlation between any of the processes in the ESG is
assumed in this section in order to avoid any misused correlations, as discussed
in section 4.1.3. In section 6.2.2 we will add the correlations as defined in section
4.1.3, in order to see if correlations have any effect on the results at all.
For the holistic balance sheets generated in the next section we add a short
rate model to the ESG, where short rate assumptions are based on the calibrations from chapter 3. The short rate assumptions in the ’base case’ we define
to be:
Table 6.1: Base Case:
Term structure: Nelson-Siegel function
Calibrated to:
Observed premiums
Method:
Jamshidian’s Decomposition
Optimization:
Levenberg-Marquardt-Fletcher
Absolute
Weights:
Note that these assumptions include the Nelson-Siegel function to interpolate
the term structure, as the differences in parameters between calibrating using
splines and the Nelson-Siegel function are in the same order of magnitude compared to the differences in parameters when calibrating using different weights.
As we will generate multiple holistic balance sheets using different weights, it
is useless to generate these results under short rate assumptions based on a
differently interpolated term structures.
The next short rate assumption we make is that we only use the calibration
parameters found while calibrating to observed premiums. Since the calibration
results when calibrating to observed premiums and observed implied volatilities is very small, the effect on the HBS will be small. Besides that, we will
only use calibration results derived using Jamshidian’s Decomposition. Note
that this choice was only available for the Hull-White one-factor model, and
as this method produces an analytical derived swaption formula, the choice
for Jamshidian’s Decomposition becomes an obvious one. The optimization
method, Levenberg-Marquardt-Fletcher or simulated annealing, produced the
same calibration results, so that we choose the faster method.
The last assumption of the ’base case’ in table (6.1), absolute weights, is
the one assumption we will change in this section. We choose absolute weights
as this assumption produced useful calibration results over calibration years
as well as short rate models. For instance, for ’absolute-relative’ and relative
weights, we have that the 2012 calibration for the Hull-White one-factor model
results in a negative value for a. Using a negative a, we have that the ESG
produces unrealistic scenarios, as the short rate process becomes mean-fleeing
and diverges to infinity.
53
Besides that, note that we cannot use the CIR++ model at all. As shown in
section 3.5.3, we have that the short rate process using the calibrated parameters
produces complex values, as the process {x} becomes negative at some point.
The corresponding ESG therefore becomes useless for valuation of the HBS, so
that we have to omit this model for analysis. This is a big downside of the
CIR++ model. Implementation of this model in IORP II is very unlikely.
The last remark we make before discussing results is the calculation of the
initial values of the assets and liabilities, A0 and L0 . Although we used an initial
funding ratio of 105%, the results will indicate a funding ratio that is smaller.
The reason behind this is that EIOPA modifies the discount rate for which A0
and L0 are valued. This modification includes the implementation of the UFR.
This method states that the long-term discount rate should converge to a fixed
rate, which is 4.2% for the Netherlands. This leads to different values for A0
and L0 .
6.2.1
Varying Short Rate Assumptions
In this section we will discuss the results when altering short rate assumptions.
As we can see in tables (E.1) and (E.2), the differences between the 2011 and
2012 HBSs when switching between different weights is very small. These differences are small when using the Hull-White one-factor model, however for the
Hull-White two-factor model these differences are even smaller. This may be
caused by the ability of the two-factor model to provide more constant calibration results compared to the one-factor model when calibrating using different
weights. Besides that, we see that the impact on the option values under the
adjustment mechanism is larger compared to the impact on the option values
for the sponsor support. This suggests that the adjustment mechanism is more
sensitive to the short rate assumptions compared to the sponsor support.
Table 6.2: Solvency capital requirements under different confidence levels based
on the Hull-White one-factor model calibrated for different weights, no correlation between state variables and an evaluation horizon of 15 years is assumed
2011
2012
Abs
Abs-Rel
Rel
Abs
SCR(99.5%) 12.56 12.60 12.76 12.84
Gross SCR 31.95
31.95
31.95
31.98
LAC −19.39 −19.35 −19.19 −19.14
SCR(97.5%)
4.89
4.94
5.09
5.16
Gross SCR 24.29
24.29
24.28
24.30
LAC −19.39 −19.35 −19.19 −19.14
SCR(95%)
1.06
1.10
1.26
1.33
Gross SCR 20.45
20.45
20.45
20.46
LAC −19.39 −19.35 −19.19 −19.14
Another conclusion we can make based on tables (E.1) and (E.2) is that
both calibration years 2011 and 2012 came up with significantly different holistic
54
Table 6.3: Solvency capital requirements under different confidence levels based
on the Hull-White two-factor model calibrated for different weights, no correlation between state variables and an evaluation horizon of 15 years is assumed
2011
2012
Abs
Abs-Rel
Rel
Abs
Abs-Rel
Abs
SCR(99.5%) 14.65 14.60 14.56 11.66 11.54 11.58
31.98
31.98
31.99
31.99
31.99
Gross SCR 31.98
LAC −17.31 −17.36 −17.41 −20.33 −20.45 −20.40
SCR(97.5%)
6.98
6.94
6.89
3.99
3.86
3.90
Gross SCR 24.30
24.30
24.30
24.31
24.31
24.31
LAC −17.31 −17.36 −17.41 −20.33 −20.45 −20.40
SCR(95%)
3.15
3.10
3.05
0.15
0.03
0.06
Gross SCR 20.46
20.46
20.46
20.48
20.48
20.48
LAC −17.31 −17.36 −17.41 −20.33 −20.45 −20.40
balance sheets. However it is difficult to conclude whether or not this is due
to the different calibrated short rate parameters per year, as the initial term
structure of interest rates also differs per year. Since the initial funding ratio
is determined by the initial term structure, this makes the steering mechanism
values dependent upon the initial term structure. We can see this in tables (E.1)
and (E.2) as well, as the values A0 and L0 are different per year.
In tables (6.2) and (6.3) the SCRs are displayed under the various short rate
assumptions. In line with their corresponding HBSs, we see that the differences
in values between weights per year and short rate model are small. These
differences are again even smaller for the Hull-White two-factor model, probably
because of the fact that the two-factor model is able to produce more constant
results over time.
An important remark however is the impact when changing between short
rate models. When we move from one-factor to two-factor model using data
from 2011, we see an increase of approximately 1.8 added to the SCR. Recall
that the value of the liabilities at initiation is 147.00, so that this suggests an
increase of around 1.2%-point when using the two-factor instead of the onefactor model. For 2012, we see a decrease of all SCRs of approximately 1.2
when using the two-factor instead of the one-factor model, which suggests a
0.80%-point decrease relative to the initial liabilities. Also note the direction of
the impact is ambiguous, as it is positive and negative when using a two-factor
instead of a one-factor model in 2011 and 2012 respectively. Therefore EIOPA
will have to give some guidance on short rate models, as arbitrage opportunities
may arise when pension funds have the choice.
6.2.2
Adding Correlations
In this section we run the same holistic balance sheets under the Hull-White
one-factor and two-factor model as in the previous section, only now with correlations implemented between the processes in the ESG. As discussed in section
55
4.1.3, we know that the correlation matrix used does have some problems. The
question in this section therefore becomes if there is any impact on the HBS
and SCR at all, when implementing some correlations. The short rate parameters used for modeling the balance sheets were derived from the short rate
calibrations using absolute weights only.
The resulting HBSs and SCRs are displayed in tables (E.3) and (6.4) respectively. When comparing these tables to the results from the previous section,
tables (E.1) and (E.2), we see that there are some differences. For the HBSs the
impact is not negative nor positive for all option values, and the impact seems
slightly larger for option values under the adjustment mechanism compared to
those option values under the sponsor support. This effect was shown in the
previous section as well, so that it indeed confirms that the adjustment mechanisms is more sensitive to short rate assumptions compared to the option values
under the sponsor support.
Table 6.4: Solvency capital requirements under various confidence levels, based
on the Hull-White one-factor and two-factor model, calibrated for absolute
weights only, correlation between state variables and an evaluation horizon of
15 years is assumed
2011
2012
HW1F
HW2F
HW1F
HW2F
SCR(99.5%) 12.09 13.93 12.65 11.29
Gross SCR 31.95
31.98
31.98
31.98
LAC −19.86 −18.06 −19.33 −20.73
SCR(97.5%)
4.43
6.25
4.98
3.61
Gross SCR 24.29
24.30
24.30
24.30
LAC −19.86 −18.06 −19.33 −20.73
SCR(95%)
0.59
2.41
1.14
0
Gross SCR 20.45
20.46
20.46
20.46
LAC −19.86 −18.06 −19.33 −20.46
When comparing the SCRs from tables (6.2), (6.3) and (6.4), we see that
the correlation structure does decrease the SCR a bit. Note that this does not
have to be the case for all correlations used. Although in this case it looks like
there is some diversification going on, note that the fund uses a static portfolio.
This means that when modeling the HBS, we do not assume the ability of the
fund to change its portfolio over the years. In other words, the diversification
in table (6.4) may just be coincidal, so that when using a correlation structure
of a different year the impact on the SCR might be opposite.
Apart from the conclusion that there is an impact on the SCR when correlations are added, note that we cannot judge the size of the impact. As discussed
in section 4.1.3, the correlation matrix used is ’broken’ because of its negative
eigenvalues. Although we fixed this matrix by setting any negative eigenvalues
positive, this solution is mathematically ungrounded. Therefore, in order to
measure the actual size of the impact when correlation are added, one needs to
find a proper solution for this ’broken’ matrix.
56
6.3
Decreasing the Evaluation Horizon
In the first part of this chapter we mainly discussed the impacts when altering the ESG. In these last two section we will alter some assumptions made
when modeling the HBS itself. In this section we will decrease the evaluation
horizon over which the option values are calculated. The expectation is that
option values become smaller, simply because the time over which the steering
mechanisms are valued becomes less.
When looking the 2011 and 2012 HBSs under the Hull-White one-factor and
two-factor models for a smaller evaluation horizon, table (E.4), we see that this
is indeed the case. The option values become smaller as the evaluated horizon
becomes smaller. The decreasing evaluation horizon also results in a decrease
of the loss absorbing capacity, as there is less time to absorb any shocks. When
comparing the SCRs calculated using a 10 year horizon, table (6.5), and the
SCRs from tables (6.2) and (6.3), we see this decrease in LAC as well. As one can
see, the impact on the SCR can become quite high, and choosing a reasonable
evaluation horizon is an important decision yet to be made by EIOPA.
Table 6.5: Solvency capital requirements under various confidence levels, based
on the Hull-White one-factor and two-factor model, calibrated for absolute
weights only, no correlation between state variables assumed and using an evaluation horizon of 10 years.
2011
2012
HW1F
HW2F
HW1F
HW2F
SCR(99.5%) 15.95 17.26 16.15 15.01
Gross SCR 31.93
31.94
31.95
31.95
LAC −15.98 −14.68 −15.80 −16.94
SCR(97.5%)
8.30
9.60
8.49
7.34
Gross SCR 24.26
24.28
24.28
24.29
LAC −15.98 −14.68 −15.80 −16.94
SCR(95%)
4.46
5.78
4.65
3.51
Gross SCR 20.44
20.44
20.45
20.45
LAC −15.98 −14.68 −15.80 −16.94
As a last remark, note that it is difficult to base the decision of the length
of the evaluation horizon on anything. Many suggested the length of the evaluation horizon to be 15 years, as most pension fund have a duration of liabilities
around 15 years. However these statements are fully independent. If the evaluation horizon becomes (too) small, the effect of the HBS and its loss absorbing
capacity will be diminished. If the evaluation horizon becomes (too) large, the
level of the SCR may become too small, causing the fund to run a higher risk of
becoming insolvent. After all, note that when we assume an evaluation horizon
of infinity, the loss absorbing capacity will be infinite as well. This will result
in pension funds holding no SCR at all.
57
6.4
Varying Policies
In table (6.6) we have stated the loss absorbing capacities implied by the various
policies, under the Hull-White one-factor and two-factor model, for 2011 and
2012. The various policies were defined in section 5.2. In this section we will
discuss the results in the table (6.6), in order to see how to LAC behaves under
different policies.
In appendix E we stated in tables (E.5) and (E.6) how the HBS behaves under various policies under the Hull-White one-factor and two-factor respectively.
Note the impact on the options when other steering mechanisms are added to
the policy. For instance, when moving from policy 2 to policy 3, we add employer contribution. Because of employer contribution, or recovery premiums,
we see that the funding ratio is more likely to increase to 105% compared to
policy 2. That is why more indexation can be given over the years, so that
the indexation option is higher under policy 3 compared to policy 2. Note that
the tables can contain some small errors due to the Monte-Carlo error or model
risk. The Monte-Carlo error arises because of an estimation error when averaging option values over all scenarios, and model risk arises because of small
errors in the model itself. An example of the latter may for instance be the
steering mechanism premium damping being greater than zero in policies 6 and
7, where it was left out of both policies initially.
Table 6.6: Loss absorbing capacities from different policies for 2011 and 2012,
based on the Hull-White one-factor and two-factor model, calibrated for absolute weights only, no correlation between state variables assumed and using an
evaluation horizon of 15 years.
2011
2012
Policy HW1F HW2F
HW1F
HW2F
1
2
3
4
5
6
7
8
0
10.53
13.61
18.16
18.21
18.46
19.48
19.39
0.03
5.94
9.39
16.39
16.39
16.79
17.48
17.31
0
6.01
12.68
17.84
17.78
17.89
19.19
19.14
0
7.05
13.68
18.88
18.89
18.95
20.34
20.33
When looking at the LACs in table (6.6), we see an increasing LAC when
more steering mechanisms are added to the policy.1 This is obvious, as the more
steering mechanisms available, the higher the ability of the fund to recover from
severe shocks. Note that the added value in the LAC when moving from one
policy to the next decreases over the policies. This indicates that the marginal
value in LAC when adding a particular steering mechanism to the policy is
higher when only a few steering mechanisms are implemented, compared to the
case where the fund already uses a lot of steering mechanisms.
1 This holds for almost all cases. In some cases the LAC decreases when another steering
mechanism is added, which can be due to the Monte-Carlo error or model risk.
58
Finally note that the LACs in table (6.6) significantly differ per short rate
model for both 2011 and 2012. Although one should keep in mind the MonteCarlo error and model risk, these are not considered to be the cause of the
varying values for the LAC under various policies. This is another indication
that the short rate models do imply different SCRs, so that EIOPA should
provide more guidance with respect to the use of various short rate models
before implementing IORP II.
59
Chapter 7
Conclusions &
Recommendations
Throughout this thesis an impression was given on the impact on the holistic balance sheet when modifying some of the various modeling assumptions.
We can conclude that EIOPA needs to give more guidance towards any of these
modeling assumptions, before an implementation of the holistic balance sheet in
IORP II is possible. In this thesis our focus was mainly on the calibration of the
various short rate models to market representative swaption premiums, and see
how HBS and SCR are affected when altering calibration assumptions. Besides
that, we also investigated the impact on the HBS and SCR when altering the
length of the evaluation horizon of the options, and when modeling under varying policies of the fund. This section sums up the most important conclusions,
after which we will end with some recommendations for future work.
The first conclusion we make is that not all short rate models can be used
for the economic scenario generator, which should value the HBS. The implementation of the Cox-Ingersoll-Ross++ model in the ESG resulted in complex
values of the short rate, so that the holistic balance sheet could not be valued
at all.
Besides that, we have seen that the choice of short rate model does impact
the holistic balance sheet. In this thesis the Hull-White one-factor and twofactor models are used for valuation of the HBS. Although the impact is not
uniformly postive or negative over all option values, we do see some significant
differences between HBSs. As a result, the corresponding SCRs also show some
differences between different short rate models. Therefore a decision by EIOPA
is necessary in order to prevent arbitrage opportunities when pension funds are
allowed to choose a short rate model. For our data the differences in SCRs
are in the order of magnitude of -0.80% of initial liabilities for 2012, to 1.28%
of initial liabilities for 2011. Note that these differences arose when using the
Hull-White two-factor model instead of the Hull-White one-factor model.
Not only did we vary the underlying short rate model, we also varied the
short rate parameters that we used for generating the economic scenarios. These
variations in calibrated parameters came from altering the weights in the optimization problem when calibrating. Therefore, the variations in parameters
basically came down to the calibrations made to different subsets of the observed
60
swaptions. As it turned out, the variations in parameter choice due to the calibration of the short rate model being based on different subsets only showed
small differences in HBSs as well as SCRs. These differences were even smaller
for the Hull-White two-factor model compared to the Hull-White one-factor,
as the calibrated parameters showed to be more consistent when calibrated to
different subsets for the two-factor model. The Hull-White two-factor model is
also able to provide a better fit over all calibrations compared to the Hull-White
one-factor model.
When adding correlations between the processes defined in the ESG, we
noticed the resulting SCRs to be lower compared to the initial SCRs without
any correlation. This looks like diversification, however since our fictitious fund
used a static portfolio this cannot be the case. Therefore the impact on the
SCR with a different correlation matrix could be positive as well. Unfortunately we cannot say anything about the size of the impact on the SCRs when
correlations are added, because our correlation matrices were ’broken’. The negative eigenvalues corresponding to these matrices resulted in unuseable matrices
for modeling correlations. By changing these negative eigenvalues into positive
eigenvalues we were able to use them, however this solution is mathematically
ungrounded. Another problem was the allocation of the correlation structure for
the two-factor models, as these models used two (correlation) processes. As the
observed correlations were stated as a single number, it was difficult to assign
two correlations impliying the observed correlation.
Another important problem for EIOPA is to determine the length of the
evaluation horizon to be used when valuing the various steering mechanisms on
the holistic balance sheet. In this thesis we modelled the HBS under an evaluation horizon for 10 and 15 years. As expected, the results show that each option
on the HBS decreases in value when we assume a smaller evaluation horizon.
As a result we get that the LAC decreases as well, so that the corresponding
SCR becomes lower. The difficulty for EIOPA here is that there is no simple
argument for choosing a particular evaluation horizon. Many suggested this
should be the length of the duration of the liabilities of a fund, typically around
15 years, however this is ungrounded as well.
For future research, we suggest to do the same research over more years.
We intended to do this, however we were forced to omit data from 2003, 2005
and 2008. The reason for this was that the term structures from these dates
were incorrect, as the observed swaption premiums could not be replicated using
Black’s formula. It would have been nice to see the impact on the HBS and
SCR for 2003, 2005 and 2008, in order to see how the HBS and SCR behaved
before and during the financial crisis.
Investigating what correlation structure to use for different models may be
interesting for future work as well. As discussed before, throughout this thesis
we found ´broken´ matrices multiple times, so that we were unable to use these
matrices for modeling. As a solution we changed the negative eigenvalues of
these ´broken´ matrices into positive eigenvalues, although this was mathematically ungrounded. Besides that, we also faced the problem of how to define the
correlation structure for a two-factor model, when a single observed correlation
was given. In order to investigate the size of the impact on the HBS and SCR
when using correlations, these two problems should be solved.
Throughout this thesis we mainly investigated the impact on the HBS for
fixed stock, wage growth and price inflation processes defined by a Black-Scholes
61
process. For future research it would be interesting as well to measure the
impact when these processes are defined by different models. The Black-Scholes
model is perhaps too simple for generating purposes, and a stock process with
a variable volatilility term may be more realistic.
The last recommendation for future reserach on this topic is to measure the
impact on the HBS and SCR using a different type of calibration for the short
rate model. In this thesis we used swaption premiums in order to calibrate
the short rate. A calibration to caplets may give an entirely different short rate
structure, which may lead to different results as well. Another popular approach
one can use is to calibrate a libor market model. This is an interest model based
on evolving LIBOR market forward rates.
It may be clear that EIOPA still needs to give more guidance on modeling
assumptions, and that IORP II is far from implementation yet. These decisions
concern the modeling assumptions of the generated risk-neutral scenarios, as
well as the assumptions for modeling the holistic balance sheet itself. Some of
the decisions may be ungrounded, such as the length of the evaluation horizon, however EIOPA does have to keep in mind the main purpose of the SCR.
This purpose is for funds to hold a sufficient buffer in order for the pension of
participants to survive any large negative shocks over time.
62
Bibliography
[1] F. Black, The Pricing of Commodity Contracts. Journal of Financial Economics, 3:167-179, 1976.
[2] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities.
Journal of Political Economy, 81:637-654, 1973.
[3] D. Brigo and F. Mercurio, Interest Rate Models - Theory and Practice (with
Smile, Inflation and Credit. 2006.
[4] J.C. Cox, J.E. Ingersoll and S.A. Ross, A Theory of the Term Structure of
Interest Rates. Econometrica, 53:385-407, 1985.
[5] EIOPA, Quantitative Impact Study on Institutions for Occupational Retirement Provision - Technical Specifications. 2012.
[6] R. Fletcher, A Modified Marquardt Subroutine for Nonlinear Least Squares.
Harwell, 1994.
[7] E. Fransen, N. Kortleve, H. Schumacher, H. Staring and J. Wijckmans, The
Holistic Balance Sheet as a Building Block in Pension Fund Supervision.
Netspar Discussion Papers.
[8] S. Gurrieri, M. Nakabayashi and T. Wong, Calibration Methods of HullWhite Model.
[9] J. de Haan, K. Janssen and E. Ponds, The Holistic Balance Sheet as the New
Framework for European Pension Supervision. Netspar Discussion Papers.
[10] P.S. Hagan, Volatility Conversion Calculators.
[11] J. Hull, Interst Rate Derivatives: Models of the Short Rate. Options, Futures and Other Derivatives, 6:657-658, 2006.
[12] F. Jamshidian, An Exact Bond Option Pricing Formula. The Journal of
Finance, 44:205-209, 1989.
[13] P. Joubert and S. Langdell, Mastering the Correlation Matrix, The Actuary,
32-33, 2013.
[14] S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Optimization by Simulated Annealing. Science, 220(4598):671-680, 1983.
[15] W. Margrabe, The Value of an Option to Exchange One Asset for Another
Journal of Finance, 33:177-186, 1978.
63
[16] J.M. Schumacher, Lecture Notes Financial Models. 2012.
[17] A. White and J. Hull, Pricing Interest Rate Derivative Securities. The
Review of Financial Studidies, 3(4):573-592, 1990.
64
Appendix A
Technical Background:
Risk-Neutral Valuation
In general, we can use three formulas to valuate some uncertain future cash flow
C. The first formula uses a so-called pricing kernel or stochastic discount factor:
P = E P (C ∗ SDF )
Here SDF stands for the corresponding stochastic discount factor. This definition is based on the assumption of a complete market. Tje stochastic discount
factor is defined uniquely and independent on the future cash flow C. In reality
the market is considered to be incomplete. Therefore when using the formula
above, pricing errors can occur. Also note that the expectation under P denotes
an expectation which is based on the real-world probability measure.
The second formula values an uncertain future cash flow using a different
formula:
P =
E P (C)
exp(R(C) ∗ maturity)
Here R(C) denotes the discount rate specific for the future cash flow, since this
rate also captures a risk premium based on the risk profile of C. Although this
method is very obvious, for simulation purposes it is very difficult to estimate
a correct risk premium.
In the third formula pricing is based on the numeraire dependent pricing
formula (NDPF). This formula is based on the fundamental theorem of asset
pricing, which states that relative price processes are martingales under a riskneutral probability measure. This fundamental theorem is a requirement in
order to have absence of arbitrage. The NDPF is defined as follows:
C
P = N0 E QN
NT
Here NT stands for the value of some traded asset, which is called the numeraire.
The risk-neutral measure is taken with respect to the numeraire, QN . More
elaboration towards the NDPF is given later this appendix. We will start by
explaining risk-neutral valuation in a discrete and continuous time setting.
65
A.1
Arrow-Debreu Securities
Consider an economy where we can only trade at time 0 and time 1. Besides
that, consider the interest rate to be equal to zero. Suppose we live at time
0, and that we have all available information at that time. The future state
at time 1 remains unknown, however we do know the future can evolve in four
different states only. Since this holds we also have that the current market is
complete. The real-world probabilities of ending at these states we define by
p(i), and that all four real-world probabilities add up to one:
4
X
p(i) = 1
i=1
An Arrow-Debreu (A-D) security pays off a certain cash flow in one particular state only. If we assume an A-D security pays off e1 in one particular state,
we know that we get a certain payoff of e1 if we aggregate all available A-D
securities in our complete market. However by aggregating all A-D securities we
are actually replicating a zero-coupon bond. According to absence of arbitrage,
the price of the aggregated A-D securities must therefore be equal to the price
of the zero-coupon bond. Since we assumed an interest rate equal to zero, both
prices must be equal to their deterministic pay offs, which is e1. Finally, by
defining a(i) as the price of a single A-D security, we have that the price of the
portfolio of all possible A-D securities must add up to one:
4
X
a(i) = 1
i=1
Again, in order for absence of arbitrage, the price of a single A-D security
equals the state price. The state price of a particular A-D security can be seen as
the risk-neutral probability of that state: q(i) = a(i). Note that this state price
is not equal to the real-world probability. Each state price is determined by the
supply and demand of that particular A-D security. Real-world probabilities do
influence state prices, however the state itself also influences state prices. For
instance e1 in a ‘bad’ state is worth more compared to e1 in a ‘good’ state.
Up till now we assumed the interest rate to be equal to zero. When the riskfree rate is not equal to zero, risk-neutral probabilities (state prices) have to be
adjusted. Again this adjustment is such that the prices of a zero-coupon bond
and the portfolio of all A-D securities is the same. Risk-neutral probabilities
are therefore adjusted according to:
q(i) = a(i)(1 + Rf )
where Rf denotes the risk-free rate. The relationship between the real-world
and risk-neutral probability measures can be expressed in terms of a RadonNikodym (R-N) derivative. We define this derivative to be a random variable
θ, such that:
E Q X = E P θX
(A.1.1)
Here Q and P denote the risk-neutral and real-world probability measure respectively. These measures must be equivalent in order for (A.1.1) to hold. This is
66
the case if both measures agree on which events can occur and which cannot.
Note that the R-N derivative is defined
by the ratio of risk-neutral probabilities
over the real-world probabilities dQ
dP . In continuous time, to be discussed in
the next section, this derivative is written as a process.
In reality we typically do not have a complete market, but an incomplete
market. In an incomplete market, not every A-D price can be constructed
uniquely. As a result, risk-neutral probabilities cannot be defined uniquely as
well.
A.2
Continuous Time
In this section we will discuss risk-neutral probability measures in a continuous
time setting. We start by defining a Radon-Nikodym process {θt }, instead of a
random variable. For every 0 ≤ s ≤ t, we have that:
EsQ Xt = EsP
θt
Xt
θs
(A.2.1)
Here Es represents an expectation based on all known information up to time s.
One may define (A.2.1) differently when considering {Xt } to be a process, where
its volatility is defined via a Brownian motion. Before we discuss the theorem in
question, first some more background information towards stochastic calculus
in continuous time is given.
A stochastic differential equation is a differential equation in which one or
more terms can be stochastic. A Brownian motion is an example of such a
stochastic term, often used in stochastic differential equations. A continuoustime process {Wt } is said to be a Brownian motion if is satisfies the following
properties:
• Normalization: W0 = 0.
• Independent increments: If t1 < t2 ≤ t3 < t4 , then Wt2 − Wt1 and Wt4 −
Wt3 are independent.
• Normally distributed increments: For any given t1 and t2 with t2 > t1 , the
distribution of the increment Wt2 − Wt1 is the normal distribution with
mean 0 and variance t2 − t1 .
A stochastic differential equation denotes an expression of the form:
dXt = µ(t, Xt ) dt + σ(t, Xt ) dWt
where µ(t, Xt ) and σ(t, Xt ) are functions of t and Xt , and represent the so-called
drift and volatilty terms of {Xt } respectively. We say that {Xt } is driven by a
Brownian motion, which means that it uses a Brownian motion as its source of
uncertainty.
In case the process {Xt } is obtained from a stochastic differential equation
driven by a Brownian motion, the conditions under which a change of measure
holds are provided by Girsanov’s theorem.
Girsanov’s theorem: Let {Wt } be a Brownian motion and {λt } a process
adapted to {Wt }. If λt satisfies some boundedness conditions, then the process
{θt } is defined by:
67
dθt = −θt λ0t dWt ,
θ0 = 1
(A.2.2)
Here {θt } denotes the Radon-Nikodym process, which is a P-martingale. In
other words, {θt } is a martingale1 under the real-world probability measure P.
We can use the process of {λt }, defined in (A.2.2), in order to derive the relation
between two Brownian motion defined each under different probabilty measures:
dW Q = dW P + λt dt,
dW0Q = 0
(A.2.3)
Here λt basically stands for the price of risk. The distributional properties of
W Q and W P do not differ as well, so that a change of measure actually becomes
a change of sdrift.
A.3
First Fundamental Theorem of Asset Pricing
In this section we will see how risk-neutral probabilities are defined according
to the first fundamental theorem of asset pricing. This theorem states some
conditions the market has to hold in order to have absence of arbitrage.
First Fundamental Theorem of Asset Pricing (FTAP): The market specified
by some real-world probability measure P is free of arbitrage, if and only if,
given any numeraire N , there is a measure QN which is equivalent to P, and
which is such that all relative price processes are QN -martingales.2
The numeraire defined in the FTAP can be any type of security or selffinancing portfolio consisting of existing securities, for which its price is always
positive. Based on the FTAP, one can price every future cash flow according
to the numeraire dependent pricing formula (NDPF). This formula, already
discussed briefly in the beginning of this appendix, prices securities using that
relative price processes follow a QN –martingale:
P (0, T )
QN P (T, T )
=E
N0
NT
where C denotes some type of security. We can use a similar application of the
FTAP in a continuous time setting. Suppose we define some generic state space
model, driven by a Brownian motion, as:
dXt
Yt
= µX (t, Xt ) dt + σX (t, Xt ) dWt
= π(t, Xt )
(A.3.1)
where Xt is a vector consisting of state variables. This vector can contain all
kind of indicators, from interest rates to unemployment rates to the amount of
rainfall in the past month. Yt typically represents security prices.
Next define a numeraire {Nt } in the general state space model (A.3.1). Also
Y
define a relative price process { N
}which is defined as follows:
d(Y /N ) = µY /N dt + σY /N dW P
1 Martingales
(A.3.2)
are processes of which its expectation in future time is equal to the current
value: Es Xt = Xs for t ≥ s.
2 Q -Martingales are martingales with respect to the probability measure Q
N
N
68
where we simplified notation a bit. If we apply Girsanov’s theorem
Y to equation
under the
(A.3.2), we get the following stochastic differential equation for N
risk-neutral probability measure QN with numeraire N :
d(Y /N ) = µY /N dt + σY /N dW QN − λN dt
= µY /N − σY /N λN dt + σY /N dW QN
Y for which the drift term must be equal to zero, as N
must be a QN -martingale
according to the FTAP. Therefore we have that the following condition must
hold in order for absence of arbitrage:
µY /N = σY /N λN
Using Itô’s lemma, one can translate the condition above into the following:
µY − rπY = σY λ
(A.3.3)
which denotes restrictions for the stochastic process of {Y } only. Here λ denotes
the price of risk, and r the risk-free interest rate. Presented next is an example
when changing the measure for a standard Black-Scholes model.
Consider the standard Black-Scholes model (under the real-world probability
measure):
dSt
= µSt dt + σSt dWtP
(A.3.4)
dBt = rBt dt
µSt
St
σSt
So that we can define µY =
, πY =
and σX =
. Since
rBt
Bt
0
r is a constant in this model, we have that according to (A.3.3):
µ−r
σ
Changing the measure using Girsanov’s theorem (A.2.3), we get:
λ=
µ−r
dt
(A.3.5)
σ
So that by substituting (A.3.5) into equation (A.3.4), the standard Black-Scholes
model under the risk-neutral probability measure Q becomes:
dW P = dW Q −
dSt
=
rSt dt + σSt dWtQ
dBt
=
rBt dt
69
Appendix B
Technical Background:
Term Structure Modeling
In this section we will see how the process of the short rate is linked with term
structure products, like the swaption. This section therefore gives some important information used for calibrating the short rate models. Preferably these
models should fulfill a number of conditions. In an addition of the condition
of absence of arbitrage, we want the short rate models to represent the current
term structure as well. Besides that, we also want to be able to reproduce current prices of term structure derivatives. The last condition we would like for a
short rate model to have is that it should be able to produce reasonable future
term structures.
In order to keep any analytical tractability, one would like to have all desirable properties described above using a minimal number of parameters. Typically over-parameterization occurs when more parameters are being used. Overparameterization describes the effect of parameters varying a lot when calibrated
to new data, so that the significance per parameter decreases. Note that one
does have to take into account the current state of economy, as parameters are
sometimes actually expected to vary a lot when calibrated during a financial
crisis.
Throughout this appendix we will start by discussing some popular term
structure derivatives, like the swaption. In addition we will discuss the basics
behind the interpretation of our data. Thereby we will discuss the derivation of
the swaption premium using a normally or lognormally distributed swap rate.
Note that the latter may also be referred to as Black’s Formula. We end this
appendix by discussing the Nelson-Siegel function, using for interpolating the
term structure, and some popular short rate models not calibrated in this thesis.
B.1
Term Structure Derivatives
We start by defining the zero-coupon bond. A zero-coupon bond pays off a fixed
amount at maturity. For instance, when defining a payout of e1 at maturity T ,
the current value of this zero-coupon bond is:
P (t, T ) = exp (−T R(t, T ))
70
(B.1.1)
where current time is t. If we consider this payout to be risk-free, R(t, T ) can
be defined as the risk-free interest rate at time t, for maturity T .
The money-market account is considered to be a risk-free deposit, where
money can be invested or withdrawn at any time. This account is considered
to be risk-free, so that there is no credit risk for instance. The interest gained
over an infinitesimal period of time is equal to the short rate r at that time.
Therefore we can write a differential equation for the movement of the process
of the money-market account {M } like:
dMt = rt Mt dt,
M0 = 1
So that by using Itô’s lemma, the solution of M (t) can be expressed as:
Z t
rt dt
(B.1.2)
M (t) = M0 exp
s=0
Say we have a zero-coupon bond paying off a certain amount at maturity
T . Suppose its price is unknown, but we want to use the NDPF to price this.
In order to do so, one can use the money-market account {Mt } as a numeraire.
This means that the expectation in the NDPF will be performed under the riskneutral measure QM .1 We get the following pricing equation when applying the
NDPF to price the zero-coupon bond:
P (t, T )
QM P (T, T )
= Et
Mt
MT
Note that P (T, T ) = 1, so that the price of a zero-coupon bond with a payoff of
e1 at maturity T equals:
Mt
P (t, T ) = EtQM
MT
By substituting in the solution of the value for the money market account,
equation (B.1.2), into the equation above, we get that:
"
!#
Z
T
P (t, T ) = EtQM exp −
rs ds
(B.1.3)
t
By definition, we have that equations (B.1.1) and (B.1.3) should match.
Therefore we can use these equations to write the term structure of interest
rates in terms of the short rate:
"
!#
Z T
1
QM
R(t, T ) = − logEt
exp −
rs ds
(B.1.4)
T
t
We used these results when calibrating the short rate models to observed swaptions. For now we will introduce the definitions of the (instantaneous) forward
rate and the swap(tion), which were also used for calibration.
A forward rate agreement is an agreement between two parties A and B,
where party A is obliged to pay party B some predetermined fixed rate of the
1 Note that the change-of-measure from the real-world measure to the risk-neutral measure
QM does not result in a change of drift in the differential equation of Mt . However the short
rate rt must be modeled risk-neutral.
71
notional. At the same time, party B is obliged to pay party A according to some
interest rate at that time. These payments will take place at a later specified
date. Note that the fixed rate should be set such that the initial value of the
contract equals zero. We can do this by calculating ft (T1 , T2 ) in the following
equation:
exp(T1 · R(t, T1 ))exp((T2 − T1 ) · ft (T1 , T2 )) = exp(T2 · R(t, T2 ))
So that the payments will take place at T1 , where the actual floating rate
is determined by the interest rate at T1 with maturity T2 − T1 . Furthermore,
we can define ft (T1 , T2 ) to be the corresponding forward rate in the following
equation:
1
P (t, T1 )
R(t, T2 )T2 − R(t, T1 )T1
=
log
ft (T1 , T2 ) =
T2 − T1
T2 − T1
P (t, T2 )
The instanteneous forward rate equals the forward rate defined above, where
the difference between T1 and T2 is infinitesimal. When rewriting, we get that
the expression for the instanteneous forward rate becomes:
ft,T
=
limMT ↓0 ft (T, T + M T )
=
)−R(t,T )
)
limMT ↓0 (R(t, T + M T ) + T R(t,T +MT
MT
= R(t, T ) + T
(B.1.5)
δR(t,T )
δT
This can also be rewritten in terms of the discount curve instead of the term
structure of interest rates by using equation (B.1.1). Equation (B.1.5) can then
be written as:
δP (t, T )
(B.1.6)
δT
One can find the inverse relationship, where the current term structure of interest rates can be derived using the instantaneous forward curve:
ft,T = −
1
R(t, T ) =
T
Z
T
ft,τ dτ
(B.1.7)
0
So that, after setting equations (B.1.4) and (B.1.7) equal to each other, we
have that the instantaneous forward rate at time t for the maturity 0 equals the
short rate at time t:
ft,0 = rt
Next we discuss swap and swaptions. These contracts are two of the most
frequently traded financial contracts. Since swaptions are traded so much, this
results in a very liquid market for swaptions. This is also one of the main
reasons swaptions are used to calibrate short rate models in this thesis. In the
remainder of this section we will explain the basics of swaps and swaptions.
By entering a swap, a party is able to hedge some interest rate risk. A swap
is a term structure product in which a variable payment stream can be converted
into a fixed stream and vice versa. These variable and fixed streams are also
called floating and fixed legs respectively, where the predetermined payment
72
dates are called tenors. If a party holds a payer swap, that party is obliged to
pay the fixed leg and receive the floating leg. For a receiver swap this is vice
versa.
In order to value a swap, one has to subtract both legs from each other.
In order to do so, one has to value each leg. Valuing the fixed leg is obvious.
As the party holding a payer swap is obliged to pay a fixed rate rsw over a
predetermined notional amount L, the value of the fixed leg at time T0 is equal
to:
L · rsw
N
X
P (T0 , Ti )
(B.1.8)
i=1
where the tenor dates are defined as T1 , T2 , . . . , TN .
In order to value the floating leg, we need to define some floating rate.
Say this rate is defined by the current term structure R(T0 , ·), so that the
corresponding discount curve P (T0 , ·) is defined by equation (B.1.1). Suppose
the tenor dates are again T1 , T2 , . . . , TN , and that we enter the swap at time
T0 . Suppose that at time T0 that initial capital of L is available. At that time
the capital can be invested in zero-coupon bonds that mature at time T1 , so
that the investment will generate an interest payment whose size is equal to the
interest rate R(T0 , T1 ) times the notional capital L. This payment is exactly
equal to the payment of the floating leg at time T1 . Now suppose at every time
Ti , i = {1, . . . , i−1}, the notional amount of L is available for investment. When
investing every time in zero-coupon bonds with a maturity one period ahead,
its interest payment will be equal to the floating rate payment of the swap. The
cost of this strategy is the difference between the value of the notional amount
at time T0 and at time TN :
L (1 − P (T0 , TN ))
(B.1.9)
So that by combining the value of the fixed leg (B.1.8) and floating leg (B.1.9)
we can express the value of a payer swap in a single equation:
Lrsw
N
X
P (T0 , Ti ) − L (1 − P (T0 , TN ))
i=1
Since the value of the swap must be zero at initiation, we have that:
Lrsw
N
X
P (T0 , Ti ) − L (1 − P (T0 , TN )) = 0
i=1
So that we can find a value for rsw for which the initial value of the swap equals
zero. We call this rate the par swap rate, which is defined explicitly as:
1 − P (T0 , TN )
FT0 = FT0 (T0 , TN ) = PN
i=1 P (T0 , Ti )
A forward swap contract is defined in the same way as a swap, except for the
fact that the start of the tenor dates will take place at a later date. Assuming
current time is t and the tenor dates defined as before, one has that the current
value of the floating leg is P (t, T0 )−P (t, TN ). The value of the fixed leg becomes
PN
rsw i=1 P (t, Ti ). Finally the value of the forward par swap rate becomes:
73
Ft = Ft (T0 , TN ) =
P (t, T0 ) − P (t, TN )
PN
i=1 P (t, Ti )
(B.1.10)
Next we will discuss the swaption. A swaption is a contract that gives the
holder the right, but not the obligation, to enter a swap contract at a specified
date with a specified strike rate. For instance define a swaption with strike rate
rK , expiry date T0 , tenor dates as before, where current time is t, t < T0 . The
corresponding payoff of the swaption at T0 can be defined as the maximum of
the value of the swap at T0 and zero:
!
N
X
sw
max 0, Lr
P (T0 , Ti ) − L (1 − P (T0 , TN ))
(B.1.11)
i=1
If this payer swaption is set to be at-the-money, strike rate rK will be set such
that the value of the underlying swap equals zero. Denote this strike rate by
AT M
rK
. In order for a payer swaption to be in-the-money, the strike rate should
AT M
be smaller than rK
. The payer swaption is out-of-the-money when the strike
AT M
. For a receiver swaption this is vice versa.
rate at initiation is larger than rK
Using the numeraire P
dependent pricing formula, we can price a swaption
N
using the annuity factor i=1 P (·, Ti ) as a numeraire. By defining π to be the
premium of a swaption, we can use the NDPF and the payoff of the swaption
at expiry, equation (B.1.11), to value the swaption at initiation:
PN πt
i=1 P (t,Ti )
EtQA
=
max(0,Lr sw
PN
)
i=1 P (T0 ,Ti )−L(1−P (T0 ,TN ))
PN
i=1 P (T0 ,Ti )
(B.1.12)
which can be rewritten into:
πt = L
N
X
P (t, Ti )EtQA [max(0, FT0 (T0 , TN ) − rK )]
i=1
where FT0 is defined as in equation (B.1.10). In the next section we will see
how a trader in practice typically prices swaptions based on an assumption of
the swap rate Ft , which will lead to the famous formula by Black. We will use
this formula for our own data as well.
B.2
Black’s Model
In this section the relationship between the lognormal volatility and the swaption premium is discussed based on Black’s formula. As this formula is used in
practice in order to price swaptions, we will see how observed lognormal volatilities fit observed premiums using Black’s formula. Per definition these should
match, however due to some conventions we will see some errors. This will be
shown later on in this appendix.
Define Ft = Ft (T0 , TN ) to be some forward swap rate. In Black’s model, Ft is
lognormally distributed. In order to have absence of arbitrage we have that the
drift term of the forward swap rate process equals zero under the risk-neutral
probability measure, so that the process of the swap rate equals:
74
dFt = σB Ft dWtQ ,
W0Q = 0
(B.2.1)
where σB is the implied lognormal volatility of the forward swap rate, which
can be observed in the market. In practice, traders also use the assumption that
the swap rate is normally distributed. The disadvantage here is that the swap
rate can become negative, which is unrealistic. Nevertheless one can find the
pricing formula for a normally distributed swap rate later on in this appendix.
Using Itô’s lemma, we can rewrite equation (B.2.1) into the following:
d(log(Ft )) =
1
1 1 2 2
1 2
dFt −
σ F dt = − σB
dt + σB dWtQ
Ft
2 Ft2 B t
2
By integrating the equation above from 0 to T , we get the following expression:
1 2
T + σB WTQ
(B.2.2)
logFT − logF0 = − σB
2
The solution for the lognormally distributed forward swap rate can be found by
taking the exponential of equation (B.2.2), so that:
1
2
Q
FT = F0 e− 2 σB T +σB WT
(B.2.3)
Next recall the pricing of a swaption using the numeraire dependent pricing
formula from the previous section:
πt = L
N
X
P (t, Ti )EtQ [max(0, FT0 (T0 , TN ) − rK )]
(B.2.4)
i=1
One can use the exact solution of the lognormally distributed swap rate to derive
an exact solution for the premium of a swaption. Therefore we implement the
solution for the lognormally distributed forward swap rate, equation (B.2.3), into
the equation above. Note that this is allowed as the probability measure used
in the expectation of equation (B.2.4) is the same as the probabilty measure of
the Brownian motion in the solution of the forward swap rate, equation (B.2.3).
As a results, we get the following:
πt = L
N
X
h
i
Q
Q
1 2
P (t, Ti )EtQ max(0, Ft e− 2 σB (T0 −t)+σB (WT0 −Wt ) − rK )
(B.2.5)
i=1
Since the Brownian motion under the risk-neutral measure WTQ is distributed
like any other Brownian motion WT , we can rewrite this to:
WT − W t =
√
T − t Z ∼ N (0, T − t)
where Z ∼ N (0, 1) is a standard normally distributed variable. When we use
this in (B.2.5), we can drop the estimation under Q and rewrite this as an
expectation conditional to time t under the real-world probability measure P:
πt = L
N
X
h
i
√
1 2
P (t, Ti )EtP max(0, Ft e− 2 σB (T0 −t)+σB T0 −tZ − rK )
i=1
As Z is standard normally distributed, one can compute the expectation by
calculating the following integral:
75
PN
πt = L i=1 P (t, Ti ) √12π
√
R ∞ − 1 z2
1 2
e 2 max(0, Ft e− 2 σB (T0 −t)+σB T0 −tz − rK ) dz
−∞
(B.2.6)
However the expression within the integral is only positive for certain values of
z. Therefore we want to bound the integral. We do that by solving for d in the
following expression:
2
1
Ft e− 2 σB (T0 −t)+σB
√
T0 −td
− rK > 0
As a result, we can bound the integral in (B.2.6) from below by –d, where d is
defined as follows:
2
(T0 − t)
log rFKt − 12 σB
√
d=
σB T0 − t
In other words, we only perform the integration in equation (B.2.6) for values
of z such that z > −d. For these values of z the integrand is non-negative, so
that we can remove the maximum-function:
πt = L
PN
=L
PN
i=1
P (t, Ti ) √12π
R∞
−d
1
2
1
2
e− 2 z (Ft e− 2 σB (T0 −t)+σB
√
T0 −tz
− rK ) dz
P (t, Ti )
√
R
R ∞ − 1 z2
1 2
1 2
∞
[ √12π −d e− 2 z Ft e− 2 σB (T0 −t)+σB T0 −tz dz − √rK
e 2 dz]
2π −d
2 (T −t) R
− 1 σB
PN
0
∞ − 1 z 2 +σB √T0 −tz
dz − rK Φ(d)]
= L i=1 P (t, Ti )[ Ft e 2√2π
e 2
−d
√
PN
= L i=1 P (t, Ti )[Ft Φ(d + σ T0 − t) − rK Φ(d)]2
i=1
where Φ(·) represents the cumulative distribution function of a standard normal
distribution. So in the end, one can use the Black-76 model to price a swaption
with expiry T0 and maturity TN according to:
πtBlack = L
N
X
P (t, Ti )[Ft (T0 , TN )Φ(d1 ) − rK Φ(d2 )]
(B.2.7)
i=1
where:
d1
d2
=
ln
Ft (T0 ,TN )
2
+ 12 σB
T0
rK
√
σ
T
B
√ 0
= d1 − σB T0
The Black-76 formula uses the lognormal volatility σB of the forward swap
rate, instead of the normal volatility. In the appendix B.5 one can see how
these volatilities are related according to Hagan [10], even though they are
defined by their own model. As the data of this thesis consisted of both types of
volatilities, it was checked whether this relationship holds. As it turns out, using
the conversion by Hagan one can replicate the lognormal volatility using the
normal volatility precisely. This relation also holds vice versa. In the following
section we define the swaption premium for the normal volatility.
76
B.3
From Lognormal Implied Volatility to Swaption Premium
Define Ft to be some forward swap rate. Unlike Black’s model in the previous
section, assume Ft is normally distributed. Therefore Ft follows the following
stochastic differential equaiton under the risk-neutral measure:
dFt = σN dWtQ
where σN is the implied normal volatilty of the forward swap rate. Note that
this volatilty differs from the Black volatility σB , however must lead to the same
premium eventually. This will be discussed in the following section.
When integrating both sides of the equation above, we get:
Z t
FT = F0 + σN
dWtQ = F0 + σN WTQ
(B.3.1)
0
Again recall the pricing formula of a swaption using the numeraire dependent
pricing formula:
πt = L
N
X
Pt (Ti )EtQ [max(0, FT0 (T0 , TN ) − rK )]
i=1
We can implement the solution of the normally distributed swap rate (B.3.1)
into the equation above. Again simplify notation by setting Ft (T0 , TN ) = Ft .
πt
=
L
PN
P (t, Ti )EtQ [max(0, FT0 − rK )]
=
L
PN
P (t, Ti )EtQ [max(0, Ft + σN (WTQ0 − WtQ ) − rK )]
i=1
i=1
Since the Brownian motion under the risk-neutral measure WTQ is distributed
like any other Brownian motion WT , one can again rewrite this to:
WT − W t =
√
T − t Z ∼ N (0, T − t)
where Z ∼ N (0, 1) is a standard normally distributed variable. When we use
this result in the numeraire dependent pricing formula, we can drop the estimation under Q and rewrite this as an expectation conditional to time t under the
real-world probability measure:
πt = L
N
X
P (t, Ti )Et [max(0, Ft + σN
p
T0 − t Z − rK )]
i=1
As Z is standard normally distributed, one can compute the expectation by
calculating the following integral:
πt = L
N
X
1
P (t, Ti ) √
2π
i=1
Z
∞
1
2
e− 2 z max(0, Ft + σN
−∞
77
p
T0 − t z − rK )dz (B.3.2)
However the expression within the integral is only positive for certain values of
z. Therefore we can bound the integral for some z > −d. We can find the value
for d by solving the following expression:
p
Ft + σN T0 − tz − rK > 0
As a result we have that:
d=
Ft − rK
√
σN T0 − t
Now we can bound the integral from equation (B.3.2) from below, so that
we remove the maximum function in the integrand:
πt
= L
PN
i=1
Pt (Ti ) √12π
√
1 2
e− 2 z (Ft + σN T0 − tz − rK )dz
R ∞ − 1 z2
e 2 dz
−d
R∞
−d
PN
t −rK
Pt (Ti )( F√
2π
√
R
1 2
σN√ T0 −t
+
∞e− 2 z z dz)
−d
2π
i∞ h
√
PN
σN√ T0 −t
− 12 z 2
= L i=1 Pt (Ti ) (Ft − rK )Φ(d) +
−e
2π
1 2 −d
√
PN
T0 −t
e− 2 d − 1
= L i=1 Pt (Ti ) (Ft − rK )Φ(d) + σN√2π
= L
i=1
where Φ(·) represents the cumulative distribution function of a standard normal
distribution. So in the end, one can use the assumption of a normal distributed
swap rate and the numeraire dependent pricing formula in order to price a
swaption using:
√
σN T0 − t − 1 d2
2
√
Pt (Ti ) (Ft − rK )Φ(d) +
πt = L
e
2π
i=1
N
X
(B.3.3)
where:
d=
B.4
Ft − rK
√
σN T0 − t
Black’s Model Fit
In this section we see how good Black’s formula and equation (B.3.3) are able
to fit observed premiums using observed implied volatilities. As discussed in
the previous sections, one can compute the premium of a swaption using the
observed lognormal and normal volatilities. Let us call this value π Black (T0 , TN ),
defined according to formula (B.2.7). The relative error between the swaption
premium according to Black’s formula and the observed swaption premium is
given in figure (B.1) for swaptions originating from ultimo 2011.
As one can see in figure (B.1), for every swaption we see that Black’s formula undervalues the premiums. However per definition these premiums should
match, as the Black volatilities observed are computed by inverting the price
formula. Nevertheless we have the pricing errors displayed in figure (B.1). This
can be due to several reasons. First of all, the observed premiums used may
have transaction costs incorporated. These transactions costs are not included
78
Figure B.1: Relative pricing errors when computing swaption premiums from
ultimo 2011
when valuing a swaption using Black’s formula. This may explain the undervaluation of all swaptions in the figure. Secondly, observed premiums used are the
average of bid and ask prices. This may cause pricing errors, even though these
errors are considered to be small. Since the evaluated swaptions are considered
to be highly liquid, bid- and ask-prices will lie very close to each other. Finally,
pricing errors may also occur due to the inter- and extrapolation of the discount
curve.
Traders nowadays often use a normally distributed swap rate in order to
price swaptions. In order to see if this premium results in the same premium
calculated using Black’s formula, all prices were computed again using equation
(B.3.3). As it turns out, the absolute difference between both premiums is 3%
at most. Again, a part of this can be explained due to the conventions discussed
above.
B.5
Lognormal and Normal Implied Volatilities
According to Black [1], one can define the process of some forward swap rate
Ft (T0 , TN ) = Ft like:
dFt = σB Ft dWtQ
where F0 is today’s forward swap rate and σB is Black’s lognormal volatility.
Like in the data, this kind of volatility typically is given in percent. However
when using a normal volatility, we can define the process of some forwards swap
rate Ft as:
dFt = σN dWt
where σN is the normal volatility.
According to Hagan [10], one can find an converison from the lognormal
volatility to the normal volatility using a Taylor expansion. In order to invert
79
this relation one can apply Newton’s scheme. The conversion from the lognormal
to normal volatilty is approximately defined as follows:
σN
=
Ft (T0 ,TN )−K
σB log(F
t (T0 ,TN )/K)
·
1
1
1
1+ 24
(1− 120
log2 (
Ft (T0 ,TN )
2 T + 1 σ4 T 2
)σB
0
K
5760 B 0
where K is the strike rate of the option. When Ft (T0 , TN ) → K, for instance
when evaluating at-the-money swaptions as in our data, the formula goes to a
“0 over 0”. In order to avoid this, Hagan uses the alternative formula:
σN ≈ σB
1
p
1 + 24
log2 (Ft (T0 , TN )/K)
Ft (T0 , TN ) · K
1 2
1
4 T2
1 + 24 σB T0 + 5760
σB
0
which can be reduced to a simpler formula when valuing at-the-money swaptions:
σN ≈
1+
σB K
1 2
1
4 2
σ
T
24 B 0 + 5760 σB T0
as for at-the-money swaptions the forward swap rate and strike rate are equal
at the initiation of the contract.
As we know, any irregularities in these volatilities can result in pricing errors
when using Black’s formula. Therefore both volatilities from the data were
compared using the formula above. As a result, one should indeed find that the
normal volatility can be found using the lognormal volatility. This conversion
is found to be exact for every swaption at every date. Using Newton’s scheme,
one can find the inverse relationship. Although Newton’s scheme approximates
the lognormal volatility, its approximation error found is negligible.
B.6
Nelson-Siegel function
Throughout this thesis we mainly used the Nelson-Siegel function to interpolate
data points, creating the term structure for initial calibration years as well as
for simulated years. The function uses four parameters in order to describe
the term structure. Determination of these parameters is often done using a
minimization of least squares. The Nelson-Siegel function becomes:
(T )
Rt
= β0,t + (β1,t + β2,t )
1 − exp(−T /τt )
− β2,t exp(−T /τt )
(T /τt )
The parameter τ represents the convergence parameter towards the longterm level β0 . The short-term and medium-term parameter are defined by β1
and β2 respectively.
B.7
Other Short Rate Models
As discussed, the process of the short rate is able to describe current and future
term structures of interest rates. In a strong simplification, we tend to describe
this process using a short rate model. In choosing a short rate model one should
keep in mind analytical tractability and realism. In this subsection we present
80
two very popular short rate models by discussing their pro’s and con’s. Note that
these models were not calibrated in this thesis, however due to their popularity
we would like to present them here after all.
The Vasicek model is a simplification of the one-factor Hull-White model.
The process of the short rate under a Vasicek model is defined according to the
following stochastic differential equation:
drt = a(b − rt ) dt + σ dWt
So that the short rate mean-reverts around the level b. Here the parameters
a, b and σ are all constant over time. Because of these few, fixed parameters, the
Vasicek model does not match realized term structures well. The short rate may
also become negative, which is not a realistic as well. The advantage however
is that the Vasicek model remains simple, so that it does have some analytical
tractability.
Black and Karasinksi (1991) solved the problem of negative short rate, by
modifying the one-factor Hull-White model for the natural logarithm of the
short rate. This way the short rate cannot have negative values. The model is
defined as follows:
d(ln(rt )) = (θ(t) − a(ln(rt ))) dt + σ dWt
which is also able to fit the current term structure like the Hull-White one-factor
model.
81
Appendix C
Proofs
C.1
Jamshidian’s Decomposition
In this section we will give proof of Jamshidian’s Decomposition. In the original
paper by Jamshidian [12], proof is only given for the Vasicek model. During this
proof we will see that every one-factor model suffices if it meets one particular
requirement.
Say we want to price a payer swaption at time t. The swaption expires at
time T0 , the underlying swap matures at time TN . The strike rate and the
notional amount of the swaption equals rK and L respectively. If one enters a
swaption contract, the payoff at time T0 becomes:
h
i+
PN
1 − P (T0 , TN ) − rK i=1 P (T0 , Ti )
h
i+
PN
= 1 − Π(T0 , TN , r(T0 )) − rK i=1 Π(T0 , Ti , r(T0 ))
h
i+
PN
= 1 − i=1 ci Π(T0 , Ti , r(T0 ))
(C.1.1)
Here we changed the future zero coupon bond price P (T0 , Ti ) into an analytical
expression Π(T0 , Ti , r(T0 )), which depends upon the short rate at time T0 . The
weights ci are defined by the strike rate rK , except for cN = 1 + rK . Jamshidian
used a simple trick in order to rewrite the expression above, which was based
on finding the solution for r∗ of the following equation:
N
X
Π(T0 , Ti , r∗ ) = 1
i=1
So that the left hand side of the equation above can be substituted into expression (C.1.1), so that:
h
i+
PN
1 − i=1 ci Π(T0 , Ti , r(T0 ))
hP
i+
N
∗
=
c
(Π(T
,
T
,
r
)
−
Π(T
,
T
,
r(T
)))
0
i
0
i
0
i=1 i
In order to make the solution r∗ unique, we assume the following:
82
(C.1.2)
δΠ(t, s, r)
<0
for all 0 < t < s
δr
Under this assumption, we can decompose (C.1.2) into:
hP
N
i=1 ci
i+
Π(T0 , Ti , r∗ ) − Π(T0 , Ti , r(T0 ))
=
h
i+
PN
∗
c
Π(T
,
T
,
r
)
−
Π(T
,
T
,
r(T
))
i
0
i
0
i
0
i=1
So that we have a combination of payoffs, weighted by our ci ’s. Note that
each payoff represents a payoff of a zero coupon bond put option (ZBP). Therefore we can write the payer swaption in terms of zero coupon bond put options:
PS(t, T0 , TN , L, K) = L
N
X
ci ZBP(t, T0 , Ti , Π(T0 , Ti , r∗ ))
i=1
C.2
Derivation theta function in HW1F Model
In this section the derivation of the deterministic function θ(t) is given of the
Hull-White one-factor model. This function is used in order to create a perfect
fit of the Hull-White one-factor model onto market data. We start of using the
stochastic differential equation of the short rate under the Hull-White one-factor
model:
drt = (θ(t) − art ) dt + σ dWtQ ,
r(0) = r0
(C.2.1)
Next we rewrite this model by defining the stochastic part using an OrnsteinUhlenbeck process. Instead of using θ(·) to fit the current term structure, we
define a function β(·) to be added to the Ornstein-Uhlenbeck process. In formula:
dxt = −axt dt + σ dWtQ
(C.2.2)
rt = xt + β(t)
Which holds for some deterministic function β(t). The parameters a and σ are
the same parameters used in equation (C.2.1). Note that the process x in (C.2.2)
follows an Ornstein-Uhlenbeck process. Next we want to price a zero-coupon
bond using the following equation:
h
R
i
T
P (0, T ) = E0QM exp − 0 r(u) du
h
R
i
R
(C.2.3)
T
T
= E0QM exp − 0 x(u) du · exp − 0 β(u) du
where the second step follows from the fact that β(t) is deterministic. In order to
compute the integral of x(u) within the equation under QM in equation (C.2.3),
we need to compute the mean and variance of the exact solution of x(u). We
need this as E[exp(K)] = exp(mK + 21 vK ), if K is a normally distributed random
variable with mean mK and variance vK . The exact solution of x(u), its integral,
and the mean and variance of that corresponding integrals are defined as follows
respectively:
83
x(t)
RT
x(u) du
T
E 0 x(u) du
R
T
Var 0 x(u) du
R
T
Var 0 x(u) du
0
R
RT
= x(0)e−at + σ 0 e−a(t−s) dW (s)
RT
RTRu
= x(0) 0 e−au du + σ 0 0 e−a(u−s) dW (s) du
RT
−aT
= x(0) 0 e−au du = x(0) 1−ea
2 RTRu
= E σ 0 0 e−a(u−s) dW (s) du
2 R T R u −a(u−s)
= E
e
dW
(s)
du
0 0
R
2
R
T
T
= σ 2 0 s e−a(u−s) du ds
2
RT
−a(T −s)
= σ 2 0 1−e a
ds
−aT
2
−2aT
= σa2 T − 1−e2a
− 2(1−ea )
(C.2.4)
However note that x(0) = 0 per definition, so that the mean of the integral of
x(u) becomes zero as well. Using the mean and variance stated above, we can
simplify equation (C.2.4) into:
h
R
i
R
T
T
P (0, T ) = E0QM exp − 0 x(u) du ∗ exp − 0 β(u) du
R
R
T
T
= exp −E 0 x(u) du + 21 Var 0 x(u) du
R
T
· exp − 0 β(u) du
R
R
T
T
= exp 21 Var 0 x(u) du · exp − 0 β(u) du
2 −aT
−2aT
σ
= exp 2a
T + 1−e2a
− 2(1−ea )
2
R
T
· exp − 0 β(u) du
Next we can use the result above to fit the initial instanteneous forward rate,
discussing in appendix B.1:
f M (0, T ) = −
δlnP (0, T )
σ2
= − 2 (1 + e−2aT − 2e−aT ) + β(T )
δT
2a
which holds per definition. After rewriting, we get that:
β(T ) = f M (0, T ) +
σ2
(1 + e−2aT − 2e−aT )
2a2
(C.2.5)
In order to find the expression for θ(t), we want to know the relation between
β(t) and θ(t). We can do this by differentiating the short rate expression in
(C.2.2):
drt
= dxt + d (β(t)) dt = (dβ(t) − axt ) dt + σ dWt
=
(dβ(t) − a (rt − β(t))) dt + σ dWt
=
(dβ(t) + aβ(t) − art ) dt + σ dWt
(C.2.6)
When comparing the drift terms from (C.2.1) and (C.2.6), we get that the
function θ(t) is defined according to:
84
δf M (0, t)
σ2
+ af M (0, t) + 2 (1 − e−at )
δt
2a
where the last step follows from differentiating (C.2.5).
θ(t) = dβ(t) + aβ(t) =
C.3
Dynamics Short Rate Integral under HW1F
Model
In this section is shown that the short rate integral under the Hull-White onefactor model follows the following normal distribution:
RT
r(u)du | Ft ∼ N B(t, T )[r(t) − α(t)]
t
(C.3.1)
P M (0,t)
1
+ ln P M (0,T ) + 2 [V (0, T ) − V (0, t)], V (t, T )
We can use stochatsic integral by parts to rewrite this to:
Z T
Z T
Z T
r(u) du = T r(T )−r(t)t−
u dr(u) =
(T −u) dr(u)+(T −t)r(t) (C.3.2)
t
t
t
where the integral on the right hand side of the equation above can be written
as:
Z T
Z T
Z T
(T − u) dr(u) = −a
(T − u)r(u) du + σ
(T − u) dW (u)
(C.3.3)
t
t
t
Again, the first integral on the right hand side of the equation above can be
written as:
RT
RT
−a t (T − u)r(u) du = −a r(t) t (T − u)e−a(u−t) du
(C.3.4)
RT
Ru
+ σ t (T − u) t e−a(u−s) dW (s) du
Rewrite the first integral on the right hand side of equation (C.3.4) into:
Z T
e−a(T −t) − 1
r(t)
−a r(t)
(T − u)e−a(u−t) du = −r(t)(T − t) −
a
t
The double integral on the right hand side of (C.3.4) can also be rewritten using
integration by parts:
Ru
RT
σ t (T − u) t e−a(u−s) dW (s) du =
i
(C.3.5)
RTh
−a(T −u)
−1
− σ t (T − u) + e
dW (u)
a
By substituting each of the expressions (C.3.3) - (C.3.5) into equation (C.3.2),
we have that:
Z T
Z
1 − e−a(T −t)
σ T
r(u) du =
r(t) +
[1 − e−a(T −u) ] dW (u)
a
a
t
t
So that we can derive (C.3.1).
85
C.4
Deriving Short Rate Solution HW2F Model
In this section the exact solution of the short rate under the HW2F model is
derived. Recall the HW2F model:
(θ(t) + ut − art ) dt + σ1 dW1,t ,
drt
=
dut
= −but dt + σ2 dW2,t
r(0) = r0
By integration both equations we get the following:
r(t)
u(t)
=
=
RT
Rt
r(s)e−a(t−s) + s θ(v)e−a(t−v) dv + s u(v)e−a(t−v) dv
RT
+ σ1 s e−a(t−v) dW1 (v)
RT
u(s)e−b(t−s) + σ2 s e−b(t−v) dW2 (v)
(C.4.1)
Assuming a 6= b, we can rewrite the second integral on the right hand side of
the expression of the short rate using the solution of u(t):
Rt
s
u(v)e−a(t−v) dv
= u(s)
Rt
e−b(v−s)−a(t−v) dv
Rt
Rv
+ σ2 s e−a(t−v) s e−b(v−x) dW2 (x) dv
s
−e−a(t−s)
+
a−b
R
t
+ σ2 e−at s e(a−b)v
= u(s) e
−b(t−s)
Rv
s
(C.4.2)
ebx dW2 (x) dv
where the last step in the expression above follows from writing out the integrals
as much as possible. The double integral on the right hand side can now be
rewritten using integration by parts:1
R t R v bx
R t (a−b)v R v bx
1
( e dW2 (x)) dv (e(a−b)v )
e
e dW2 (x) dv = a−b
s s
s
s
R
R t (a−b)v
R v bx
1
(a−b)t t bx
= a−b e
e dW2 (x) − s e
dv ( s e dW2 (x))
s
(C.4.3)
R
R
t
v
1
[e(a−b)t − e(a−b)v ] dv ( s ebx dW2 (x))
= a−b
s
R t at−b(t−v)
1
= a−b
[e
− eab ] dW2 (v)
s
By substituting (C.4.2) and (C.4.3) into equation (C.4.1), we can create a single
expression for the short rate under a Hull-White two-factor model:
Rt
Rt
r(t) = r(s)e−a(t−s) + s θ(v)e−a(t−v) dv + σ1 s e−a(t−v) dW1 (v)
(C.4.4)
R t −b(t−v)
−b(t−s)
σ2
−e−a(t−s)
−a(t−v)
+
[e
−
e
]
dW
(v)
= u(s) e
2
a−b
a−b s
In the next section expression (C.4.4) will be matched to the exact expression of
the short rate under the G2++ model in order to find the equivalance between
parameters.
1 Integration
by parts:
R
R
u(x)v 0 (x) dx = u(x)v(x) − u0 (x)v(x) dx
86
C.5
Dynamics short rate integral G2++ Model
In this section we will derive the mean and variance of the integral defined in
section (3.4.7):
T
Z
[x(u) + y(u)] du
I(t, T ) =
(C.5.1)
t
where x(t) and y(t) are defined under the risk-adjusted measure QM , which we
already derived as:
Rt
x(t) = x(s)e−ā(t−s) + σ̄ s e−ā(t−u) dW1 (u)
Rt
y(t) = y(s)e−b̄(t−s) + η̄ s e−b̄(t−u) dW2 (u)
The mean and variance of (C.5.1) will be found by deriving the exact expression
for I(t, T ) first. We can simplify I(t, T ) by splitting up the integral:
T
Z
T
Z
[x(u) + y(u)] du =
Z
T
x(u) du +
t
t
y(u) du
t
So that if we compute one integral, we can find the other integral as well by
substituting in the corresponding parameters. Therefore, we would like to have
RT
RT
the exact expression for t x(u)du and t y(u)du. Fortunately, we already
derived the dynamics of the short rate integral under the HW1F model written
as an Ornstein-Uhlenbeck process, which resembles processes {x} and {y} in
the G2++ model. Therefore we can use equation (C.2.4) and replace a by ā, b
by b̄, σ by σ̄ and η by η̄. We get the following:
RT
t
RT
t
x(u) du =
1−e−ā(T −t)
x(t)
ā
+
σ̄
ā
y(u) du =
1−e−b̄(T −t)
y(t)
b̄
+
η̄
b̄
RT
[1 − eā(T −u) ] dW1 (u)
t
RT
[1 − eb̄(T −u) ] dW2 (u)
t
So that I(t, T ) can be written as:
I(t, T )
=
1−e−ā(T −t)
x(t)
ā
RT
[1 − eā(T −u) ] dW1 (u)
t
RT
−b̄(T −t)
y(t) + η̄b̄ t [1 − eb̄(T −u) ] dW2 (u)
+ 1−e b̄
+
σ̄
ā
One can now simply take the mean and variance, M (t, T ) and V (t, T ) respectively, of the expression above:
M (t, T )
=
1−e−ā(T −t)
x(t)
ā
V (t, T )
=
σ̄ 2
ā2
+
1−e−b̄(T −t)
y(t)
b̄
1 −2ā(T −t)
3
T − t + ā2 e−ā(T −t) − 2ā
e
− 2ā
h
2
+ η̄b̄2 T − t + 2b̄ e−b̄(T −t) − 21b̄ e−2b̄(T −t) −
+2ρ̄ σ̄āη̄b̄ [T − t +
+e
−b̄(T −t)
b̄
−1
−
87
e−ā(T −t) −1
ā
e−(ā+b̄)(T −t) −1
ā+b̄
3
2b̄
i
C.6
Exact Swaption Formula under G2++ Model
In this section we will derive the price of the swaption under the two-additivefactor Gaussian model. The proof originates from the book by Brigo and Mercurio [3]. We start with the numeraire dependent pricing formula. Instead of
using the annuity factor as numeraire as in equation (B.1.12), we will use a
zero-coupon bond maturity at the option expiry:


PN
max
1
−
P
(T
,
T
)
−
r
P
(T
,
T
),
0
0
N
K
0
N
i=1
πt

(C.6.1)
= ET 
P (t, T0 )
P (T0 , T0 )
where the denominator on the right hand side is equal to 1. Note that we are
valuing a payer swaption at time t with strike rate rK , expiry date T0 and
tenor dates Ti , i = {1, . . . , N }. The expectation is taken with respect to the
forward T -measure, indicating that the chosen numeraire is the zero-coupon
bond maturity at T0 . Equation (C.6.1) can be written into the following:
"
!#
N
X
πt = P (t, T ) E T max 1 −
ci P (T0 , Ti ), 0
(C.6.2)
i=1
where we define the weights ci as:
ci
=
rK
cN
=
1 + rK
We can rewrite the expection in equation (C.6.2) by applying a double integral with respect to x and y. We also insert expression (3.4.12) for P (T0 , Ti ) in
order to rewrite (C.6.2) into:
πt
= P (t, T )
RR∞
[(1 −
−∞
PN
−B(a,T0 ,Ti )x−B(b,T0 ,Ti )y +
) ]
i=1 ci A(T0 , Ti )e
· f (x, y) dy dx
(C.6.3)
where f (x, y) is the probability density function of a bivariate normally distributed random vector (x(T ), y(T )), which is defined as follows:
2
2
y−µy
(x−µx )(y−µy )
x−µx
1
−
2ρ
+
])
exp(− 2(1−ρ
2 )[
xy
σ
σ
σ
σ
x
x y
y
xy
q
f (x, y) =
2
2πσx σy 1 − πxy
By fixing x in the integrand of the equation above, and integrating this over y
from −∞ to +∞, we get the following:
!
Z ∞
N
X
2
1−
λi e−B(b,T0 ,Ti )y γeE+F (y−µy )−G(y−µy ) dy
(C.6.4)
ŷ(x)
i=1
88
where
λi (x)
γ
= ci A(T0 , Ti )e−B(ā,T0 ,Ti )x
=
2πσx σy
1
√
E
1
= − 2(1−ρ
2
F
=
G =
1−ρ2xy
xy )
ρxy x−µx
1−ρ2xy σx σy
1
2(1−ρ2xy )σy2
x−µx
σx
2
Note that instead of calculating the integral in (C.6.3) from −∞ to +∞, we
can bound it from below as the integrand of this equation becomes monotone
decreasing when x is fixed, Therefore the lower bound ŷ(x) is found for y when
the integrand of (C.6.3) is equal to zero. In formula, the lower bound ŷ(x) is
found for:
N
X
λi (x)e−B(b,T0 ,Ti )y =
i=1
N
X
A(T0 , Ti )e−B(a,T0 ,Ti )x−B(b,T0 ,Ti )y = 1
i=1
as A(T0 , Ti ) was defined in equation (3.4.13). Next we decompose the integral
in (C.6.4):
R∞ PN
2
−B(b,T0 ,Ti )y
1
−
λ
e
γeE+F (y−µy )−G(y−µy ) dy
i
i=1
ŷ(x)
R∞
2
(C.6.5)
= ŷ(x) γ eE+F (y−µy )−G(y−µy ) dy
PN R ∞
2
− i=1 ŷ(x) γ λi e−B(b,T0 ,Ti )y+E+F (y−µy )−G(y−µy ) dy
So that we get two expressions. We can rewrite each expression using the
following trick:
Rb
a
e−Az
2
√
B2
π 4A
e
dz = √A
h √
√
· Φ b 2A − √B2A − Φ a 2A −
+Bz
√B
2A
i
(C.6.6)
We start with the first integral on the right hand side of (C.6.5):
R∞
2
γ eE+F (y−µy )−G(y−µy ) dy
ŷ(x)
2 R∞
2
= γ eE−F µy −Gµy ŷ(x) e(F +2µy G)y−G µy dy
2
√
F2
2
π 4G +µy F +µy G
e
= γ eE−F µy −Gµy √G
h
i
√
F +2µ G
· Φ(∞) − Φ ŷ(x) 2G − √2Gy
h
i
√
√
2
π E+ F
= γ √G
e 4G 1 − Φ (ŷ − µy ) 2G − √F2G
where the second-to-last step in the derivation above follows from using the trick
in equation (C.6.6), where A = G and B = F + 2µy G. The second integral on
the right hand side of equation (C.6.5), for any i = {1, . . . , N }, can be rewritten
89
as follows:
R∞
2
γ λi e−B(b,T0 ,Ti )y+E+F (y−µy )−G(y−µy ) dy
ŷ(x)
2 R∞
2
= γ λi eE−F µy Gµy ŷ(x) e(F +2µy G−B(b,T0 ,Ti ))y−Gµy dy
2
√
F2
2
π 4G +µy F +µy G−B(b,T0 ,Ti )µy
= γ λi eE−F µy Gµy √G
e
h
i
√
F +2µy G−B(b,T0 ,Ti )
0 ,Ti )−2F ]
√
+ B(b,T0 ,Ti )[B(b,T
1
−
Φ(ŷ(x)
)
2G
−
4G
2G
√
√
2
F
,T )
π E+ 4G
√ 0 i
= γ √G
e
1 − Φ (ŷ − µy ) 2G − F −B(b,T
2G
where the second to last step followed from using the trick in equation (C.6.6),
where A = G and B = F + 2µy G − B(b, T0 , Ti ). Now that we have rewritten
both expression on the right hand side of (C.6.5), we can rewrite this as well:
R∞ PN
2
−B(b,T0 ,Ti )y
1
−
λ
e
γeE+F (y−µy )−G(y−µy ) dy
i
i=1
ŷ(x)
h
i
√
√
2
π E+ F
= γ √G
e 4G 1 − Φ (ŷ − µy ) 2G − √F2G
h
i
√
√
2
PN
,T )
π E+ F
√ 0 i )
− i=1 γ √G
e 4G 1 − Φ((ŷ − µy ) 2G − F −B(b,T
2G
So that by substituting the equation above into (C.6.4), we get the exact formula
for pricing payer swaptions under the G2++ model:
π0G2++
=
x 2
R ∞ − 12 ( x−µ
σx )
L · P (0, T0 ) −∞ e σ √2π
x
h
i
PN
· Φ(−h1 (x)) − i=1 λi (x)eκi (x) Φ(−h2 (x)) dx
where the following functions are defined in the equation above:
ŷ−µy
√
−
2
ρxy (x−µx )
√
σx 1−ρ2xy
h1 (x)
=
h2 (x)
=
h1 (x) + B(b̄, T0 , Ti )σy
λi (x)
=
ci A(T0 , Ti )e−B(ā,T0 ,Ti )x
κi (x)
=
−B(b̄, T0 , Ti )
i
h
x
· µy − 12 (1 − ρ2xy )σy2 B(b̄, T0 , Ti ) + ρxy σy x−µ
σx
σy
1−ρxy
q
1 − ρ2xy
and where the following expressions were used in (C.6.7):
µx
= −MxT0 (0, T0 )
µy
= −MyT0 (0, T0 )
q
−2āT
= σ̄ 1−e2ā 0
q
−2b̄T
= η̄ 1−e2b̄ 0
h
i
η̄
−(ā+b̄)T0
= (ā+ρ̄σ̄
1
−
e
b̄)σx σy
σx
σy
ρxy
for MxT0 (0, T0 ) and MyT0 (0, T0 ) defined as follows:
90
(C.6.7)
MxT (s, t)
=
σ̄ 2
ā2
MyT (s, t)
=
η̄ 2
b̄2
C.7
+ ρ̄ σ̄āη̄b̄ [1 − eā(t−s) ] − 2āσ̄2 e−ā(T −t) − e−ā(T +t−2s)
h
i
ρ̄σ̄ η̄
−b̄(T −t)
−b̄T −āt+(ā+b̄)s
− b̄(ā+
e
−
e
b̄)
h
i
σ̄ η̄
+ ρ̄ āb̄ [1 − eb̄(t−s) ] − 2āσ̄2 e−b̄(T −t) − e−b̄(T +t−2s)
h
i
ρ̄σ̄ η̄
−ā(T −t)
−āT −b̄t+(ā+b̄)s
e
−
e
− ā(ā+
b̄)
Equivalence G2++ and HW2F Model
In this subsection the equivalence between the two-additive-factor Gaussian
model and the Hull-White two-factor model will be shown. The proof will be
given by setting the exact expressions for the short rate for both model equal to
each other. Like in chapter 3 will parameters from the G2++ model be barred.
Brownian motions from the HW2F model will be denoted by the letter Z, also
in order to keep distinctive notation. The expressions for the short rate from
the HW2F and G2++ model are respectively:
rHW2F (t)
=
rG2++ (t)
=
RT
Rt
θ(v)e−a(t−v) dv + σ1 0 e−a(t−v) dZ1 (v)
R t −b(t−v)
σ2
e
− e−a(t−v) dZ2 (v)
+ a−b
0
Rt
Rt
σ̄ 0 e−ā(t−v) dW1 (v) + η̄ 0 e−b̄(t−v) dW2 (v) + φ(t)
r0 e−at +
0
Assuming a > b, we define the following parameters:
q
σ22
σ1 σ2
σ12 + (a−b)
σ3 =
2 + 2ρ b−a
σ
dZ3 (t)
=
σ4
=
2 dZ (t)
σ1 dZ1 (t)− a−b
2
σ3
σ2
a−b
Using these newly defined parameters, we can rewrite the short rate under the
HW2F model as follows:
RT
rHW2F (t) = r0 e−at + 0 θ(v)e−a(t−v) dv
i
h
Rt
σ2
+ σ1 0 e−a(t−v) σ1 dZ1 (v) + b−a
dZ2 (v)
R t −b(t−v)
σ2
(C.7.1)
+ a−b
e
− e−a(t−v) dZ2 (v)
0
R
T
= r0 e−at + 0 θ(v)e−a(t−v) dv
Rt
Rt
+ σ3 0 e−a(t−v) dZ3 (v) + σ4 0 e−b(t−v) dZ2 (v)
where the correlation between Z2 (t) and Z3 (t) can be found as follows:
σ1 dZ1,t − σ4 dZ2,t
σ1 ρ − σ4
dZ2,t dZ3,t = dZ2,t
=
σ3
σ3
Since dZ1,t dZ3,t = ρdt. We can derive the relationship between parameters
by setting the expression for the short rate under the G2++ model and the
91
rewritten short rate expression under the HW2F model, equation (C.7.1), equal
to each other.
ā = a
b̄
σ̄
= b
q
σ12 +
=
η̄
=
ρ̄ =
φ(t)
=
σ22
(a−b)2
1 σ2
+ 2ρ σb−a
σ2
a−b
σ1 ρ−η
σ̄
r0 e−at +
Rt
0
(C.7.2)
θ(v)e−a(t−v) dv
By inverting all relations in (C.7.2), we can express the parameters of the
HW2F model as parameters of the G2++ model:
a
= ā
b
= b̄
p
σ̄ 2 + η̄ 2 + 2ρ̄σ̄ η̄
=
= η̄ ā − b̄
σ1
σ2
ρ
=
σ̄ ρ̄+η̄
σ1
θ(t)
=
dφ(t)
dt
92
+ a φ(t)
Appendix D
Calibration Results
D.1
Hull-White one-factor Model
Table D.1: Calibration results of the HW1F model, using the SMM approximation, based on absolute weights
Term Structure #1: Interpolating Data Points EONIA using Nelson-Siegel
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2011 0.14891 0.00584
0.15053
0.00011
2012 0.14122 0.00505
0.16292
0.00009
Term Structure #2: Interpolating Data Points EONIA using Splines
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2011 0.09974 0.00451
0.11792
0.00010
2012 0.00969 0.00231
0.16457
0.00028
Term Structure #3: Calculated by inverting Black’s Formula
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2003 0.19634 0.00192
0.09487
0.00001
2005 0.16289 0.00412
0.09090
0.00005
2008 0.11248 0.00458
0.11291
0.00009
93
94
Var
-
Note that it is impossible to calibrate the model for 2003, 2005 and 2008 using observed implied volatilities and a calculated term structure. The calculated
term structure cannot be derived as we need the term structure itself in order to compute swaption premiums using Black’s formula.
Term Structure #3: Calculated by inverting Black’s Formula
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
2003 0.10160 0.02772
0.67484
0.00378
2005 -0.04467 0.00658
0.41633
-0.00048
2008 0.08774 0.01719
0.71518
0.00168
-
Term Structure #2: Interpolating Data Points EONIA using Splines
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
Var
2011 0.07396 0.01498
0.44816
0.00152 0.07402 0.01494
0.44723
0.00151
2012 0.04033 0.01066
0.24212
0.00141 0.04024 0.01061
0.24183
0.00140
Table D.2: Calibration results of the HW1F model, calibrated analytically, based on absolute weights
Term Structure #1: Interpolating Data Points EONIA using Nelson-Siegel
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
Var
2011 0.05608 0.01422
0.39639
0.00180 0.05610 0.01418
0.39587
0.00179
2012 0.00660 0.00901
0.30027
0.00615 0.00645 0.00897
0.30270
0.00623
95
Term Structure #3: Calculated by inverting Black’s Formula
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2003 0.17047 0.00169
0.09533
0.00001
2005 0.15115 0.00399
0.09139
0.00005
2008 0.11751 0.00476
0.11314
0.00010
Term Structure #2: Interpolating Data Points EONIA using Splines
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2011 0.9960 0.00458
0.11831
0.00011
2012 0.01018 0.00234
0.16474
0.00027
Table D.3: Calibration results of the HW1F model, using the SMM approximation, based on absolute-relative weights
Term Structure #1: Interpolating Data Points EONIA using Nelson-Siegel
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2011 0.15511 0.00611
0.15110
0.00012
2012 0.13333 0.00492
0.16313
0.00009
96
Var
-
Note that it is impossible to calibrate the model for 2003, 2005 and 2008 using observed implied volatilities and a calculated term structure. The calculated
term structure cannot be derived as we need the term structure itself in order to compute swaption premiums using Black’s formula.
Term Structure #3: Calculated by inverting Black’s Formula
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
2003 0.08511 0.02477
0.69280
0.00360
2005 -0.04096 0.00679
0.41711
-0.00056
2008 0.07216 0.01522
0.73230
0.00161
-
Term Structure #2: Interpolating Data Points EONIA using Splines
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
Var
2011 0.06733 0.01421
0.45383
0.00150
0.06726 0.01417
0.45295
0.00149
2012 0.02753 0.00974
0.25538
0.00172
0.02701 0.00967
0.25585
0.00173
Table D.4: Calibration results of the HW1F model, calibrated analytically, based on absolute-relative weights
Term Structure #1: Interpolating Data Points EONIA using Nelson-Siegel
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
Var
2011 0.04089 0.01277
0.41336
0.00200
0.04075 0.01273
0.41306
0.00199
2012 -0.01536 0.00764
0.34204
-0.00190 -0.01596 0.00758
0.34560
97
Term Structure #3: Calculated by inverting Black’s Formula
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2003 0.14686 0.00148
0.09669
0.00001
2005 0.13439 0.00378
0.09363
0.00005
2008 0.12239 0.00495
0.11390
0.00010
Term Structure #2: Interpolating Data Points EONIA using Splines
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2011 0.09928 0.00466
0.11990
0.00011
2012 0.01105 0.00239
0.16556
0.00026
Table D.5: Calibration results of the HW1F model, using the SMM approximation, based on relative weights
Term Structure #1: Interpolating Data Points EONIA using Nelson-Siegel
Method:
SMM approximation
Data used:
Observed volatilities
Year:
a
σ
RMSE (vol)
Var
2011 0.16332 0.00649
0.15353
0.00013
2012 0.12859 0.00487
0.16373
0.00009
98
Var
-
Note that it is impossible to calibrate the model for 2003, 2005 and 2008 using observed implied volatilities and a calculated term structure. The calculated
term structure cannot be derived as we need the term structure itself in order to compute swaption premiums using Black’s formula.
Term Structure #3: Calculated by inverting Black’s Formula
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
2003 0.05397 0.01983
0.80467
0.00364
2005 -0.03149 0.00726
0.42540
-0.00084
2008 0.06341 0.01377
0.79654
0.00150
-
Term Structure #2: Interpolating Data Points EONIA using Splines
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
Var
2011 0.06267 0.01357
0.47433
0.00147
0.06223 0.01350
0.47346
0.00146
2012 -0.00046 0.00812
0.35578
-0.07204 -0.00173 0.00802
0.36104
-0.01861
Table D.6: Calibration results of the HW1F model, calibrated analytically, based on relative weights
Term Structure #1: Interpolating Data Points EONIA using Nelson-Siegel
Method:
Jamshidian’s Decomposition
Data used:
Observed premiums
Observed implied volatilities
Year:
a
σ
RMSE (prem)
Var
a
σ
RMSE (prem)
Var
2011 0.02326 0.01132
0.46458
0.00276
0.02274 0.01125
0.46565
0.00279
2012 -0.04614 0.00612
0.51032
-0.00041 -0.04752 0.00603
0.51907
-0.00038
99
D.2
Table D.7: Calibration results of the HW2F model, calibrated analytically, based on absolute weights
Term Structure: EONIA-curve, intepolated using Nelson-Siegel function
Method:
Exact Swaption Formula under G2++ Model
Data used:
Observed Premiums
Year (Model):
a / ā
b / b̄
σ1 / σ̄
σ2 / η̄
ρ / ρ̄
RMSE(prem)
Var
2011 (HW2F):
0.31805 0.08928 0.01032 0.00567 0.11011
0.38586
0.00161
2011 (G2++):
0.31805 0.08928 0.02580 0.02481 -0.91748
0.38586
0.00161
2012 (HW2F):
0.31112 0.08501 0.00919 0.00555 -0.62012
0.20078
0.00135
2012 (G2++):
0.31112 0.08501 0.03111 0.02457 -0.97277
0.20078
0.00135
Hull-White two-factor Model
100
Table D.9: Calibration results of the HW2F model, calibrated analytically, based on relative weights
Term Structure: EONIA-curve, intepolated using Nelson-Siegel function
Method:
Exact Swaption Formula under G2++ Model
Data used:
Observed Premiums
Year (Model):
a / ā
b / b̄
σ1 / σ̄
σ2 / η̄
ρ / ρ̄
RMSE(prem)
Var
2011 (HW2F):
0.31318 0.09719 0.00968 0.00545 0.33920
0.35616
0.00148
2011 (G2++):
0.31318 0.09719 0.02375 0.02521 -0.92360
0.35616
0.00148
2012 (HW2F):
0.31029 0.08298 0.00980 0.00560 -0.70369
0.22082
0.00139
2012 (G2++):
0.31029 0.08298 0.03228 0.02462 -0.97646
0.22082
0.00139
Table D.8: Calibration results of the HW2F model, calibrated analytically, based on absolute-relative weights
Term Structure: EONIA-curve, intepolated using Nelson-Siegel function
Method:
Exact Swaption Formula under G2++ Model
Data used:
Observed Premiums
Year (Model):
a / ā
b / b̄
σ1 / σ̄
σ2 / η̄
ρ / ρ̄
RMSE(prem)
Var
2011 (HW2F):
0.31398 0.09385 0.01092 0.00545 0.26415
0.35610
0.00155
2011 (G2++):
0.31398 0.09385 0.02429 0.02477 -0.90108
0.35610
0.00155
2012 (HW2F):
0.39727 0.11300 0.01144 0.00789 -0.91643
0.18324
0.00112
2012 (G2++):
0.39727 0.11300 0.03852 0.02777 -0.99292
0.18324
0.00112
D.3
Cox-Ingersoll-Ross++ Model
Table D.10: Calibration results of the CIR++ model, calibrated analytically,
based on absolute weights
Term Structure: EONIA-curve, intepolated using Nelson-Siegel function
Method:
Exact Swaption Formula under CIR++ Model
Data used:
Observed Premiums
Year (Model):
k
θ
σ
x0
RMSE(prem)
Var
2011:
0.03092 0.00153 0.05440 0.06440
0.25353
0.00007
2012:
0.00206 0.65066 0.05105 0.03040
0.26476
0.41137
Table D.11: Calibration results of the CIR++ model, calibrated analytically,
based on absolute-relative weights
Term Structure: EONIA-curve, intepolated using Nelson-Siegel function
Method:
Exact Swaption Formula under CIR++ Model
Data used:
Observed Premiums
Year (Model):
k
θ
σ
x0
RMSE(prem)
Var
2011:
-0.01060 -0.00160 0.07452 0.02244
0.21604
0.00042
2012:
0.00307
0.93157 0.04967 0.02563
0.30675
0.37409
Table D.12: Calibration results of the CIR++ model, calibrated analytically,
based on relative weights
Term Structure: EONIA-curve, intepolated using Nelson-Siegel function
Method:
Exact Swaption Formula under CIR++ Model
Data used:
Observed Premiums
Year (Model):
k
θ
σ
x0
RMSE(prem)
Var
2011:
0.00572 0.03317 0.05593 0.03940
0.29243
0.00907
2012:
0.00364 4.16890 0.04816 0.01650
0.49004
0.46756
101
Appendix E
Results on Simulated
Holistic Balance Sheets
102
103
Varying Short Rate Assumptions
17.11
6.80
10.73
−0.66
0.25
14.66
178.78
V0SPS
V0EC
V0EG
V0PC
V0PD
V0Deficit
AHBS
0
178.96
14.85
17.11
6.79
10.74
−0.64
0.24
147.00
179.29
15.16
17.13
6.76
10.75
−0.60
0.23
147.00
189.91
15.40
17.95
7.04
11.31
−0.63
0.23
156.58
179.71
20.49
V0Surplus
LHBS
0
11.03
19.63
0.46
−14.46
5.40
148.14
V0AM
V0IND
V0SUR
V0BC
V0CAT
LTrad
0
179.51
20.46
10.93
19.53
0.49
−14.48
5.39
148.14
179.80
21.08
10.59
19.13
0.61
−14.55
5.40
148.14
190.71
22.73
11.99
20.85
0.76
−15.25
5.63
156.01
Note that by ’Abs’, ’A-R’ and ’Rel’ we mean absolute, absolute-relative and relative weights respectively. In order to see what steering mechanism each
abbreviation stands for, see table (5.2).
147.00
ATrad
0
Table E.1: Holistic balance sheets based on the Hull-White one-factor model calibrated for different weights, no correlation between state
variables assumed
Calibration Year(weight)
Calibration Year(weight)
Assets
Liabilities
Abs
A-R
Rel
Abs
Abs
2011
2011
2011
2012
2011
2011A-R 2011Rel
2012Abs
E.1
Table E.2: Holistic balance sheets based on the Hull-White two-factor model
calibrated for different weights, no correlation between state variables assumed
Assets
Year(weight)
2011Abs 2012Abs
ATrad
0
147.00
156.58
LTrad
0
148.14
156.01
V0SPS
V0EC
V0EG
V0PC
V0PD
16.51
7.26
10.29
−1.70
0.66
18.70
7.14
11.80
−0.34
0.09
V0AM
V0IND
V0SUR
V0BC
V0CAT
15.34
23.95
1.43
−15.46
5.43
16.04
25.04
0.23
−14.85
5.63
V0Deficit
21.56
13.23
V0Surplus
19.91
16.85
AHBS
0
185.08
188.74
LHBS
0
183.39
188.90
2011A-R
2012A-R
2011A-R
2012A-R
ATrad
0
147.00
156.58
LTrad
0
148.14
156.01
V0SPS
V0EC
V0EG
V0PC
V0PD
16.51
7.28
10.29
−1.71
0.66
18.78
7.24
11.80
−0.34
0.09
V0AM
V0IND
V0SUR
V0BC
V0CAT
15.36
24.04
1.39
−15.46
5.41
16.34
25.31
0.18
−14.81
5.66
V0Deficit
21.49
12.75
V0Surplus
19.78
16.29
AHBS
0
185.00
188.35
LHBS
0
183.29
188.64
2011Rel
2012Rel
2011Rel
2012Rel
ATrad
0
147.00
156.58
LTrad
0
148.14
156.01
V0SPS
V0EC
V0EG
V0PC
V0PD
16.54
7.29
10.29
−1.70
0.66
18.70
7.14
11.81
−0.33
0.09
V0AM
V0IND
V0SUR
V0BC
V0CAT
15.40
24.08
1.35
−15.44
5.41
16.08
25.04
0.24
−14.81
5.61
V0Deficit
21.50
13.20
V0Surplus
19.69
16.84
AHBS
0
185.03
188.71
LHBS
0
183.23
188.91
Liabilities
Year(weight)
2011Abs 2012Abs
Note that by ’Abs’, ’A-R’ and ’Rel’ we mean absolute, absolute-relative and relative weights
respectively. In order to see what steering mechanism each abbreviation stands for, see
table (5.2).
104
105
Including Correlations
17.09
6.75
10.71
−0.59
0.23
13.51
177.60
V0SPS
V0EC
V0EG
V0PC
V0PD
V0Deficit
AHBS
0
181.38
17.76
16.63
7.26
10.38
−1.85
0.84
147.00
188.26
13.68
18.03
7.04
11.36
−0.56
0.20
156.58
184.65
9.20
18.88
7.01
11.94
−0.11
0.05
156.58
177.99
18.26
V0Surplus
LHBS
0
11.59
19.94
0.50
−14.31
5.46
148.14
V0AM
V0IND
V0SUR
V0BC
V0CAT
LTrad
0
In order to see what steering mechanism each abbreviation stands for, see table (5.2).
147.00
ATrad
0
184.55
18.46
17.95
17.95
2.16
−15.05
5.45
148.14
189.23
20.19
13.04
21.18
1.18
−15.03
5.70
156.01
185.21
13.31
15.90
23.86
0.41
−14.11
5.74
156.01
Table E.3: Holistic balance sheets based on the Hull-White one-factor and two-factor model, calibrated for absolute weights, correlation
between state variables is assumed
2011
2012
2011
2012
Assets
Liabilities
HW1F
HW2F
HW1F
HW2F
HW1F
HW2F
HW1F
HW2F
E.2
106
Smaller Evaluation Horizon
9.85
3.60
6.34
−0.15
0.08
14.45
171.30
V0SPS
V0EC
V0EG
V0PC
V0PD
V0Deficit
AHBS
0
176.10
19.53
9.59
3.83
6.10
−0.76
0.41
147.00
181.91
15.20
10.14
3.66
6.58
−0.16
0.08
156.58
179.44
12.63
10.24
3.60
6.73
−0.14
0.06
156.58
171.10
18.96
V0Surplus
LHBS
0
4.00
12.89
0.08
−12.39
3.43
148.14
V0AM
V0IND
V0SUR
V0BC
V0CAT
LTrad
0
In order to see what steering mechanism each abbreviation stands for, see table (5.2).
147.00
ATrad
0
175.64
21.46
6.04
15.00
0.56
−13.14
3.61
148.14
182.01
21.23
4.79
14.13
0.25
−13.04
3.44
156.01
180.28
17.35
6.91
15.66
0.08
−12.29
3.46
156.01
Table E.4: Holistic balance sheets based on the Hull-White one-factor and two-factor model, calibrated for absolute weights, no correlation
between state variables assumed and using an evaluation horizon of 10 years
2011
2012
2011
2012
Assets
Liabilities
HW1F
HW2F
HW1F
HW2F
HW1F
HW2F
HW1F
HW2F
E.3
E.4
Varying Policies
Table E.5: Holistic balance sheets based on the Hull-White one-factor using
data from 2011, calibrated for absolute weights, no correlation between state
variables assumed
Policy
Value:
1
2
3
4
5
6
7
8
V0SPS
V0EC
V0PC
V0PD
V0EG
0
0
0
0
0
0
0
0
0
0
7.36
7.36
0
0
0
7.40
7.40
0
0
0
7.40
7.40
0
0
0
6.88
7.40
−0.55
0.03
0
16.91
6.80
−0.64
0.03
10.73
17.11
6.80
−0.66
0.25
10.73
V0Deficit
93.93
32.41
26.86
18.24
18.24
18.25
14.66
14.66
AHBS
0
240.91
179.40
181.23
172.63
172.64
172.11
178.58
178.78
V0AM
V0IND
V0BC
V0CAT
V0SUR
88.96
88.96
0
0
0
15.36
15.36
0
0
0
15.89
15.89
0
0
0
7.45
17.51
−15.23
5.16
0
7.90
17.54
−15.24
5.16
0.44
7.81
17.51
−15.25
5.16
0.36
10.98
19.63
−14.48
5.40
0.43
11.03
19.63
−14.46
5.40
0.46
V0Surplus
10.50
16.46
17.61
17.51
17.10
16.69
20.00
20.14
LHBS
0
247.59
179.96
181.64
173.14
173.15
172.63
179.11
179.30
Note that the initial values of assets and liabilties for 2011 are 147.00 and 148.14 respectively, where we assume an initial funding
ratio of 105%. In order to see what steering mechanism each abbreviation stands for, see table (5.2).
107
108
81.86
81.86
0
0
0
14.55
244.55
V0AM
V0IND
V0BC
V0CAT
V0SUR
V0Surplus
LHBS
0
186.45
19.00
19.31
19.31
0
0
0
187.69
40.69
0
0
0
0
0
187.90
20.01
19.75
19.75
0
0
0
189.45
34.74
7.71
7.71
0
0
0
178.70
19.21
11.35
22.08
−15.98
5.25
0
180.18
25.40
7.79
7.79
0
0
0
178.90
17.89
12.88
22.20
−15.99
5.25
1.43
180.36
25.58
7.79
7.79
0
0
0
177.66
17.00
12.53
22.09
−16.01
5.25
1.20
179.05
25.61
6.44
7.80
−1.45
0.09
0
182.96
19.64
15.19
23.91
−15.46
5.43
1.31
184.60
21.55
16.05
7.26
−1.60
0.10
10.30
183.39
19.91
15.34
23.95
−15.46
5.43
1.43
185.08
21.56
16.51
7.26
−1.70
0.66
10.29
Note that the initial values of assets and liabilties for 2011 are 147.00 and 148.14 respectively, where we assume an initial funding
ratio of 105%. In order to see what steering mechanism each abbreviation stands for, see table (5.2).
244.05
97.05
V0Deficit
AHBS
0
0
0
0
0
0
V0SPS
V0EC
V0PC
V0PD
V0EG
Table E.6: Holistic balance sheets based on the Hull-White two-factor using data from 2011, calibrated for absolute weights, no correlation
between state variables assumed
Policy
Value:
1
2
3
4
5
6
7
8