Modeling the dynamics of equity index
option implied volatilities in a real world
scenario set
Stefan N. Singor, Alex Boer
and Cornelis W. Oosterlee
Methodological Working Paper No. 2014-02
November 19, 2014
OFRC WORKING PAPER SERIES
Modeling the dynamics of equity index
option implied volatilities in a real world
scenario set
Stefan N. Singor*,+,1,2 , Alex Boer1 , and Cornelis W. Oosterlee2,3
Methodological Working Paper No. 2014-02
November 19, 2014
Ortec Finance Research Center
P.O. Box 4074, 3006 AB Rotterdam
Boompjes 40, The Netherlands, www.ortec-finance.com
Abstract
This article proposes a method of analyzing and modeling the real world dynamics
of equity put/call option implied volatilities using the risk neutral Heston model with
specific parameter restrictions. In our modeling approach, we construct a stable and
accurate method for calibrating the Heston model to historic market data. In this way,
the risk neutral Heston model is embedded in a real world scenario generator and can
be used to generate implied volatility structures, evaluate option investment strategies
or to construct hedging strategies. The proposed methodology results in a stable valuation of embedded options, which is in practice preferred by, among others, insurance
companies and pension funds.
Keywords: Heston model, implied volatility, risk neutral scenarios, real world scenarios,
risk management.
JEL Classification: C02, C10, C13, C52, C53, C58, C63, C80, G12, G13, G17, G22,
G23, G29.
*
Email corresponding author: [email protected]
c 2014 Ortec Finance bv. All rights reserved. No part of this paper may be reproduced, in any form or by any means, without
Copyright permission from the authors. Short sections may be quoted without explicit permission provided that full credit is given to the source.
The views expressed are those of the individual author(s) and do not necessarily reflect the views of Ortec Finance bv.
1
Ortec Finance, Boompjes 40, 3011 XB Rotterdam, The Netherlands.
2
Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands.
3
CWI - National Research Institute for Mathematics and Computer Science. Science Park 123, 1098 XG Amsterdam, The Netherlands.
+
Contents
1 Introduction
2
2 Real world and risk neutral scenarios in risk management
4
2.1 Modeling the risk neutral Q measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.1 Calibration of the Heston model to option market data . . . . . . . . . . . . . . . . . .
6
2.1.2 Implied volatilities in the Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2 Modeling the real world P measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3 Relevant literature
9
4 Model description and properties
10
4.1 Historical S&P-500 index IV data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
4.2 The VIX-Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4.2.1 The Heston-Benchmark model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.2.2 Calibration results of the VIX-Heston model . . . . . . . . . . . . . . . . . . . . . . . .
14
4.3 Principal component analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.4 Analytical tractability of the VIX-Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4.4.1 Example: application of the COS method . . . . . . . . . . . . . . . . . . . . . . . . .
19
5 Validation of the VIX-Heston model
20
5.1 Out-of-sample test for testing stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
5.2 Hedge test: Historical comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5.3 The VIX-Heston model in a risk management environment . . . . . . . . . . . . . . . . . . . .
21
6 Conclusions
23
References
23
Appendix A
24
1
1 Introduction
In the wake of the credit crisis, risk management is of highest importance for, among others, insurance companies and pension funds. Insurance companies and pension funds use Asset Liability Management (ALM)
as a tool for risk management, see Zenios and Ziemba (2007). In ALM calculations, real world and risk neutral economic scenarios form the basis of all computations, such as computing the value of the liabilities,
investments, Solvency II risks, etc. The real world and risk neutral worlds are often referred to as the P world1
and the Q worlds respectively, which are the corresponding probability measures.
In ALM, implied volatilities (IVs) are important for computing option values for investment or hedging strategies2 and calibration of risk neutral model(s), that are used to compute embedded option values. These
options are present on the balance sheet of pension funds or insurance companies. Due to the European
Solvency II regulation, insurance liabilities and pension funds are required to value their embedded options
on the balance sheet at market value and ideally also through time. ALM results depend on the value of the
investments and liabilities and, thus, on the value of (embedded) options. Embedded options are rights in
(life) insurance policies or pension contracts that can provide a profit to policy holders and never a loss.
In real world economic scenarios the simulation of equity index IVs is a challenging and important problem,
because ALM computations depend on it. Implied volatilities are values of the volatility parameters in the
Black-Scholes model (Black and Scholes (1973)), which match the model price with observed market prices.
We denote the Black-Scholes IV by σ := σ(t, K, τ), where t denotes time in the real world measure, K the
strike level and τ := T − s > 0 time to maturity, where T the maturity of the option and s denotes time measured
as of (real world) time t (0 ≤ s ≤ T ). The mapping (K, T ) → σ(t, K, T ) represents the IV surface at time t.
Available values for K and T depend on the liquidity of the market at time t. To obtain the corresponding
Black-Scholes put/call option values at time t, σ is substituted into the Black-Scholes formula:
C BS := C BS (t, σ, r, q, S t+s , K, τ, ω) = ω S t+s e−qτ N(ω d1 ) − K e−rτ N(ω d2 ) ,
(1.1)
where r := r(t, T ) denotes the deterministic interest rate
attime t and for maturity T , q the
√
√ fixed dividend yield,
S t+s the observed equity price index at time t + s, d1 = log SKt+s + (r − q + 12 σ2 )τ / σ τ and d2 = d1 −σ τ.
Furthermore, N(·) denotes the cumulative distribution function of the normal distribution and ω = 1 for a call
option and ω = −1 for a put option.
Even though it is widely recognized that the Black-Scholes model is not valid, the formulas are still used to
quote prices in terms of volatilities. IVs can thus be interpreted as transformations of prices that make options
with different maturities and moneyness easily comparable. Although IVs are not strictly ‘volatilities’, they
move closely together with other measures of volatility of the underlying asset. The complexity of simulation
of IVs is due to the non-constant behavior over time, see figure 1.1a, the high dimensionality, i.e. IVs depend
on option maturity and moneyness and the complex dynamics between IVs skewness and smiles, see figure
1.1b for an illustration. We note that sensitivities of the implied volatility in option portfolios are measured
inaccurately, when the above features of IVs are not properly modeled. The latter could lead to incorrect
conclusions/decisions in risk management. In Fengler (2012) more information can be found on stylized facts
regarding IVs.
To give an impression of the historical evolution of a number of S&P-500 equity index options and the IV
surface at 31/12/2012 we show figures 1.1a and 1.1b. Strike levels are given in terms of moneyness, i.e. 80%
refers to an absolute strike equal of 80% times the initial strike level. ATM denotes the at-the-money level, i.e.
100% moneyness.
In figure 1.1a a high correlation between 3 different time series is observed, although the level of the IVs
is different. In figure 1.1b we observe a smile pattern in the IV surface in the strike dimension. The IV surface
is highly skewed in the maturity dimension.
1
2
The real world measure is also known as the physical measure.
Option values of plain index options are computed by the Black-Scholes model of which the implied volatility is an input parameter.
2
Figure 1.1: Historical overview of 1 year IVs and IV surface as of 31/12/2012. Source: Bloomberg.
(b) IV surface as of 31/12/2012
(a) IVs (monthly data (30/06/2005 - 31/03/2014))
60
40
Implied volatility (%)
Implied volatility (%)
50
90 days, ATM
720 days, ATM
360 days, 80%
40
30
20
10
30
20
10
0
80
0
Jun05 Sep06 Dec07 Mar09 Jun10 Sep11 Dec12 Mar14
100
120
Strike (%)
0.5
1
1.5
2
Maturity (year)
The central topic of this article is the approximation of the mapping (K, T ) → σ(t, K, T ). We approximate
σ(t, K, T ) by using the risk neutral Heston model (see Heston (1993)), where we model the Heston parameters
conditional on the VIX index3 . The Heston model is widely used in practice for modeling risk neutral scenarios
and is flexible and accurate in modeling IVs. Since the VIX index can be incorporated in a real world scenario
model, we are able to embed the Q measure in the P measure. The advantages of the proposed methodology
are as follows:
• At each time point of a real world scenario a calibrated risk neutral Heston model is obtained, which
can be used for valuation purposes. Our modeling approach can be combined with real world scenario
models that can include the VIX index.
• Empirical facts are replicated, amongst them is that the resulting IV surfaces generated are arbitrage
free.
• The resulting Heston calibrations through time are more stable than calibrations without parameter
restrictions (e.g. no limits need to be imposed).
Extensive research is done to the modeling of IVs based on (semi) parametric models. In the book
Fengler (2006) multiple semi parametric methods are described on the modeling of IVs. This book fills a
gap in the financial literature by bringing together advances in the theory of implied volatility, semi parametric
estimation strategies and dimension reduction methods for functional surfaces. In Section 3 we will explore
relevant literature on the modeling of IVs.
Furthermore, there is much literature available on the pricing under the Heston model. In
Fang and Oosterlee (2008) the so-called COS method is constructed, which is a valuation method based on
Fourier-cosine series expansions, to price European type of options. The computational speed is high due to
the exponential convergence of the error. Furthermore, in Andersen (2008) a detailed discussion is found on
the simulation of the Heston model, which can be used in a Monte Carlo (see Glasserman (2004)) framework.
The so-called Quadratic Exponential (QE) discretization scheme is in practice often the default choice.
The remainder of this paper is organized as follows. In Section 2 we discuss the importance of modeling
IVs with respect to risk management and the modeling of real world and risk neutral scenarios. In Section 3
we provide an overview of relevant literature. In Section 4 we discuss our proposed VIX-Heston model. We
apply the method to S&P-500 index implied volatility data, but we note that the methodology can easily be
3
VIX is a popular measure of the implied volatility of short maturing S&P 500 index options. It represents one measure of the market’s
expectation of equity market volatility over the next 30 day period.
3
applied to other index options. In Section 5 we validate our approach using a number of applications. We
conclude in Section 6.
2 Real world and risk neutral scenarios in risk management
The main objective of ALM is to support the choice of risk and return appetite of stakeholders and specify
an integral ALM policy that given the specified risk limits maximizes the ambition. Thus, ALM represents a
decision making framework under uncertainty.
To determine the future average balance sheet or profit and loss account (and related value-at-risk (VaR)
measures) or to find the desired strategic asset mix of a pension fund in terms of the objectives and constraints
of the stakeholders, ALM studies apply the technique of scenario analysis. The base measure for ALM
computations is the P measure. Scenarios are future trajectories modeling the relevant risk factors such as
inflation, interest rates, currencies, the returns of the various investment categories, and the development
of options. In ALM studies the consequences are calculated of the policy intentions for all stakeholders
involved. This is done by taking into account the relevant characteristics of all individual participants, namely
the dynamics regarding long-life, career and disability etc., and how these characteristics are translated, given
the pension scheme, into premiums, indexations and funding ratios.
Remark. In our modeling framework we assume P := Pt=0 . I.e. we use all relevant information available at
current time t = 0 to construct P.
It is however quite common for pension funds and insurance companies to have an (equity or interest rate)
option portfolio in their investment portfolio for speculation or hedging of risks. Life insurers for example often
make use of replicating portfolios to hedge/model their embedded options. To obtain the market value of the
invested capital or replicating portfolio at a certain time point in a generated real world scenario, the option
portfolio must be valued at market value. For the latter we typically need the risk neutral Black-Scholes model
of which the market implied volatility is input to transform the implied volatility for a certain strike and maturity
to an option price.
Option portfolios can also consist of more complex option products, which are not traded on a market.
Hence, the generated implied volatility surface is not sufficient to obtain the corresponding market value of an
option portfolio. Instead, one needs a risk neutral scenario set, which is calibrated to the generated implied
volatility surface, in order to use a Monte Carlo simulation to obtain the market value. Thus, at a certain time
point in the P measure a risk neutral Q measure, which is dependent on t, is used to obtain the risk neutral
value of the option portfolio.
We propose in this article a modeling approach where the dynamics of IVs over time are modeled, so that
the Black-Scholes formula can be used to obtain the corresponding option prices of plain vanilla equity options.
Furthermore, within the modeling framework we automatically obtain a calibrated risk neutral scenario set to
the generated IVs. This scenario set can be used to value all kinds of complex options.
Remark (Nested simulation problem). Related valuation topics are embedded options on the liability side of
the balance sheet of a life insurer, economic capital/Solvency II computations for insurance companies, the
holistic balance sheet approach for pension funds (see Haan, Janssen, and Ponds (2012)). All risk neutral
valuations, for which Monte Carlo simulations are needed, suffer from the curse of dimensionality, i.e. the socalled nested simulation problem. Consider for example a simulation horizon of 1-year and 1.000 real world
scenarios for all kinds of economic variables. In the case of complex options, Monte Carlo simulations are
used to obtain the market value. Assume we need 10.000 scenarios to obtain a fair price of the options, then
in every real world scenario such a valuation has to be performed, which is rather computationally expensive.
In figure 2.1 we illustrate the relation between the measures P and Qt , where t denotes time in a real
world scenario. In this example we assume current market conditions as of June 2011, i.e. this is the starting
point of the real world scenario set. By generating real world scenarios we generate multiple possible market
4
conditions, thus different interest rates, equity indices, dividends etc. A possible market condition is illustrated
as of December 2013. The market data at that specific point is then input for a risk neutral model to value
(embedded) options as of December 2013.
Figure 2.1: Real world versus risk neutral scenarios.
Figure 2.1 illustrates that a series of risk neutral Qt (for every time t in a real world scenario) measures
should indeed be modeled within a generated real world scenario (generated under the P measure):
· · · , Qt=−2 , Qt=−1 , Qt=0 , Qt=1 , Qt=2 , · · · , where t denotes time in a certain generated real world scenario.
2.1 Modeling the risk neutral Q measure
In a risk neutral world all individuals are indifferent to risk and expect to earn on all assets a return equal
to the (instantaneous) risk free short rate. Assuming that the world is risk neutral facilitates the valuation of
options: the option payoffs can simply be discounted along the path of the short rate4 for each scenario. It is
also important to note that risk neutral valuation gives the fair price of an option in all worlds, not just in the
risk neutral world. For more information about risk neutral modeling we refer to Hull (2011).
In the academic world much research is devoted to risk neutral models and their applications. As a result,
multiple risk neutral models exist for all kinds of economic variables: interest rates, equity indices, inflations,
exchange rates etc. In this article we restrict ourselves to the modeling of equity indices. Popular risk neutral
equity models are: the Schöbel-Zhu model and its extensions (see
Grzelak, Oosterlee, and Van Weeren (2012) and Van Haastrecht, Lord, and Pelsser (2009)), the Heston model
and its extensions (See Grzelak and Oosterlee (2011)).
All these models have their own advantages and disadvantages. For our purposes the well known (plain)
Heston model (see Heston (1993)) is a well suited model to model long term behavior of skews and smiles of
equity indices. The Heston model characterizes itself by its analytical tractability and capability of modeling
4
The path of the short rate is depending on the modeling approach. In case of a fixed interest rate, this path is fixed and deterministic.
5
different sorts of IV surfaces. The latter enables one, in general, with a good calibration fit to option IVs observed in the market.
We model the risk neutral evolution of the equity index, denoted by S , and the coupled stochastic variance
factor ν, under the nominal risk neutral economy spot measure5 Q. Assuming the probability space is given
by (Ω, F , Q), the dynamics of the Heston model are given by:



 dS (t) = (r − q) S (t) dt +


 dν(t) = κ (ν̄ − ν(t)) dt +
p
ν(t) S (t) dW S (t), S (0) = S 0 ≥ 0,
p
γ ν(t) dW ν (t), ν(0) = ν0 ≥ 0,
(2.1)
where t represents time in the Q measure, r is the deterministic flat interest rate, q is the dividend yield, κ
is a mean-reversion parameter, γ a volatility parameter and ν̄ denotes the long-term variance level. The two
Wiener processes, dW S (t) and dW ν (t) are correlated with correlation parameter ρ.
In the Heston model the variance process is modeled by a so-called Cox Ingersoll Ross (CIR) process.
The density function of the variance process is known in advance. It turns out that the non-stationary solution,
conditional on F s (0 ≤ s < t ≤ T ), is a constant C times a non-central chi-squared distribution, with d degrees
of freedom and non-centrality parameter λ, where
C=
γ2 1 − e−κ(t−s)
4κ
,
d=
4κν̄
,
γ2
λ=
The mean and variance of ν(t), conditional on F s , are given by
E [ν(t) | F s ]
=
Var (ν(t) | F s )
=
4κe−κ(t−s) ν(s)
.
γ2 1 − e−κ(t−s)
ν̄ + (ν(s) − ν̄) e−κ(t−s) ,
ν̄γ2 2
ν(s)γ2 e−κ(t−s) 1 − e−κ(t−s) +
1 − e−κ(t−s) .
κ
2κ
(2.2)
(2.3)
(2.4)
The constant C is positive and a non-central chi-squared random variable has positive support for strictly
positive parameters d and λ, which implies that the variance process is always well defined because in realistic
applications the parameters of the CIR model are strictly positive, resulting in strictly positive values of d and
λ. To ensure that the variance process remains strictly positive at time t with 0 < t ≤ T and conditional on ν0
(> 0), we need
2κν̄ > γ2 ,
(2.5)
which is called the Feller condition. When the Feller condition is not fulfilled, equity volatilities can attain zero.
In practical applications the Feller condition is often not fulfilled which implies that a simulation scheme, such
as the Euler scheme, can break down due to the square root process. Therefore, we use the simulation
scheme of Andersen (2008) to generate the Monte Carlo scenarios of the Heston model.
2.1.1 Calibration of the Heston model to option market data
In general, the Heston model is calibrated to stock option market data using an optimization procedure to determine the model parameters in such a way that (relevant) option market prices are replicated
by the model
n
o
as good as possible. Hence, a calibration procedure consists of the computation of minΩHeston kC BS − C Heston k ,
where C BS denotes the (Black-Scholes) option market price, C Heston the option Heston model price, ΩHeston =
{κ, ν0 , ν̄, γ, ρ} the set of parameters (including constraints) and k · k some norm. We note that instead of using
option prices in the optimization procedure, one can also use implied volatilities, which leads to more accurate
results, but is more time consuming.
The market option prices are obtained by substituting the observed IV σ in the market into the BlackScholes formula, Eq (1.1). Market prices of plain vanilla index options are often used for calibration. The
corresponding IV of C Heston is obtained by numerically solving σHeston := σHeston (t, T, K, ΩHeston ) from
5
In the nominal economy this measure is generated by the nominal money-savings account M(t), which evolves according to dM(t) =
rM(t)dt.
6
C BS (t, σHeston , r, q, S t+s , K, τ, ω) = C Heston . In this way one is able to compare σHeston to σ.
We minimize the Euclidean norm of the difference between market and model option prices. We solve
this minimization problem iteratively using a numerical algorithm. We first sample random starting points and
then refine this solution using the well-known Levenberg-Marquardt least-squares algorithm, which is a local
minimization method. This procedure is repeated and the best solution is kept.
The computation of index option prices using the Heston model can be done in several ways. Typical
for calibration purposes a fast valuation method is needed so that a calibration is done within acceptable
computation time limits. Therefore, we use the semi-analytical approximation of Fang and Oosterlee (2008).
This approximation is efficient in terms of accuracy and computation time and thus suitable for calibration
purposes.
2.1.2 Implied volatilities in the Heston model
The flexibility of modeling the IV surface is provided in terms of five parameters in the Heston model. Within
the Heston model the IV surface is computed by (see Section 2.5 of Gatheral (2006)):
σHeston (t, T, K, Ω) =
p
EQ [ν(T ) | S (T ) = K].
(2.6)
√ out that for T ↓ 0 the forward ATM IV is equal
√Eq. (2.6) is investigated in detail in Gatheral (2006). It turns
to ν(0) and when T → ∞ the forward ATM IV is equal to ν̄. Hence, the parameters κ, ν0 and ν̄ affect the
modeling of IVs for different maturities, i.e. the term structure of IVs. The mean reversion, κ, indicates the
convergence between the short and long term variance.
The volatility parameter γ affects the kurtosis (peak) of the probability distribution function of equity (log)
returns. The lower the volatility of variance parameter, the higher the kurtosis (peak). The correlation parameter ρ affects the skewness of the probability distribution function of equity (log) returns. The higher the
correlation parameter (between the stock index and the variance process), the higher the skewness (heavy
tails to the right). Hence, these parameters affect the modeling of IVs in the strike dimension. In practice, ρ is
often negative and therefore skewness will be negative (more probability mass to the left-hand side).
In Gatheral (2006) it is noted that the short term skew is asymptotically independent from κ and T and
ργ
is approximately equal to 2 . Thus, increasing either ρ or γ implies an increase in the skew. The long term
skew is proportional to the inverse option maturity T1 , so the skew decreases approximately linearly when T
increases.
2.2 Modeling the real world P measure
Where risk neutral scenarios are typically used for valuation applications, real world scenarios are used in risk
management for (investment) decision making problems. An important issue in time series modeling is how to
describe the relevant empirical behavior/stylized facts as good as possible for the specific problem at hand. It
is known that the empirical behavior of economic and financial variables is typically different for different time
horizons (centuries, decades, years, months, etc.) and different observation frequencies (annual, monthly,
weekly, etc.). Think about typical and well known economic phenomena such as long term trends, business
cycles, seasonal patterns, stochastic volatilities, etc. For example, on a 30 year horizon with an annual observation frequency, long term trends and business cycles are important while on a 1 year horizon with a monthly
observation frequency, seasonal patterns need to be taken into account and on a 1 month horizon with a daily
observation frequency modeling stochastic volatility becomes a key issue. A second insight in the relevance
of the horizon and observation frequency is by thinking about the so called term structure of risk and return.
This means that expected returns, volatilities and correlations of and between asset classes are different at
different horizons. For example, the correlation between equity returns and inflation rates is negative on short
(for example 1 year) horizons while the same correlation is typically positive on long (for example 25 year)
7
horizons.
Hence, modeling real world scenario dynamics is a complex field of research, which we do not further
explore in this article. More information on this topic regarding ALM can be found in Steehouwer (2005). In
the academic literature it is quite common to apply a change of measure to the risk neutral model in Q to
obtain the dynamics in P (or vice versa). Although this is theoretically appealing, such models are often not
suited to model the typical stylized facts in P6 . In this modeling approach, the concept of market price of risk
(see Hull (2011)) is introduced, which provides for a (theoretical) consistent modeling of the real world, w.r.t.
the traditional risk neutral world. We note that our approach can be combined with any real world scenario
generator that includes scenarios of the VIX index. How the VIX index is incorporated in our modeling approach is outlined in the next section.
To understand the connection between the real world and risk neutral worlds, we show the following
example where the stock index S P under the real world probability measure P is modeled by the Heston
model:


µ S P (t) dt +

 dS P (t) =


 dν (t) = κ(ν̄ − ν (t)) dt +
P
P
p
νP (t) S P (t) dW S P (t), S P (0) = S P,0 ≥ 0,
p
γ νP (t) dW νP (t), νP (0) = νP,0 ≥ 0,
(2.7)
where µ is the expected return of S P . Thus, by specifying the Heston’s parameters we are able to generate
scenarios for {S P , νP }t . Here, νP is the realized volatility corresponding to S P . Next, by applying the Girsanov’s
theorem we obtain the dynamics under the risk neutral measure:


r S Q (s) ds +

 dS Q (s) =


 dνQ (s) = (κν̄ − (κ + ξν ) νQ (s)) ds +
p
νQ (s) S Q (s) dW S Q (s), S Q (0) = S P,0 ≥ 0,
p
γ νQ (s) dW νQ (s), νQ (0) = νP,0 ≥ 0,
(2.8)
where the equity risk premium is equal to µ − r and ξν is the volatility risk premium7 . In
Panigirtzoglou and Skiadopoulos (2004) implied risk neutral distributions are investigated. These implied risk
neutral distributions are also indirectly investigated in this article, where the (implied) risk neutral world is modeled by the Heston model (but with time-varying parameters). In risk management one starts by generating
real world scenarios of the stochastic processes {S P , νP }t . At time t = 0 the corresponding risk neutral set is
specified with initial values S Q (0) = S P,t=0 and νQ (0) = νP,t=0 , as shown in Eq. (2.8).
Next, at time t = 1 the parameters of the risk neutral model in Eq. (2.8) are implied by the market
{S P , νP }t=1 . In this case only the initial values of the implied risk neutral model are implied by S Q (0) = S P,t=1
and νQ (0) = νP,t=1 . The related implied risk neutral measure is then Qt=1 . In this way the implied risk neutral
dynamics are obtained for t = 0, 1, . . .8 .
The deterministic interest rate (and other relevant variables) in Eq. (2.8) is also implied by the market.
Although in Eq. (2.7) we have not specified an interest rate model, such a model is often part of a real world
scenario generator. Then, the interest rate9 r is implied by the simulated interest rate, hence r := rt . In our
modeling approach, we assume a deterministic interest rate in the risk neutral measure.
By using a statistical time series model to model the real world probability measure, parameter relations
between the real world model and the risk neutral model are more difficult to determine, since different models are used. However, in Section 4.2 (Eq. (4.2)), where we introduce the VIX-Heston model, the ‘parameter
relation’ between real world and risk neutral world, that we will use, is shown.
Comparison of risk premia is still possible and important for validation (prior to an ALM study for example).
The risk premium at time t is then computed by the difference of the expected returns on a certain equity
6
Examples are time varying risk premia, term structure of risk and return, etc.
See Bollerslev, Gibson, and Zhou (2011) for more information about the estimation of the volatility risk premia.
8
In this context, the period t = 1 is especially important for insurers. W.r.t. Solvency II the 99.5% value at risk measure after a 1-year
period is used to compute risk charges for determining the required capital.
9
For example the short interest rate.
7
8
variable and the risk free rate. For more information about risk premiums we refer to
Dimson, Marsh, and Staunton (2006), Campbell (2008) and Lettau and Van Nieuwerburgh (2008).
3 Relevant literature
In Homescu (2011) an overview is found on recent research activities in the field of IV modeling. We here
highlight some of the important contributions in our context. In Cont et al. (2002) a study is performed to
historical time series of option prices on the S&P-500 and FTSE indices. They study the dynamics of the IV
surfaces and show that they may be represented as a randomly fluctuating surface driven by a small number
of random factors. Next, they investigate the dynamics of these factors and propose a factor model, based
on a so-called Karhunen-Loève decomposition, which is compatible with the empirical observations. This
model extends and improves the ‘constant smile’ or ‘sticky delta’ model, often used by practitioners. Furthermore Cont et al. (2002) remark that the approach bridges the gap between risk-neutral approaches and the
empirical work on historical dynamics of implied volatility and allows automatic adjustment to today’s option
prices which are simply calibrated to the initial condition. The latter is true, but restrictive, since the practioner
needs to additionally calibrate a risk neutral model for valuation purposes. On top of this, for valuation at
time t calibrations need to be performed to the generated scenarios of the IVs. This results in high computation times. We discuss three proposed extensions of Cont et al. (2002), i.e. empirical findings based on a
stochastic volatility model (the Heston model), generation of scenarios of option portfolios and incorporation
of the dynamics of the VIX index in our modeling approach.
In Duan and Yeh (2011) a particle-filter estimation method is developed and applied to the S&P 500 index
and the VIX term structure jointly. They start with the following general dynamics under the real world measure
P:
p


dS (t) =










 dν(t) =
(r − q + δ0 + δ1 ν(t)) S (t− )dt +
ν(t)S (t− )dW S (t)
1 2
+(e J(t) − 1)dN(t) − (λ0 + λ1 ν(t))(eµ j + 2 γ j − 1)dt,
κ(ν̄ − ν(t))dt + γν(t)γ dW ν (t),
(3.1)
where S (t− ) denotes the left time limit of S (t); r is the deterministic flat interest rate, q is the dividend yield, κ
is a mean-reversion parameter, γ a volatility parameter and ν̄ denotes the long-term variance level. The two
Wiener processes, dW S (t) and dW ν (t) are correlated with correlation parameter ρ. N(t) is a Poisson process
with time-varying intensity λ0 + λ1 ν(t) and is independent of dW S (t) and dW ν (t); J(t) is an independent normal
random variable with mean µ J and standard deviation γ J . The term δ0 + δ1 ν(t) is the combined risk premium to
compensate for the diffusion and jump risks. This model is used to obtain the corresponding model under the
risk neutral Q measure. They remark that their estimation method takes advantage of the VIX term structure
information, and the model’s estimation no longer needs to rely on valuing individual options. Applying the
estimation method leads to the conclusion that the risk-neutral volatility dynamics are stationary and evolve
around a level that is higher than the physical volatility level.
In Lee (2005) the following three relevant questions are discussed. Does implied volatility admit a probabilistic interpretation? How does implied volatility behave as a function of strike and expiry? How does
implied volatility evolve as time rolls forward? With respect to the third issue it is mentioned that there are
two modeling approaches for IVs: direct implied volatility models, also known as market models, by enforcing
arbitrage free conditions or statistical models. An example of a statistical model is Cont et al. (2002), where
arbitrage free conditions are not used as a ‘stylized fact’. In Schönbucher (1999) the IV for a certain K and T ,
σ(t) = σ(t, K, T ), is directly modeled by
dσ(t) = ηt σ(t)dt + λt dW S (t) + φdW(t),
(3.2)
where W S and W are independent Brownian motions. The spot equity price under Q has the Black-Scholes
dynamics:
dS (t) = rS (t)dt + σS S (t)W S (t),
(3.3)
where r denotes the risk-free (flat) interest rate and σS denotes the volatility of the log returns. By using the
fact that the call price is a martingale under Q, drift restrictions can be derived in terms of partial differential
9
equations and in that way the functions ηt , λt and φ are derived. We note that in our modeling methodology,
the generated IV surfaces are (by definition) arbitrage-free.
In Carr and Wu (2010) a partial differential equation (PDE) approach is presented to model the dynamics
of IVs, which fills the gap between the two modeling approaches explained by Lee (2005). They directly model
both the drift and the martingale component of the implied volatility dynamics, and derive the dynamic-noarbitrage implication of the assumed dynamics on the shape of the implied volatility surface. Their approach
guarantees the dynamic consistency between the implied volatility dynamics and the derived implied volatility surface shape. They consider two parametric specifications for the implied volatility dynamics, which both
lead to very easy implied volatility surface constructions. Under both specifications, the whole implied volatility
surface becomes a solution to a quadratic equation, which is fast to evaluate. As a result, constructing implied
volatility surfaces based on the models in Carr and Wu (2010) can be much faster than with traditional models. It remains open how to guarantee static no-arbitrage among many options across different strikes and
maturities. More research is also needed on how to link the implied volatility dynamics to the instantaneous
variance rate dynamics.
In Fouque et al. (2004) the IV surface is modeled by their so-called log-moneyness-to maturity ratio (LMMR)
model:
log (K/S t )
+ b,
σ=a √
T −t
(3.4)
where a and b are estimated using market data. The goal is there to analyze Eq. (3.4) w.r.t. in-sample
estimates and stability over time. They analyze the system in the real world measure
 ǫ

dS (t) =





ǫ



 dY (t) =
µS ǫ (t)dt + f (Y ǫ (t))S ǫ (t)dW S (t),
p
1
1
α(t)(m(t) − Y ǫ (t))dt + √ ν(t) 2α(t)dW Y (t),
ǫ
ǫ
(3.5)
where Y is the volatility process, dW S and dW Y are correlated Brownian motions with correlation function ρ(t),
α(t), m(t), ν(t) are time-dependent parameters, the volatility function f is bounded, µ is an expected return
parameter and ǫ > 0 is small. The asymptotic case ǫ ↓ 0 is analyzed to gain insight in the local size of volatility
fluctuations. They determine the corresponding dynamics under the risk neutral measure, which they use for
the valuation of derivatives by asymptotic expansions. By using the numerical expansion of the option price
they determine a corresponding expansion of the IV, which is then linked to the a and b parameters of the
LMMR model in Eq. (3.4).
4 Model description and properties
We start in Section 4.1 with an overview of the historical data we use for the calibrations. In Section 4.2 we
introduce the VIX-Heston model. In Section 4.3 we show results of a principal components analysis, which
we apply to historical IV data and the IVs resulting from the VIX-Heston model.
4.1 Historical S&P-500 index IV data
In our analysis, we use the most liquid option data. Multiple S&P-500 index options are quoted with strike
levels (moneyness) ranging from 30% − 300% and maturities ranging from 30 − 720 days. For the construction
of the IV surfaces a data filtering is used to obtain the most liquid options. To gain more insight in the liquidity
of the IV data two aspects of the data are of interest, bid/ask spreads and availability of historical data. In the
data filtering process unreliable data is filtered out and the most liquid data is considered. For each maturity,
the bid/ask data is considered to be of good quality if at least 10% of the quotes and at least 3 options have
bid/ask prices. We choose to use the IVs for moneyness levels ranging from 50% to 150% and for maturities
(days) 90, 180, 360, 540, 720. Historical data for moneyness levels 80% − 120% and all maturities is available
as of June 2005 and historical data for other moneyness levels, for all maturities, as of November 2010; data
is collected until March 2014. In practice, it is common to smooth the gathered IV market data to correct for
10
outliers. We use the non-smoothed data as input for calibrations.
Statistics of the IV data are summarized in table 6.1 in Appendix A. Based on table 6.1 and figure 4.1a we
observe the following:
• For a certain maturity, the IVs decrease on average when the moneyness level increases. This is also
the case for the maximum and minimum IV, the skewness and kurtosis increase when the moneyness
level increases.
• For a certain moneyness level, the IVs become less volatile when the maturity of the option increases.
The skewness and kurtosis decrease when the maturity increases.
In figures 4.1a and 4.1b we have visualised the average historical IV surface of S&P-500 index options
and the autocorrelation function of the 1-year ATM option.
Figure 4.1: Average IV surface and autocorrelation function of 1-year ATM option.
(b) Autocorrelation function of 1-year ATM option
(a) Average IV surface of S&P-500 index options
1
Implied volatility (%)
50
Autocorrelation
40
30
20
0.5
0
10
0.5
2
1.5
1
1
0.5
1.5
Strike
0
−0.5
Maturity (years)
0
5
10
15
20
Lag
We observe in figure 4.1b high autocorrelation for different lags, which indicates large mean reversion.
This autocorrelation/mean reversion should be captured by the underlying factors in the real world scenario
generator for modeling the IV surfaces, this is however outside the scope of this article. By modeling the
dynamics of IVs one must be aware of existing empirical ‘facts’, which must be replicated (as good as possible) in the model at hand. In Kamal and Gatheral (2010) a short overview is provided of some empirical
findings: IV surfaces dynamically change over time, see figure 1.1a. IVs should fulfill arbitrage free conditions
in the risk neutral Q measure, i.e. the discounted call price should be a martingale, which implies no arbitrage
restrictions to the IV surface. IVs of long maturing options are less volatile than of short maturing options.
See table 6.1. The skew in the strike dimension in IVs converges to zero when the maturity increases and
no arbitrage requirements in implied volatility surface, see Roper (2010) and Carr and Madan (2005). The
volatility of volatility is level dependent. The higher the (average) level of the IV, the higher the volatility of
volatility, i.e. the IV surface is more skewed. This is due to the facts that investors sell their call options and
buy put options for protection in case of high volatility, i.e. in crisis times.
For a given maturity, IVs increase as strike levels decrease. Possible explanations for this phenomenon
are the (see Kamal and Gatheral (2010)) negative correlation between asset returns and volatility changes
(leverage effect). Big jumps in the asset (spot) price tend to be downwards rather than upwards. There is a
typical negative skewness in stock returns on the short term (daily or monthly) and positive skewness on the
long term (due to cumulative return effects). The risk of default: there is a probability for the price of a stock
to collapse if the issuer defaults. In Section 4.2 we account for the replication of these empirical facts by the
VIX-Heston model.
11
4.2 The VIX-Heston model
In Section 2.1 we have outlined the calibration procedure of the Heston model given the market IVs at a
certain time t. We note, however, that such a numerical method is computationally too expensive to apply in
every time step of a generated real world scenario set. Consider, for example, the case of determining the
Heston model parameters, to model the risk neutral measure, at t = 1 in the P measure, where we use 1.000
scenarios. Given the 1.000 simulated IV surfaces we have to perform 1.000 numerical calibrations to obtain
the corresponding Heston model parameters. Assuming one calibration takes 10 minutes, the total time for
calibration at t = 1 would be 10.000 minutes ≈ 167 hours, which is not acceptable in practice. We propose a
model that is able to model IV surfaces without calibration.
From a computational point of view it is more convenient to invert this calibration method, i.e. can we
choose the Heston model parameters in such a way that the resulting IV surface,
σHeston := σHeston (t, ΩHeston , T, K) matches the Black-Scholes IV, σ? This question is investigated in this section, where the answer leads to the VIX-Heston model. In order to do so, we consider time-dependent Heston
parameters (a 5-dimensional parameter space), ΩtHeston , because market conditions differ at each time in a
real world scenario. In this way we are able to evaluate σHeston through time given ΩtHeston (see Section 2.1.1).
We are, thus, interested in the function at time t, ΩtHeston → σHeston (t, ΩtHeston , K, T ). We note that this approach
leads, by definition, to arbitrage-free IV surfaces and it provides an arbitrage free interpolation of IVs in the
strike and maturity dimensions, see Homescu (2011) for more information.
The Heston model parameters are not observable in the market, so that we cannot substitute the true parameters in the model to ensure a good correspondence with the IV surface. Hence, we have to fit the Heston
model to the observed option prices / IVs. The goal is to dynamically model the Heston parameters ΩtHeston in
a real world scenario generator. Besides the empirical facts in Section 4.1, we assume the following requirements for σHeston : the methodology should be analytically tractable and ideally depend on observable market
indices, so that it can easily be applied within a real world scenario generator. The historical IV surfaces,
see Section 4.1, must be replicated well enough, i.e. σHeston should match σ sufficiently well. We measure
error by using the sum squared error (S S E ), the R-squared error (R2 )10 and the average absolute error (|E|),
and the Heston parameters should not fluctuate too much over time, which is especially important for stable
out-of-sample option valuations.
Unlike traditional methods in the literature, we directly model, at each time t, in the P measure the risk
neutral probability measure Q using the Heston model, see Eq. (2.8), by incorporating the simulated real
world market circumstances. We start by assuming that we are given an univariate process for theVIX index:
(V IXt )t≥0 and we denote the multivariate stochastic process of the Heston parameters by: ΩtHeston , where
t≥0
ΩtHeston
= κt , ν0,t , ν̄t , γt , ρt .
(4.1)
Typical examples of modeling the process V IXt are GARCH, ARCH and CIR models11 . We assume the
processes V IXt , κt , ν0,t , ν̄t and γt take on values in R+ and ρt takes on values in [−1, 1]. Next, we define a
mapping Φ : V IXt → ΩtHeston in Eq. (4.2) that produces a stochastic process of the Heston parameters from
a process for the VIX index. We therefore denote the image of Φ by ΩtHeston . Hence, conditional on time t,
Φ maps R+ to R+ × R+ × R+ × R+ × [−1, 1]. We assume that Φ is twice differentiable as in the requirements
for Ito’s lemma. The function Φ is constructed in such a way that it reflects the empirical facts as good as
possible, see Eq. (4.2). Next, the real world processes (V IXt )t≥0 imply a risk neutral Heston model12 at time
t, see Eq. (2.8) with corresponding risk neutral probability measure Qt .
Our main contribution in this article is the construction of the mapping Φ within the VIX-Heston model,
which is given by:
10
The R-squared error measures the goodness of fit of a statistical model to a set of data points with. More specifically, it compares
the unexplained variance (variance of the model’s errors) with the total variance (of the data).
11
See for example Fernandes, Medeiros, and Scharth (2014)
12
See Section 2.2 for the link between real world and risk neutral.
12

κt







ν0,t






ν̄t






γt




 ρ
t
=
κ,
=
2
κ ∈ R+ ,
(aν0 · V IXt + bν0 ) ,
aν0 , bν0 ∈ R,
ρ,
ρ ∈ [−1, 1].
2
= (aν̄ · V IXtrendt + bν̄ ) ,
=
aγ · V IXt + bγ ,
=
(4.2)
aν̄ , bν̄ ∈ R,
aγ , bγ ∈ R,
The stochastic process (V IXtrendt )t≥0 is obtained by the mapping φ : V IXt → V IXtrendt , see Section
4.2.1.The derivations of the relations in Eq. (4.2) are found in Section 4.2.2.
Since we can directly observe the VIX index, we can calibrate the 8 parameters of Eq. (4.2) by fitting the
IV surfaces (through time) to the observed ones. This results in the following calibration procedure:


2 
X X X σ(t, K, T ) − σHeston (t, ΩtHeston , K, T )  ,
min 
X
n
t
o
K
(4.3)
T
where X = κ, aν0 , bν0 , aν̄ , bν̄ , aγ , bγ , ρ is used via Eq. (4.2) to determine ΩtHeston , i.e. the Heston parameters at
time t. The optimal parameters of the VIX-Heston model are here given by
κ = 0.6949,
aγ = 1.035,
aν0 = 0.8952,
bγ = 0.3437,
bν0 = 0.0049, aν̄ = 0.9123,
ρ = −0.7287.
bν̄ = 0.1202,
(4.4)
We note that within the VIX-Heston model, IV surfaces dynamically change over time since there is a link
with the VIX index. The IV surface is arbitrage-free, because it is generated by the risk neutral Heston model.
Since the long term volatility parameter is linked to a trend component of the VIX, the IVs of long maturing options are less volatile than short maturing options. In the Heston model the skew in the IV surface converges
linearly in the maturity to zero, the volatility of volatility is time dependent since it depends on the VIX index,
the higher the level of the IV the higher the volatility of volatility. There is a negative correlation between the
equity returns and volatility changes.
In Section 4.2.1 we first fit a Heston model to each historic IV surface, so that we obtain a set of time series
for the Heston model parameters ΩtHeston . We call this model the Heston-Benchmark model. We then refine
this model, where we impose more structure to the calibration procedure of the Heston-Benchmark model by
introducing some simplifying assumptions on ΩtHeston , see Eq. (4.2).
4.2.1 The Heston-Benchmark model
To model ΩtHeston and ensure a good fit with the IV surfaces at each time t, we start with a full calibration of
a Heston model to each historical period (see Section 4.1 for more information about the data), without any
parameter restrictions. In other words, we determine ΩtHeston , where t runs over the historical IV surfaces, by
computing


2 
X X X  .
Heston
Heston

,
K,
T
)
σ(t,
K,
T
)
−
σ
(t,
Ω
min
t

Heston
Ωt
t
K
(4.5)
T
Hence, at each time t we calibrate a new set of Heston parameters in order to achieve the best fit possible
within this modeling framework. The domain Dom of ΩtHeston at time t is given by:
Dom := ∀t : κt ∈ [0, 10], ν0,t ∈ [0, 0.36], ν̄t ∈ [0, 0.36], γt ∈ [0, 2], ρt ∈ [−1, 1] .
(4.6)
R2 = 0.97,
(4.7)
In this way we obtain a time series for all Heston model parameters, ΩtHeston , which we further analyze. The
minimization problem is solved numerically as outlined in Section 2.1.1. This calibration serves as benchmark,
since a more accurate calibration does not exists in our modeling framework. We call this model the HestonBenchmark model. The calibration results are summarized in Eq. (4.7).
S S E = 0.96,
13
|E| = 0.69%.
We observe a satisfactory fit with the historical IV data. The resulting small error is due to model errors (limitation of the Heston model). In figures 4.2a and 4.2b the calibrated Heston model parameters are visualized.
For reference, the VIX index is also shown.
Figure 4.2: Unrestricted calibrated Heston model parameters.
(b) Heston κt and ρt parameters
(a) VIX index and Heston parameters ν0,t , ν̄t , γt
100
√
√ν0,t
ν̄t
−0.75
0.38
−0.755
0.36
−0.76
0.34
Correlation
γt
IV
60
40
Mean reversion
VIX
80
20
−0.765
0.32
Jun05 Sep06 Dec07 Mar09 Jun10 Sep11 Dec12 Mar14
0
Jun05 Sep06 Dec07 Mar09 Jun10 Sep11 Dec12 Mar14
We observe however an irregular pattern in the time series of the Heston parameters, although the correlation and mean reversion parameter are approximately constant (see figure 4.2b). The correlation parameter
is on average equal to −0.76. Hence, the leverage effect is correctly incorporated in the model. Figure 4.2a
shows a high correlation between the initial volatility parameter and the VIX index of 0.99. The long term
volatility shows a less volatile pattern compared to the VIX index. We further observe in figure 4.2a that γt
depends on the IV level because it shows a high correlation with the VIX index of 0.76.
4.2.2 Calibration results of the VIX-Heston model
By using the parameter relations in Eq. (4.2), we obtain the calibration results in Eq. (4.8).
S S E = 2.17,
R2 = 0.93,
|E| = 1.14%.
(4.8)
The calibration results are still rather good w.r.t. the Heston-Benchmark model (R2 : 0.97 → 0.92). The major
advantage of the VIX-Heston model over the Heston-Benchmark model is that the parameters are directly
linked to observable market indices. This enables one with efficient modeling in a real world scenario generator. In the paragraphs below we discuss the imposed parameter relations in more detail.
To gain insight in the parameter structure over time we show figures 4.3a and 4.3b, where we compare
the volatility parameters
√ of the Heston-Benchmark model with the the VIX-Heston model. In figure 4.3a we
√
compare ν0,t and ν̄t and in figure 4.3b we compare γt .
We observe in figure 4.3a that the initial volatility parameter is approximated rather well. The fits to the
long term volatility and volatility of volatility parameter is less accurate. This is due to the imposed structure
which enables one with more stable out-of-sample tests. The calibration fit to historical IVs is still very good.
Constant parameters κ and ρ We observe in figure 4.2b an almost constant time series for κt and ρt .
Hence, an obvious direction to impose structure to the calibration procedure is to investigate the case where
κt = κ and ρt = ρ, with κ ∈ [0, 10]13 and ρ ∈ [−1, 1], i.e. no time-dependency in the P measure. Keeping
these parameters constant also facilitates a pragmatic modeling in a real world scenario generator, since it is
difficult to link them to observed indices in the market.
13
We have chosen a realistic subset of R+ .
14
Figure 4.3: Comparison of
100
Volatility (%)
80
√
ν0,t and
√
ν̄t
(b) Comparison of γt
100
√
√ν0,t - Heston-Bench.
√ν̄t - Heston-Bench.
√ν0,t - VIX-Heston
ν̄t - VIX-Heston
80
Volatility (%)
(a) Comparison of
√
√
ν0,t , ν̄t and γt .
60
40
20
60
40
20
γt - Heston-Bench.
γt - VIX-Heston
0
Jun05 Sep06 Dec07 Mar09 Jun10 Sep11 Dec12 Mar14
0
Jun05 Sep06 Dec07 Mar09 Jun10 Sep11 Dec12 Mar14
Parameter ν0 - A linear relation to the VIX index In the Heston model the initial volatility parameter is a
representation of the IV of (ultra) short maturing options, so there is a direct link with the VIX index. The time
series of the initial volatility parameter are highly correlated with the VIX index, i.e. the Pearson’s correlation
√
coefficient is equal to 0.99. This implies a linear relation between the VIX index and ν0,t , which is also
confirmed by estimations in Kimmel et al. (2007). A scatter plot of the initial volatility and the VIX index is
shown in figure 4.4.
Figure 4.4: Scatterplot of
√
ν0,t and VIX index
60
VIX index (%)
50
40
30
20
10
0
0
10
20
30
√
40
ν0,t (%)
50
60
Parameter ν̄ - filtering trending behavior Coupling the long term volatility parameter to the VIX index is
less intuitive since the long term volatility parameter models the ATM IV of long maturing options, see Section
2.1.2. Because market data is only available until maturity 2-years14 , assumptions (expert opinions) have to
be imposed to model this (ultra) long implied volatility. Imposing structure to this long term volatility parameter
gives rise to stable valuations of exotic derivatives.
The topic of modeling the long maturing IV is less researched. It is likely that the (ultra) long ATM IV will
converge to a constant level. The strong convergence in the IV data confirms this. Historical data show that
the long term IV is stochastic and is dependent on the level of the VIX index, i.e. a high VIX index implies a
14
This in contrary to interest rates where maturities go up to 60 years in some cases.
15
high long term IV and vice versa. The latter is illustrated in figure 4.5a, where we illustrate the level dependency for different moneyness levels.
A way of modeling the long term ATM IV is by trends in the VIX index. The advantage is a direct link with
the VIX index which enables one for an direct implementation in a real world scenario generator. Traditionally,
trends are computed by moving averages. However, this leads to delayed behavior and loss of data points.
Kalman filters are well-known and popular method to extract trends from data without delayed behavior. See
for example Harvey (1990) and Durbin and Koopman (2012). We determine the trend component of the VIX
index, which we call√ V IXtrend, by using the Kalman filter in such way that the linear relation with the calibration time series ν̄t of the Heston-Benchmark model is optimal. The smoothing parameter in the Kalman
filter is determined in such a way that the Pearson’s correlation coefficient is maximized. The ‘optimal’
trend
√
component is visualized in figure 4.5b. The resulting correlation parameter with the time series ν̄t is 0.74.
The estimation of the trend component should preferably be done within the calibration procedure. However, this is computationally heavy. Therefore √
we apply a two step procedure. We first estimate the trend
component such that the linear relation with ν̄t is optimal. This relation is included into the calibration
procedure and the parameters are calibrated subsequently.
Figure 4.5: Modeling ν̄t by extracting trends of the VIX index.
(b) VIX index, V IXtrend and
(a) Level dependency of 2-year IVs w.r.t. the VIX index
45
70
40
VIX index
VIXtrend
√
60
ν̄t
35
50
30
Volatility (%)
2-year IV (%)
√
ν̄t
25
20
15
80%
90%
10
0
0
10
110%
120%
10
20
30
40
50
60
30
20
ATM
5
40
0
Jun05 Sep06 Dec07 Mar09 Jun10 Sep11 Dec12 Mar14
70
VIX index
√
The time series ν̄t shows a (rather) irregular pattern. This indicates that the long term volatility parameter
in the Heston model cannot be calibrated in a stable way, which is probably due to the lack of long term implied
volatility data (up to 2-years). Given the estimated trend we hence impose structure and a more realistic
relation to the long term volatility parameter. Since the filtered trend component V IXtrend is highly correlated
with ν̄t , we impose a linear relation (see Eq. (4.2)).
√
Parameter γ - A linear relation to the VIX index Since we model ν0,t by a linear combination of the
VIX index, it seems obvious to use the so-called VVIX index15 to model γt . It turns out that the relation
between the VVIX and γt is weak, which is probably due to the fact that VIX options cannot be valued by
the Heston model because many of the strikes are in the tail of the volatility process’s distribution, as remarked by Papanicolaou and Sircar (2013). Therefore, we do not proceed in this direction. It turns out that
a model for consistent pricing of both VIX and SPX options should have a volatility process that has a distribution which distributes probability mass over a broader range than a standard square-root process. In
Papanicolaou and Sircar (2013) the addition of a regime-switching process to the Heston model is proposed.
15
The VVIX Index is an indicator of the expected volatility of the 30-day forward price of the VIX. This volatility drives nearby VIX
option prices. CBOE also calculates a term structure of VVIX for different VIX maturities. The VVIX or any point on its term structure is
calculated from a portfolio of VIX options (VVIX portfolio) using the same algorithm used to calculate the VIX.
16
In figure 4.2a we observe an irregular pattern for γt and a dependency on the IV level; a correlation of 0.76
with the VIX index. Since short maturing options are more skewed than long maturing options and within the
Heston model γ typically models this skew in IV, an IV level of short maturing options seems appropriate to
impose a relation to. The relation between γt and the VIX index is visualized in figure 4.6.
Figure 4.6: Modeling γt .
100
γt (%)
80
60
40
20
0
0
10
20
30
40
50
VIX (%)
Given the relative high correlation with the VIX index and that this index is directly observable in the market,
we impose a linear combination to the VIX index (see Eq. (4.2)).
4.3 Principal component analysis
By performing a principal component analysis (PCA) to the historical S&P-500 equity index IV market data and
the corresponding IVs generated by the VIX-Heston model, we investigate whether the principal component
factors are comparable. Principal component factors are orthogonal (zero correlation) linear combinations of
a time series that describe the largest parts of the total variance. These factors are determined by assigning
a weight (loading) on each of the input time series. The weights are determined such that the resulting factors
describe largest part of the joint movements (correlations) of the input time series.
Remark. It turns out that the principal component factors are in general hard to interpret. As the number of
factors increases, the factors are capturing a smaller part of the joint movements, correlations with variables
decrease and the link between factors and variables quickly gets unclear. Therefore, methods have been
devised to improve the interpretation without compromises on the optimality of the factors. Most applied
is the so-called varimax method from Kaiser (1958). This method ‘rotates’ the factors in order to shift the
correlations between factor and variables as much as possible towards 1, 0 or −1 rather than remaining at
intermediate values. In this article we investigate the pure PCA results.
We start by performing a PCA to the (uncorrected) historical S&P-500 equity index IV market data (see
Appendix A for more information). Hence, we apply the PCA to 25 time series since we use the option
maturities 0.25, 0.50, 1.00, 1.50 and 2.00 years and strike levels 80%, 90%, 100%, 110% and 120%. It turns
out that the first component accounts for 97.6% and the second component for 1.5%. Since the first component
accounts for the most variance, this component describes the largest part of the joint movement of the time
series. The principal component coefficients (loadings) of the first 2 components are summarized in figures
4.7a and 4.7b.
In figure 4.7a we observe that the most significant skews are observed for out-of-the-money strike levels
(for put options) and short maturing options. We observe an almost flat/linear implied volatility structure for
the in-the-money strike levels (for put options) and long maturing options.
Next, we perform the same PCA, but instead of using historical S&P-500 equity index IV market data, we
use the corresponding IVs generated by the VIX-Heston model. In this case the first components accounts
17
Figure 4.7: PCA of historical S&P-500 equity index IV market data.
(b) Second principal component
120
120
110
110
Strike (%)
Strike (%)
(a) First principal component
100
90
80
100
90
0.5
1
1.5
80
2
0.5
1
Maturity (year)
0.18
0.19
0.2
0.21
1.5
2
Maturity (year)
0.22
0.23
0.24
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
for 98.0% and the second component for 1.4%. The principal component coefficients (loadings) of the first 2
components are summarized in figures 4.8a and 4.8b.
Figure 4.8: PCA of S&P-500 equity index IV approximated by the VIX-Heston.
(b) Second principal component
120
120
110
110
Strike (%)
Strike (%)
(a) First principal component
100
90
80
100
90
0.5
1
1.5
80
2
0.5
1
Maturity (year)
0.18
0.19
0.2
0.21
0.22
1.5
2
Maturity (year)
0.23
0.24
−0.2
−0.1
0
0.1
0.2
0.3
0.4
By comparing the PCA results of the historical S&P-500 equity index IV market data and the corresponding
IVs generated by the VIX-Heston model, we conclude that the first components, which account for most of
the variance, are very similar (97.6% vs. 98.0%). From these results we conclude that the VIX-Heston model
is able to model the dynamics in the historical IV surfaces rather well, which is important for risk management
purposes.
4.4 Analytical tractability of the VIX-Heston model
In the academic literature many numerical methods exist to compute C Heston . They take advantage of the fact
that the characteristic function is available in closed form and evaluate the probability density function (PDF).
The PDF f (x), and the characteristic function φ(x), are related via:
f (x) =
1
2π
Z
∞
e−iwx φ(ω)dω, φ(x) =
−∞
18
Z
∞
−∞
eiwx f (x)dx.
(4.9)
The characteristic function of the Heston model φHeston , is given by (see Albrecher et al. (2006) for more
information)
1−e−Dτ
ν0,t
(κ−iργω−D)
φHeston (ω) = eiω(r−q)τ e γ2 1−Ge−Dτ
q
κ−iργω−D
where D = (κ − iργω)2 + ω2 + iω γ2 and G = κ−iργω+D .
ν̄t
e γ2
τ(κ−iργω−D)−2 log
1−Ge−Dτ
1−G
,
(4.10)
Within risk management applications we opt using the VIX-Heston model to generate IVs. We generate
real world scenarios for the relevant economic variables, such as interest rates, stock indices, VIX and VVIX.
We use these scenarios to derive ΩtHeston for a certain time t in P. We denote the generated PDF for ΩtHeston
by fΩtHeston . In the VIX-Heston model we keep the mean reversion and correlation parameters constant, which
reduces the dimension of fΩtHeston from 5 to 3 Heston parameters. However, for generality we keep working with
fΩtHeston .
Since ΩtHeston is stochastic, C Heston is stochastic. The cumulative distribution function (CDF), FC : Dom →
[0, 1], where Dom denotes the domain of ΩtHeston , is given by
FC (x|t) =
Z
x
Dom
C Heston (t, ΩtHeston , T, K) fΩtHeston (ΩtHeston )dΩtHeston .
(4.11)
In general it is rather difficult to obtain a closed form expression of Eq. (4.11). This is due to the fact that
the underlying models for the parameters ν0 , ν̄ and γ have complex structures in order to model the (realistic)
stylized facts. In Section 4.4.1 a worked out example of Eq. (4.11) is presented.
4.4.1 Example: application of the COS method
In Fang and Oosterlee (2008) an efficient, and easy to implement, pricing method is constructed which we
further analyze in this section. Besides the distributional properties of the option price, one is also interested
in the Greeks corresponding to the option price such as the Delta and Gamma sensitivities. The expressions
are given by:
eHeston (t, ΩtHeston , T, K)
C Heston ≈ C
∆Heston ≈ e
∆Heston (t, ΩtHeston , T, K)
ΓHeston ≈ e
ΓHeston (t, ΩtHeston , T, K)
where
P
!
kπ
) ,
(4.12)
k=0
b−a
!
XN−1 Vk
ikπ ikπ x−a
kπ
= e−rτ
ℜ
e b−a φHeston (
) ,
(4.13)
k=0 S 0
b−a
b−a



XN−1 Vk  ikπ !2
kπ 
ikπ  ikπ x−a


−rτ


) , (4.14)
−
ℜ 
= e
 e b−a φHeston (
k=0 S 2
b−a
b − a
b−a 
Ke−rτ ℜ
=
XN−1
x−a
Uk eikπ b−a φHeston (
0
mean that the first term (k = 0) is multiplied by 12 , Uk and Vk are found in Fang and Oosterlee (2008).
Next, the corresponding distribution functions FCe, Fe∆ and FeΓ are given by
!
kπ
) fΩtHeston (y)dy ,
φHeston (
k=0
b−a
Dom
!
Z
XN−1 Vk
ikπ ikπ x−a
kπ
−rτ
b−a
= e
ℜ
e
) f Heston (y)dy ,
φHeston (
k=0 S 0
b−a
b − a Ωt
Dom



XN−1 Vk  ikπ !2
 ikπ x−a Z

ikπ
kπ


−rτ
−
) fΩtHeston (y)dy ,
φHeston (
ℜ 
= e
 e b−a
k=0 S 2
b−a
b−a
b−a
Dom
0
FCe =
Fe∆
FeΓ
Ke−rτ ℜ
XN−1
x−a
Uk eikπ b−a
Z
(4.15)
(4.16)
(4.17)
where fΩtHeston models uncertainty of the Heston model parameters at time t. These expressions all have the
following integral term in common:
Θ(ω) =
Z
Dom
φHeston (ω) fΩtHeston (y)dy,
19
(4.18)
From a computational efficiency perspective we are interested in the analytical properties of the function
Θ in Eq. (4.18). Popular computation programs to investigate analytical properties are Mathematica and
Maple. We observe in Eq. (4.10) rather complex non-linear terms with respect to parameters κ, γ and ρ and
integration techniques need to be applied to evaluate Eq. (4.18). On the other hand, the log characteristic
function log φHeston is proportional to ν0 and ν̄, i.e.
φHeston (ω) ∼ eν0 C1,ω +ν̄C2,ω +C3,ω ,
(4.19)
where C1,ω , C2,ω and C3,ω are constants. This might facilitate for efficient computation of the distribution
function given the assumptions of fΩtHeston (y).
5 Validation of the VIX-Heston model
In this section we validate the VIX-Heston model. In Section 5.1 we perform an out-of-sample test for testing
stability of the estimated model parameters and the fit to the IV data. In Section 5.2 we perform a historical
hedge test. We dynamically buy put and call options within the available historical period, and compare the
computed portfolio market value with the true (realized) market value. in Section 5.3 we apply the VIX-Heston
model in a risk management environment. We generate a set of scenarios and analyze the properties with
respect to historical dynamics.
5.1 Out-of-sample test for testing stability
In this section we perform an out-of-sample test to the VIX-Heston model for computing IVs. We perform two
experiments to show the stability of the estimated parameters and R2 error. In the first experiment we estimate
the model parameters in Eq. (4.2) to the IV surfaces of the first 82 months of the total historical period (see
Section 4.1 for more information on the used historical data). We then measure the R2 error of the last two
years and of the total historical period. In the second experiment we estimate the model parameters based
on the first 94 months of the total historical period and then measure the R2 error of the last year and of the
total historical period.
We note that the parameters of the VIX-Heston model, i.e. based on the full historical period, are shown
in Eq. (4.4). The estimated parameters of the first experiment are
κ = 0.6432, aν0 = 0.8897, bν0 = 0.0107, aν̄ = 0.9570,
aγ = 0.9474, bγ = 0.3585,
ρ − 0.7458.
bν̄ = 0.1059,
(5.1)
The estimated parameters of the second experiment are
κ = 0.7175, aν0 = 0.8863, bν0 = 0.0151, aν̄ = 0.9561,
aγ = 0.9727, bγ = 0.3569,
ρ = −0.7424.
bν̄ = 0.1031,
(5.2)
First of all, we observe a small change in the parameters of the two experiments compared to the VIX-Heston
model, which is beneficial for out-of-sample applications. Next, the in-sample R2 of the first and second experiment is equal to 0.9355 and 0.9271, respectively. We recall that the R2 of the VIX-Heston model is equal
to 0.9236. The reason that the R2 of the two experiments are higher than the R2 of the VIX-Heston model is
due to the calibration to less historical data, which (in this case) automatically leads to a better fit. The R2 of
the total historical period of the first and second experiment is equal to respectively 0.9186 and 0.9200. These
fits are still satisfactory compared to the fit of the VIX-Heston model.
The R2 of the VIX-Heston model to the last and last two years is equal to respectively 0.8686 and 0.8829.
The R2 of the first experiment to the last and last two years is equal to respectively 0.8243 and 0.8519. The R2
of the second experiment to the last and last two years is equal to respectively 0.8300 and 0.8630. Based on
these results we observe a small decrease of accuracy compared to the VIX-Heston model.
20
5.2 Hedge test: Historical comparison
In this section we compare the market value of a hedging option portfolio based on the VIX-Heston model
with the true market value. In order to do so, we perform a hedging experiment based on historical data and
check whether the VIX-Heston model is sufficiently accurate. There are many option strategies that can be
computed, for example delta hedging, gamma hedging, etc. Without loss of generality, we assume a fixed
option portfolio, which is bought at fixed times. In this way we are able to test at each timestep whether the
VIX-Heston model is accurate enough.
More specifically, we compare the market Black-Scholes option values C BS in Eq. (1.1), with the approximated option values from the VIX-Heston model C Heston (see Section 4.2). Time t runs over the available
historical period 06/2005-03/2014 (see Section 4.1), where we use a monthly frequency. The equity index S
denotes the S&P-500 index. Without loss of generality we set the dividend equal to zero.
We apply two experiments: the first experiment (Experiment I) is based on buying put options (long puts)
as an insurance for negative equity returns, the second experiment (Experiment II) is based on selling call
options (short calls) to take advantage of the premium as insurance for negative equity returns. Experiments
I and II are respectively based on moneyness levels 80% − 100% and 100% − 120%. For both experiments we
use maturities 0.25, 0.5, 1, 1.5, 2 years. This (fixed) option portfolio is bought/sold every 6 months. In figures
5.1a and 5.1b we show scatter plots for Experiments I and II to visualize the accuracy in terms of market
values.
Figure 5.1: Scatter plots of Experiment I and II.
(b) Experiment II
40
80
35
70
Approximated market value
Approximated market value
(a) Experiment I
30
25
20
15
10
5
0
60
50
40
30
20
10
0
5
10
15
20
25
30
35
0
40
True market value
0
10
20
30
40
50
60
70
80
True market value
It turns out that the fit in terms of market value is rather accurate. For Experiment I we measure an R2 error
of 0.998 and for Experiment II of 0.999. Changing the frequency of buying options does not have a significant
impact on the accuracy. We conclude that the VIX-Heston model is well suited to model IVs for applications
in risk management in this experiment.
5.3 The VIX-Heston model in a risk management environment
In this section we apply the VIX-Heston model in a risk management environment. I.e., we start by generating
scenarios of the relevant real world economic variables and we use the VIX-Heston model in Section 4.2 to
incorporate the simulated market circumstances in the risk neutral world (in each time step of a scenario). We
model real world scenarios for the VIX index16 . The smoothed VIX level, V IXtrend, of the VIX-Heston model
(see Eq. (4.2)) are computed conditionally on the VIX index. Since we do not wish to incorporate a view of the
16
Without loss of generality we assume a fixed and flat interest rate curve.
21
future behavior of the VIX index (or IV surfaces), we are especially interested in the replication of historical IV
surfaces dynamics in the generated scenarios.
We use the ARMA(P, Q) conditional mean model extended by a GARCH(p, q) component to model the
conditional variance of the VIX index (see Francq and Zakoian (2011))17 . We use the econometric toolbox in
Matlab to estimate the model parameters. Some historical statistics of the VIX index are compared with the
scenario statistics in Table 5.1.
Table 5.1: Summary of historical and scenario VIX index statistics
Average (%)
St.dev. (%)
Skewness
Kurtosis
Minimum (%)
Median (%)
Maximum (%)
1/31/1990 - 31/12/1997
16.84
4.96
1.24
1.53
10.63
15.68
35.09
1/31/1990 - 31/12/2003
20.19
6.45
0.80
0.84
10.63
19.63
44.28
1/31/1990 - 31/12/2012
20.43
7.77
1.59
4.04
10.42
19.38
59.89
Scenarios
18.39
6.48
1.42
8.86
2.40
17.53
92.96
We conclude from Table 5.1 that the (unconditional) scenario statistics are generally in line with the historical statistics. Based on the generated scenarios of the VIX index we are able to compute IV surfaces
based on the VIX-Heston model. To investigate these surfaces we compare the first principal component to
the historical one. The results are shown in figures 5.2a and 5.2b.
Figure 5.2: First principal component of historical and scenario IVs.
(b) Scenario IV data
120
120
110
110
Strike (%)
Strike (%)
(a) Historical IV data
100
90
80
100
90
0.5
1
1.5
80
2
0.5
1
Maturity (year)
0.18
0.19
0.2
0.21
0.22
1.5
2
Maturity (year)
0.23
0.24
0.14
0.16
0.18
0.2
0.22
0.24
0.26
It turns out that the first principal component from the generated IV surface scenarios contributes 98.8%
of the total variance. We know from Section 4.3 that the first principal component from the historical data
contributes 97.6% of the total variance. In figures 5.2a and 5.2b we observe some differences in the shape
of the first principal components. This is due to the the generated VIX scenarios, i.e. the generated statistics
(see Table 5.1) are do not fully coincide with the historical ones, which leads to a (slightly) different behavior
of the first principal component.
17
Other (more advanced) models can be used as well.
22
6 Conclusions
In this article we have proposed a method to generate equity index implied volatility surfaces using the risk
neutral Heston model. This method can be combined with any real world scenario generator that models
the VIX index. The main advantage of the modeling approach is that a calibrated risk neutral scenario set is
automatically obtained at a certain time t in a real world scenario set. In this way the implied risk neutral probability measure is embedded in the P probability measure. Computations where the risk neutral Q measure is
embedded in the P measure are important for risk management in the financial industry.
An advantage of the approach is that the optimal parameters of the Heston model are either linked to
the well known VIX index or set to fixed values. This makes the calibration of the parameters stable over
time and facilitates out-of-sample valuations. Furthermore, by applying a principal component analysis to
the generated scenarios, historical dynamics are replicated rather good, which is an important feature in our
modeling approach.
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Appendix A
Historical statistics of the S&P-500 index IV data
In this section we provide an overview of the implied volatility data, which is used for the calibrations. In
Table 6.1 the historical data for a number of options is summarised for the historical timespan 23/06/2005 31/12/2012 and a daily frequency.
24
Table 6.1: Summary of historical S&P-500 index IV (%) statistics
Average
St.dev.
Minimum
Maximum
Skew
Kurtosis
90D
180D
360D
540D
720D
29.51
28.17
26.82
26.21
25.93
9.11
8.03
7.18
6.55
6.24
11.88
10.32
13.98
14.61
15.11
73.63
62.49
53.91
50.00
47.47
1.20
0.87
0.54
0.46
0.40
2.43
1.32
0.43
0.19
0.06
90D
180D
360D
540D
720D
25.31
24.60
24.10
23.85
23.87
8.57
7.58
6.76
6.30
6.01
11.88
10.31
13.98
14.61
15.11
68.39
58.40
51.15
47.65
45.45
1.44
1.14
0.85
0.70
0.60
2.88
1.78
0.91
0.52
0.33
90D
180D
360D
540D
720D
20.24
20.88
21.40
21.63
21.93
8.40
7.39
6.58
6.14
5.87
10.13
10.32
12.54
13.01
13.43
63.45
54.62
48.56
45.48
43.58
1.62
1.32
1.03
0.87
0.75
3.28
2.14
1.27
0.85
0.62
90D
180D
360D
540D
720D
16.48
17.77
19.04
19.68
20.22
7.84
7.15
6.38
5.96
5.70
7.38
8.69
10.59
11.46
12.08
58.91
51.16
46.15
43.50
41.86
1.91
1.47
1.19
1.03
0.90
4.26
2.55
1.67
1.21
0.94
90D
180D
360D
540D
720D
15.54
16.10
17.24
18.11
18.80
6.76
6.54
6.04
5.71
5.48
7.37
7.91
9.29
10.51
10.83
54.88
48.09
43.93
41.67
40.28
2.16
1.68
1.42
1.21
1.06
5.59
3.30
2.31
1.68
1.33
80%
90%
100%
110%
120%
25