Simulating the dynamics of the
risk neutral distribution
d-fine
Kellogg College
University of Oxford
A thesis submitted in partial fulfillment of the requirements for the
MSc in
Mathematical Finance
Trinity Term 2003
Simulating the dynamics of the risk neutral distribution
MSc in Mathematical Finance
Trinity Term 2003
Abstract
Knowledge about the dynamics of the risk neutral distribution is necessary for the valuation of complex options on financial assets. We present
and explore a formalism (the twin formalism) that allows to simulate how
the risk neutral probability density function evolves with time. The main
idea of the twin formalism is to describe the risk neutral distribution as
a mixture of possible future distributions, and to simulate its time development in a simplified (,,twin”) model space. This approach is model
independent in the sense that we may apply it to a large variety of distributions (or processes): The formalism will automatically take care of
the necessary no-arbitrage conditions between the future and today’s risk
neutral distribution. As an illustration, we apply the twin formalism to
several models and to the valuation of different options, in particular in
connection with the problem of so-called smiles in implied volatility. The
modeling framework is formulated in terms of probability distributions,
rather than processes. This means that one does not have to specify
the dynamics (i.e. stochastic differential equations) of the underlying processes. It suffices to determine the distributions only at those points of
time that are relevant for the option one wants to price.
Contents
1 Introduction
1.1 Notation and general assumptions . . . . . . . . . . . . . . . . . . . .
1.2
1
2
The risk neutral distribution . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
The concept of risk neutral valuation . . . . . . . . . . . . . .
3
1.2.2
1.2.3
The use of the risk neutral pdf . . . . . . . . . . . . . . . . . .
Some technical properties of the risk neutral pdf . . . . . . . .
6
7
1.2.4
Forward start call . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Skews and smiles
9
2.1
2.2
Implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stylized empirical facts . . . . . . . . . . . . . . . . . . . . . . . . . .
9
10
2.3
Modeling the smile . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3.1
Stochastic volatility . . . . . . . . . . . . . . . . . . . . . . . .
13
2.3.2
2.3.3
Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reduced dependence on the price level . . . . . . . . . . . . .
14
15
3 The twin formalism
18
3.1
Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2
3.3
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connection between market and model . . . . . . . . . . . . . . . . .
20
21
3.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.5
Conditions for the model distributions . . . . . . . . . . . . . . . . .
23
3.5.1
3.5.2
23
23
Conditions for the time dependence . . . . . . . . . . . . . . .
Conditions for the X-dependence of g . . . . . . . . . . . . . .
4 Properties
4.1
25
Properties of XT
4.1.1
4.1.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Existence of XT . . . . . . . . . . . . . . . . . . . . . . . . . .
Case t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
25
i
4.2
4.3
4.1.3
Case t = T . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.1.4
Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
Properties of Xt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Existence of Xt . . . . . . . . . . . . . . . . . . . . . . . . . .
27
27
4.2.2
Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Properties of ft,T . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.3.1
4.3.2
Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . .
No arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
4.3.3
Explicit form of ft,T
. . . . . . . . . . . . . . . . . . . . . . .
30
4.3.4
Possible and impossible events . . . . . . . . . . . . . . . . . .
30
4.3.5
4.3.6
Tail behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time homogeneity . . . . . . . . . . . . . . . . . . . . . . . .
31
31
5 Numerical evaluation schemes
33
5.1
Evaluation of the mapping S ↔ X
. . . . . . . . . . . . . . . . . . .
33
5.2
5.3
Evaluation of the future distribution . . . . . . . . . . . . . . . . . .
Valuation of a call at time t . . . . . . . . . . . . . . . . . . . . . . .
34
34
5.4
Valuation of a forward start call . . . . . . . . . . . . . . . . . . . . .
35
6 Applications
6.1
6.2
6.3
36
Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
37
6.1.2
Call at time t . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
6.1.3
Forward start call . . . . . . . . . . . . . . . . . . . . . . . . .
38
Black-Scholes model with a volatility jump . . . . . . . . . . . . . . .
6.2.1 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
39
6.2.2
Call at time t . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
6.2.3
Forward start call . . . . . . . . . . . . . . . . . . . . . . . . .
43
Displaced diffusion model . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
44
6.3.2
Call at time t . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
6.3.3
Forward start call . . . . . . . . . . . . . . . . . . . . . . . . .
48
6.3.4
6.3.5
Negative values of ST . . . . . . . . . . . . . . . . . . . . . . .
Comparison to market data . . . . . . . . . . . . . . . . . . .
48
50
ii
7 Summary and outlook
52
7.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
7.2
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
A The displaced diffusion distribution
55
B Analytical solutions for M
59
C Proof of the Breeden Litzenberger formula
62
Bibliography
63
iii
Chapter 1
Introduction
The risk neutral distribution is an important mathematical tool for the valuation of
options on financial assets. Options with a complex structure can be priced only if
the future form of the risk neutral distribution is known. Therefore, it is necessary
to simulate the dynamics (i.e. the time development) of the risk neutral distribution.
The standard (Black-Scholes) approach to this problem is to assume that the risk
neutral distribution is log-normal, and remains so during its time development. One
advantage of this approach is that the log-normal distribution automatically ensures
important no-arbitrage conditions. The disadvantage, however, is that the assumption of log-normality contradicts empirical evidence. If one tries to move beyond
log-normality in order to make the distribution more realistic one needs a formalism
that nevertheless ensures the no-arbitrage conditions.
The purpose of this paper is to present and explore such a formalism. The basic
concepts of this formalism are due to [Skiadopoulos and Hodges, 2002]. Its main
idea is to describe the risk neutral probability density function (pdf) as a mixture
of possible future distributions, and to simulate its time development in a simplified
(,,twin”) model space. For this reason we call it twin formalism. This approach is
model independent in the sense that we may ,,plug in” a large variety of distributions
(or processes): The twin formalism will automatically take care of the necessary noarbitrage conditions between the future and today’s risk neutral distribution. As an
illustration, we apply the twin formalism to several models, in particular in connection
with the problem of so-called smiles in implied volatility. The modeling framework is
formulated in terms of probability distributions, rather than processes. This means
that one does not have to specify the dynamics (i.e. stochastic differential equations)
of the underlying processes. It suffices to determine the distributions only at those
points of time that are relevant for the option one wants to price.
1
The thesis is organized as follows. Chapter 1 contains information on the concept
and use of the risk neutral probability distribution. Chapter 2 describes the problem
of so-called smiles in implied volatility. We use this problem later on as a case study
to illustrate how the twin formalism works in practice. The twin formalism itself
is presented in Chapter 3. In Chapter 4 we derive several useful analytical results,
whereas we suggest numerical evaluation schemes in Chapter 5. Several applications
of the twin formalism are presented in Chapter 6. The thesis is concluded by a
summary and an outlook (Chapter 7), and by several Appendices.
1.1
Notation and general assumptions
We use the following notation: The sign ≡ symbolizes ,,equal by definition” (with the
definiendum usually on the left hand side). marks the conclusion of a proof. We
use the acronyms ATM = at the money, OTM = out of the money, cdf = cumulative
probability density function, pdf = probability density function, iff = if and only if,
w.r.t. = with respect to, r.h.s. = right hand side.
Furthermore, we use the following notation for the normal and lognormal distribution
"
#
2
1
−
(x
−
x
)
0
n x x0 , σ 2 ≡ √
exp
,
2σ 2
2πσ
Z x
2
N x x0 , σ
dξ n ξ x0 , σ 2 ,
≡
and
"
#
2
2
1
1
−
[log
(x/x
)
+
σ
/2]
0
logN x x0 , σ 2 ≡ √
exp
,
2σ 2
2πσ x
Z x
2
LogN x x0 , σ
≡
dξ logN ξ x0 , σ 2 .
Throughout this paper we deal with a single underlying S, and we use forward values
for this underlying and options written on it. This allows us to interpret the underlying in two different ways. Firstly, St can be interpreted as the forward price from
time t until time T of a non-dividend paying stock
−1
St ≡ Dt,T
st ,
where st is the stock’s spot value and D is a discount factor (a zero bond). This
underlying is tradable, and consequently (given completeness of the market and no
arbitrage) it is a martingale under the risk neutral measure, see Sec. 1.2.
2
Secondly, St can be interpreted as the value of a forward interest rate at time t,
for a time period from time T until T 0
−1
Dt,T Dt,T
0 − 1
St ≡
.
0
T −T
Although a rate itself is not tradable, St times the zero bond Dt,T 0 is. If we use
this bond as numeraire, St Dt,T 0 /Dt,T 0 = St is a martingale under the corresponding
measure (the forward neutral measure). Consequently, the expectation values we
need to evaluate have the same form as in the case of stocks.
The formalism we are going to develop does in no way restrict us to this choice of
forward values. The advantage of our approach, however, is that we can treat equities
and interest rates on the same footing, and that we can dispense with discount factors
altogether. In this context one should remember that we are dealing with forward
probability distributions (i.e. distributions of forward prices), with forward volatilities
(i.e. volatilities of forward prices), and with forward option prices. For instance, we
will use call prices Ct,T and forward start call prices V0,t,T defined by
−1
Ct,T ≡ Dt,T
ct,T ,
−1
V0,t,T ≡ D0,T
v0,t,T .
ct,T is the spot price of a call (or caplet) at time t with expiry T , and v0,t,T is the spot
price of a forward start call (or forward start caplet) at time 0, with fixing time t and
expiry T , see Sec. 1.2.4. In addition, we should stress that the conditions in these
options apply for forward prices as well - for instance, the strike in the forward start
call is determined by the forward price (instead of the spot price) of the underlying.
This may not necessarily coincide with market practice.
Finally, all prices shall be understood as being divided by appropriate basic units
(dollars, say, or basis points in the case of interest rates). As a consequence, we do
not need to take account of units.
1.2
1.2.1
The risk neutral distribution
The concept of risk neutral valuation
The concept of risk neutral valuation can be nicely illustrated by studying the behaviour of bookmakers for a horse race [Baxter and Rennie, 1996]. A bookmaker has
two types of information available to set his odds for a race. On the one hand, there
are ,,objective” or ,,real world” factors like statistical data on the performance of
3
horses and jockeys. This information may be translated into (real world) probability
distributions of a horse’s chance to win the race. On the other hand, there are the
bets actually made so far. If the bookmaker sets his odds according to the real world
factors, he will make a profit in the long run (but risk substantial short-term losses).
If, however, he chooses his odds depending on the ratios of the bets actually made so
far, he can adjust them in a way that will guarantee him a profit already in the next
race, no matter what the actual outcome of this race is going to be. This setup of
odds is sometimes called a Dutch Book [Kyburg and Smokler, 1980]. It is important
to notice that, to make his bets riskless, the bookmaker has to choose other than
real world probabilities (i.e. odds). Since in choosing his probabilities the bookmaker
does not care about the risk that a particular horse might not win the race, these
probabilities could be called ,,risk neutral”. Let us now find out to which degree these
considerations apply to financial markets.
The risk neutral distribution is a mathematical tool for the valuation of options
on financial assets. As we have just seen in the example of the bookmaker, it is
important not to confuse the risk neutral distribution with the actual (,,real world”)
distribution of the underlying. This real world distribution tells us how the underlying is going to behave in the long run. Consequently, knowledge of the real
world distribution allows to calculate an option price that lets us break even in the
long run. This means that if we buy the option for this price, and keep doing so
many times, we will eventually break even. There is, however, another mechanism
at work, a mechanism that forces an option price at every instance. This mechanism is arbitrage, see for instance [Baxter and Rennie, 1996], [Deutsch, (2001)] or
[Wilmott, Howison, and Dewynne, (1995)]. Arbitrage is the chance to make a profit
within a given time span, without own investment and without risk to make a loss.
At least in principle (and in contrast to horse races) financial markets allow its participants to take both sides of a trade. For this reason it is a common assumption
in finance theory that arbitrage is absent from financial markets. If we furthermore
assume that a market is complete, i.e. that all options in this market are redundant
in the sense that they can be replicated from other assets in the market, we obtain an
arbitrage-free price for every option. Assume that we buy such an option at a different price (for instance at the price that would make us break even in the long run).
Then, using replication, the seller of the option will pass on her risk to somebody else
in the market and keep a risk-free profit for every trade (at the long term expense of
the person who took the risk from the option seller). Whereas the ,,long run” price is
4
the expectation under the real world probability distribution, the arbitrage-free price
is the expectation under the risk neutral probability distribution.
We can sum up these considerations in a somewhat more technical way. Define
a complete market as one in which every derivative can be replicated, i.e. imitated
with a portfolio of other market assets and using a pre-visible and self-financing strategy. Pre-visible means that at any point in time the strategy only uses information
collected up to this time. Self-financing, on the other hand, means that changes
in the portfolio value are due only to changes in the asset prices (but not due to
changes in the composition of the portfolio). Call a market arbitrage-free if it does
not allow to make a risk-free profit without own investment. Add to this the concept
of discounting which can be generalized to the division by any tradable asset (with
non-vanishing value), the so-called numeraire. Finally, define a martingale to be any
stochastic process for which the expected value coincides with the conditional value
(i.e., essentially, which is drift-free). Then [Harrison and Pliska, 1981]
Theorem 1 In a complete and arbitrage-free market, the price V of any derivative
is the expectation of the corresponding (discounted) payoff profile π (S) under the
measure Q that makes the option’s underlying S a martingale
Z
V = dS π (S) fQ (S) .
This risk neutral measure Q is unique.
An important point to notice here is the separation of probability measure and
process. While the replicating strategy will depend on the process that the underlying
follows, the real world measure P of this underlying is irrelevant for the pricing. How
can we find the risk neutral measure Q? If we are more specific about the underlying’s
process, Girsanov’s Theorem will give us the answer
Theorem 2 If S is a Brownian motion with drift µ under the real world measure P ,
then S is a Brownian motion under Q as well, but with a different drift γ.
Thus, in this case the step from the real ,,world” into the risk neutral ,,world”
(i.e. the change of measure from P to Q) is a simple change of drift. This gives
another motivation to choose the name ,,risk neutral” for the measure Q: under Q, all
tradeable assets have the same growth rate, irrespective of the risk they are carrying.
Finally, we would like to notice that the name ,,risk neutral” is sometimes reserved for
the measure that obtains when the numeraire is a money account, whereas the name
5
,,forward neutral” is used if the numeraire is a zero bond [Deutsch, (2001)]. In this
thesis we are going to use the name ,,risk neutral”, irrespective of which numeraire is
actually chosen.
1.2.2
The use of the risk neutral pdf
As we have seen in Sec. 1.2.1, knowledge of today’s risk neutral pdf f0,T (ST ) allows
us to calculate the price of every derivative with a payoff profile that depends only
on the underlying’s value ST at expiry T . This is a lot of information. Interestingly,
in order to obtain this information, we do not need to know how the risk neutral pdf
will look in the future. Why, then, should we care to simulate the time development
of the risk neutral pdf? The answer is: We need to know the future risk neutral pdf
if we want to price options that are conditional on values of the underlying before
expiry. One example is a forward start call, i.e. a call with a strike that will be
set in the future, depending on the future value of the underlying (Sec. 1.2.4). In
addition to the pricing of complex derivatives, the risk neutral pdf can also be used
as a diagnostic tool for policy makers like central banks. For instance, in the field of
interest rates, it allows the policy maker or monetary authorities to [Bahra, 1997]:
1. Assess monetary conditions (i.e. market opinion on future events).
2. Assess monetary credibility (i.e. the difference between market expectation and
government targets).
3. Assess the effectiveness of monetary actions (ex ante: assess the market’s likely
reaction; ex post: assess if a government action − like a rate cut − was already
,,priced in”).
4. Identify market anomalies (e.g. if a market crash was due to market failure or
macroeconomic effects).
[Bliss and Panigirtzoglou, 2002] contains further references to papers that use the
risk neutral distribution as indicator of market sentiment. Finally, we stress that in
all of our considerations we should remember that our results hold for the risk neutral
pdf, and will therefore deviate from the real pdf. In some markets, however, there
may be only small differences between these two distributions [Rubinstein, 1994].
6
1.2.3
Some technical properties of the risk neutral pdf
Suppose we know today’s (t = 0) risk neutral probability density function (pdf)
f0,T (ST |S0 , θ0 ) for the event that the price of asset S at time T will be ST . This
pdf is conditional on today’s value S0 of the asset, and (maybe) on today’s value of
other stochastic parameters, symbolized by θ0 . Our aim is to model the (as of today
unknown) risk-neutral pdf ft,T (ST |St , θt ) at a future time t (0 < t < T ). This pdf
may depend on the future value θt of parameter θ at time t, and we want it to fulfill
the following (no-arbitrage) conditions:
1. ft,T is a mixture of possible future distributions. Another way of putting this
is to say that integration over all possible paths has to lead us back to f0,T
Z Z
dSt dθt ft,T (ST |St , θt ) p0,t (St , θt |S0 , θ0 ) = f0,T (ST |S0 , θ0 ) ,
(1.1)
where p0,t is the probability that St and θt obtain (we will have to model this
probability as well). Note that we do not assume that St and θt are independent.
2. ST is a martingale with respect to ft,T
Z
dST ST ft,T (ST |St , θt ) = St .
(1.2)
The assumption that today’s distribution is a mixture of possible future distributions has some general implications. Firstly, mixing property Eq.(1.1) extends to
expectation values. For any function φ (ST ) define the expectation values
Z
E0,T [φ |S0 , θ0 ] ≡
dST φ (ST ) f0,T (ST |S0 , θ0 ) ,
Z
Et,T [φ |St , θt ] ≡
dST φ (ST ) ft,T (ST |St , θt ) .
(1.3)
(1.4)
Then,
Z
E0,T [φ |S0 , θ0 ] =
dST φ (ST ) f0,T (ST |S0 , θ0 )
Z
=
dST φ (ST )
Z Z
dSt dθt ft,T (ST |St , θt ) p0,t (St , θt |S0 , θ0 ) .
This means that the present expectation value is a mixture of the possible future
expectation values, i.e.
Z Z
E0,T [φ |S0 , θ0 ] =
dSt dθt Et,T [φ |St , θt ] p0,t (St , θt |S0 , θ0 ) .
(1.5)
7
Secondly, since we regard the present distribution as a mixture of possible future
distributions, the mean variance of these future distributions will be smaller than the
variance of the present distribution. Define the variances
var0,T (S0 , θ0 ) ≡ E0,T ST2 |S0 , θ0 − (E0,T [ST |S0 , θ0 ])2 ,
vart,T (St , θt ) ≡ Et,T ST2 |St , θt − (Et,T [ST |St , θt ])2 .
(1.6)
(1.7)
Then,
Z Z
dSt dθt Et,T ST2 |St , θt p0,t (St , θt |S0 , θ0 )
Z Z
2
−
dSt dθt Et,T [ST |St , θt ] p0,t (St , θt |S0 , θ0 )
Z Z
≥
dSt dθt Et,T ST2 |St , θt p0,t (St , θt |S0 , θ0 )
Z Z
−
dSt dθt (Et,T [ST |St , θt ])2 p0,t (St , θt |S0 , θ0 )
Z Z
=
dSt dθt Et,T ST2 |St , θt − (Et,T [ST |St , θt ])2
var0,T (S0 , θ0 ) =
p0,t (St , θt |S0 , θ0 ) ,
or
Z Z
var0,T (S0 , θ0 ) ≥
dSt dθt vart,T (St , θt ) p0,t (St , θt |S0 , θ0 ) .
(1.8)
Note that this result does not exclude that specific future distributions have a larger
variance than today’s distribution − they just have a smaller weight p0,t .
1.2.4
Forward start call
Knowledge of the future risk-neutral pdf is necessary for the valuation of products
that are conditional on future values of the underlying asset. One example is a
European forward start call. Consider a call option on S with a strike that sets at
time t to K = φ (St ), with a deterministic function φ, and pays {ST − φ (St )}+ ≡
max [ST − φ (St ) , 0] at time T . At time 0, its value is
Z Z
V0,t,T (S0 , θ0 ) =
dSt dθt Ct,T [St , φ (St ) , θt ] p0,t (St , θt |S0 , θ0 ) ,
(1.9)
where Ct,T [St , φ (St ) , θt ] is the option’s value at time t, given that St and θt obtain
Z
Ct,T [St , φ (St ) , θt ] = dST {ST − φ (St )}+ ft,T (ST |St , θt ) .
(1.10)
Eq.(1.10) is the value at time t of a plain vanilla call with strike φ (St ).
8
Chapter 2
Skews and smiles
The main goal of this thesis is to discuss a modeling framework for the simulation
of the risk neutral distribution. To illustrate the practical importance of such a
framework we are going to implement models that have been designed to tackle the
problem of so-called smiles in implied volatility. This chapter describes the smile
problem in some detail.
2.1
Implied volatility
Nowadays, most plain vanilla options are quoted in terms of implied volatility. Implied volatility σimp is the value of volatility σ one has to put into a log-normal
[Black, 1976a] model to reproduce the market prices. Thus, the market uses Black’s
formula only as a metric to express prices as volatilities. This convention has several
advantages: Since the influence of strike and expiry on the option price is removed
from σimp , implied volatility allows to compare the relative cheapness of options with
different strikes and expiries. Furthermore, volatility usually is a number in the range
of 10-40%, whereas prices can lie in a very wide range. The mapping of option prices
onto implied volatilities is possible for two reasons. First, because there is sufficient
consensus among market participants about the other parameters of Black’s formula
(like the interest rate). Second, because Black(-Scholes) call (or caplet) prices increase
strictly monotonically with σ. Consequently, there is a one–to–one relationship between prices and volatilities: Given call (or caplet) prices, σimp can be determined
uniquely, and vice versa. Notice that implied volatility is not only a property of
the underlying, but also carries information about the derivative (as it is implied
from derivative prices). Therefore, we cannot expect implied volatility to be equal
to the actual volatility of the underlying (as measured directly from market prices).
For instance, supply and demand for options will influence the implied volatility
9
[Wilmott, 1998]. Nevertheless, due to their forward-looking nature, derivative prices
(and, hence, implied volatilities) encapsulate market perceptions about underlying
asset prices in the future. For instance, in the case of interest rates, the bandwidth
of confidence intervals can be seen as an indicator of credibility of monetary policy
[Jondeau and Rockinger, 2000], see also Sec. 1.2.2.
Suppose a given underlying is log-normally distributed. Then, at a given time
and for a given price of the underlying, σimp for an option on this underlying should
be independent of the expiry T and the strike K of the option. Unfortunately, this
is usually not the case: the implied volatility is found to depend on both quantities:
σimp = σimp (T, K). Thus, implied volatility is the ,,wrong number to put in the
wrong (Black) formula to get the right price” [Rebonato, 2002]. A more benign view
is an analogy to solid state physics, where one puts an (imaginary) effective electron
mass into a model of free electrons to describe the (actual) electronic properties of a
crystal [Ashcroft and Mermin, 2000].
2.2
Stylized empirical facts
It certainly still makes sense to quote derivatives in terms of implied volatilities, even
if these depend on expiry and strike. However, this dependence indicates that a lognormal model does not correctly describe the behaviour of the underlying. It is generally accepted that a log-normal distribution lacks kurtosis (,,fat tails”) and skewness
compared to the distributions observed in the market [Das and Sundaram, 1999]. The
good news is that the functional form of the empirical σimp (T, K) can tell us something about what has to be done to design a more realistic model for the underlying
process (and, consequently, for the prices of derivatives issued on this underlying).
The following basic forms of K dependence can be observed:
• Skew: out–of–the–money calls have a lower value, in–the–money calls a higher
value of σimp
• Smile: both out–of–the–money and in–the–money calls have a higher value of
σimp
• Frowns and other more complex forms
Historically, smiles in equity-derivatives have been observed as early as 1987. In
the FX world, volatility smiles have been known for a longer period of time as well.
In the field of interest rates, however, only about 1996 did the implied volatility
10
for caplets of a given maturity begin to assume a marked monotonically decreasing
behaviour as a function of strikes [Rebonato, 2002]. Examples are the Japanese Libor
market, and the US and German markets [Andersen and Andreasen, 2000]. During
1998, the implied volatility took the form of a more complex smile [Rebonato, 2002].
50%
45%
40%
0.45-0.5
0.4-0.45
0.35-0.4
0.3-0.35
0.25-0.3
35%
30%
1m
4m
25%
1y
5y
4523 5088
5371 5654
7y
5936 6219
6785
Figure 2.1: Empirical evidence: Smile surface for calls on the FTSE as observed
on 18 August 1998. The x-axis displays the strike K in index points (spot at this
day was 5648), the y-axis displays time to maturity T (in years y or months m),
and the z-axis displays the corresponding implied volatilities σimp . Data taken from
[Rebonato, 1999].
Figure 2.1 shows equity market data in the form of an implied volatility surface, i.e. σimp (T, K) as a function of expiry T and strike K. Data are taken from
[Rebonato, 1999]. Notice the typical behaviour of a skew that increases as time to
maturity decreases. This should be contrasted with evidence from other markets. In
the interest rate market, for instance, the skew is found not to increase, but rather to
display a hump-shaped dependence on T [Rebonato, 1999]. Other forms can be found
in foreign exchange markets as well. There are also differences in the dependence of
smiles on spot values. In the case of equities, smiles in implied volatility are found
11
Equities
K-dependence
T -dependence
mostly skewed
skew increases
with decreasing T
IR
slight smile
FX
pronounced smile
form of smile does
not change, but
absolute level is
hump-shaped
function of T
mixed evidence
S0 -dependence
,,floating”: smile
migrates with the
actual equity price
,,sticky”: smile is
independent of the
actual interest rate
mixed evidence
Table 2.1: Stylized empirical facts on smile phenomena in different markets: Dependence on strike K, expiry T and spot S0 .
to be a function of moneyness (K − S0 ) /S0 , i.e. they migrate with changing spot
prices S0 . This is called a ,,floating” smile. In the field of interest rates, on the other
hand, smiles tend to be a function of K alone, i.e. they are independent of the actual
value of the interest rate. This is called a ,,sticky” smile. These empirical observations are summarized as ,,stylized empirical facts” in Table 2.1, cf. [Rebonato, 1999],
[Jondeau and Rockinger, 2000].
Which economic reasons lead to these phenomena? One possible answer might be
that people like to use slightly out–of–the–money equity options for hedging (which
leads to higher prices for these options), while they tend to realize their profit by
selling options that are just in the money (which leads to lower prices for in–the–
money options). Notice that OTM puts are a ,,crash insurance”, and that fear of a
crash will lead to OTM put prices that are higher than OTM call prices. (Due to
their payoff profile European calls measure the upper tail, European puts the lower
tail of the risk neutral pdf.) Furthermore, if one wants to keep the same level of
crash insurance, one has to buy puts that are similarly OTM (i.e. puts with the same
moneyness, depending on the actual spot value), which would explain the floating
nature of the equity smile. However, in this paper I will not further speculate on
the reasons of skews in empirical implied volatility, but rather try to reproduce these
empirical findings within a model framework.
2.3
Modeling the smile
There are several ways to extend the lognormal model in order to allow for skews and
smiles in implied volatility (one might contemplate to use several of them simultane-
12
ously [Rebonato, 2002]). These approaches can be grouped according to the property
of the lognormal model that they relax. These properties are:
1. Volatility behaves deterministically (see Sec. 2.3.1).
2. Price changes are continuous (see Sec. 2.3.2).
3. Price changes are directly proportional to the current level (see Sec. 2.3.3).
All these extensions of the lognormal model are usually carried out at the level
of stochastic processes (for the underlying price or for its volatility). Alternatively,
we might also change the form and time behaviour of the distributions. We will
follow this line of thought in the present paper. However, one should not forget that
knowledge of the underlying stochastic process can be important for other reasons.
For instance, imagine that a given distribution could be produced by either some
Brownian motion, or by a jump-diffusion process with random amplitudes. While
the underlying has the same distribution in both cases, no perfect hedging is possible
in the latter case [Rebonato, 1999].
2.3.1
Stochastic volatility
If property 1 (deterministic volatility) were true, life in the financial markets would be
a lot easier. In reality, however, volatility (as measured directly from market prices)
is an unpredictable function of time. Thus, it is natural to model it as a random
variable. This leads to so-called ,,fully stochastic volatility models”, in which there
are two sources of randomness (usually Brownian motions): one for the underlying
price, and one for its volatility. This should be contrasted with ,,restricted stochastic volatility models” in which volatility is a deterministic function of the (stochastic) underlying price (cf. Sec. 2.3.3). Examples are given by [Amin and Ng, 1993],
[Melino and Turnbull, 1990] and [Hull and White, 1987], to name but a few.
Stochastic volatility in the future leads to smiles in today’s implied volatilities
[Hull and White, 1987]. This can be made plausible as follows. Suppose volatility is
stochastic with mean σ and distribution k (σ |σ ). Let us expand Black-Scholes call
prices C as a Taylor series around σ
2 ∂C 2 1 ∂ C
C (σ) = C (σ) + (σ − σ)
+ (σ − σ)
+ ...
∂σ σ
2 ∂σ 2 σ
13
Consequently, if the underlying price and its volatility are uncorrelated, the expectation value of the call price is
Z
E [C (σ)] ≡
dσ C (σ) k (σ |σ )
Z
2 ∂C 2 1 ∂ C
=
dσ C (σ) + (σ − σ)
+ (σ − σ)
+ . . . k (σ |σ )
∂σ σ
2 ∂σ 2 σ
Z
1 ∂ 2 C dσ (σ − σ)2 k (σ |σ ) + . . .
= C (σ) +
2 ∂σ 2 σ
1 ∂ 2 C var (σ) + . . .
(2.1)
= C [E (σ)] +
2 ∂σ 2 σ
where var(σ) is the variance of the volatility. What do we know about the behaviour
of Black-Scholes call prices as a function of volatility? At the money (ATM), C is
a concave function of volatility (∂ 2 C/∂σ 2 < 0). On the other hand, if the call is
significantly in or out of the money (and σ is not too large), C is a convex function
of volatility (∂ 2 C/∂σ 2 > 0). Thus, stochastic volatility will reduce call prices in a
region around ATM, and increase call prices far in our out of the money. In terms of
implied volatilities, this translates into a volatility smile.
However, fully stochastic volatility models have a big drawback: in sharp contrast
to what is observed in the markets (see Fig. 2.1), they tend to produce large smiles
for long maturities and shallow smiles for short maturities, see [Rebonato, 1999] and
references cited therein. Again, there is a simple intuitive explanation for this observation. With decreasing time to maturity the variance of the volatility will decrease due
√
to the properties of Brownian motion (standard deviation decreases as T ). Eq.(2.1)
implies that this will flatten out the smile in implied volatility. Consequently, we need
some deviation from the lognormal model that will not vanish with time to maturity.
One possibility is jump-diffusion.
2.3.2
Jumps
Property 2 (continuous price changes) seems in particular violated in the stock markets, where at times share prices may change (usually drop) very swiftly. One way
to relax the continuity of price changes is to allow for random jumps in addition to a
lognormal diffusion with a constant volatility
dSt = λ (1 − α) St dt + σ St dWt + (α − 1) St dNt ,
where Wt is a Brownian motion and Nt is a Poisson process that has value 1 with
probability λdt (and value 0 otherwise). α is the jump amplitude which has been
14
assumed to be constant (it could itself be random). The probability density for the
event that the asset price at time T will be ST , given that it was S0 at time 0 and
given that n jumps have occurred in the meantime, is
f0 (ST | S0 , n) = logN ST S0 αn exp (λ (1 − α) T ) , σ 2 T
1
1
= √
S
2πT
" σ T
2
− [log (ST / (S0 αn )) + λ (1 − α) T + σ 2 T /2]
exp
2σ 2 T
#
.
Examples for Jump-diffusion models are, amongst others, [Amin, 1993], [Bates, 1991]
and [Merton, 1976]. Jump-diffusion models lead to volatility smiles that decrease
with increasing time to maturity. Unfortunately, however, the smiles flatten out far
too quickly [Das and Sundaram, 1999].
2.3.3
Reduced dependence on the price level
Property 3 (price changes directly proportional to spot) implies that, if an underlying’s price St is very small, in absolute terms it will almost not change anymore
dSt = σ St dWt ,
where Wt is a Brownian motion. This is clearly not what has been observed, for
instance, in Japanese interest rates during the 1990’s. Thus, it is natural to try and
reduce the dependence of the process on the current level (i.e. to increase the left
tail of the distribution). What if we go all the way and make the process completely
independent of the current level? This holds true for a normal process
dSt = σ dWt .
One obvious disadvantage of this process is that it allows for negative values of
the underlying price. Therefore, we would like to somehow interpolate between, or
,,blend” a lognormal process and a normal process. There are several ways to do this,
for instance:
1. CEV (constant elasticity of variance) model [Cox and Ross, 1976]: One can
control the lognormality of a process using an exponential factor
dSt = σ (St )α dWt ,
with 0 ≤ α ≤ 1.
15
2. Double CEV model: This model goes a step further by using a linear combination of two contributions
h
α
β
dSt = σ (St ) + b (St )
i
dWt ,
with 0 ≤ α ≤ 1 < β, and with a positive b.
3. Displaced diffusion [Rubinstein, 1983]: Another possibility is to make a linear
interpolation between a lognormal and a normal process
dSt = σ [(1 − γ) St + γ] dWt ,
where γ controls the skew of the volatility, and σ its level. Here, S is driven by
a mixture of a lognormal and a normal responsiveness to the same influence.
4. Mixture of lognormals [Brigo and Mercurio, 2001] and references cited therein:
A discrete mixture of (log)normal densities is known to provide for good data
fits in stock markets, since it has fatter tails than a single (log)normal density
[Kon, 1984]. It leads to a process of the form
dSt = v (λ1 , . . . λN , σ1 , . . . σN , t, St ) dWt ,
where v is some deterministic function, σi is the volatility of the ith lognormal,
and λi is its weight.
The CEV models exclude negative values for the price, but absorption at S = 0
is still possible [Andersen and Andreasen, 2000]. Furthermore, they have the disadvantage that, for most values of α (and β), vanilla option prices have to be calculated
numerically [Andersen and Andreasen, 2000]. This is a serious drawback, as efficient
calibration of the model to market prices is important for its usefulness in practical
applications. For displaced diffusion, on the other hand, vanilla options can still be
priced analytically for the following reason. Due to the linearity of the interpolation,
the process Yt ≡ (1 − γ) St + γ is still lognormal, with volatility (1 − γ) σ
dYt
d [(1 − γ) St + γ]
=
Yt
(1 − γ) St + γ
1−γ
=
dSt
(1 − γ) St + γ
1−γ
=
[(1 − γ) St + γ] σ dWt
(1 − γ) St + γ
= (1 − γ) σ dWt .
16
However, displaced diffusion allows for negative values of the price (with small probability, if γ is small). We will show a way to avoid this shortcoming in the twin
formalism, albeit sacrificing analytical vanilla option prices, see. Sec. 6.3.4. Furthermore, displaced diffusion leads only to skews (and cannot create smiles). Finally,
strong skews are impossible to recover [Brigo and Mercurio, 2001]. The mixture of
lognormals avoids all of these problems, albeit at the cost of a higher number of
parameters.
Finally, notice that all approaches in this subsection may also be interpreted
as follows: They amount to making the volatility a deterministic function of the
(stochastic) underlying value − i.e. they are so-called ,,restricted stochastic volatility models”. For instance, a purely normal model can be written as a lognormal
model with a volatility σ (St ) = 1/St . This functional dependence mirrors the wellknown empirical observation that ,,when stocks go down, volatilities seem to go up”
[Black, 1976b].
Summing up, a wide variety of models has been proposed to account for smiles
in implied volatility. Since none of them seems able to fully describe all empirically
observed facts, the most realistic approach should be a combination of several of them
[Das and Sundaram, 1999], [Rebonato, 2002].
17
Chapter 3
The twin formalism
This chapter presents a formalism, the twin formalism, that allows to simulate how
the risk neutral probability density function (pdf) evolves with time. The twin formalism is the centerpiece of this thesis, and it will be investigated, implemented
and illustrated in later chapters. Our approach builds on the foundation laid by
[Skiadopoulos and Hodges, 2002]. We substantially improve and deepen these author’s proposal, for instance by taking account of the conditional dependencies of
all distributions, and by ensuring martingale property Eq.(1.2). The basic idea of
the twin formalism is to describe the risk neutral pdf as a mixture of possible future
distributions, and to simulate its time development in a simplified (,,twin”) model
space. We start with a description of market quantities (Sec. 3.1) and model quantities (Sec. 3.2). Then, we connect market and model (Sec. 3.3), summarize, and list
conditions that the model distributions have to fulfill (Secs. 3.4 and 3.5). The twin
formalism provides a general framework in which a wide variety of models can be
evaluated. By design, the twin formalism ensures no-arbitrage conditions Eqs.(1.1)
and (1.2) between the future and today’s risk neutral pdf. The modeling framework
is formulated in terms of probability distributions, rather than processes. This means
that one does not have to specify the dynamics (i.e. stochastic differential equations)
of the underlying processes.
3.1
Market
First step of the twin formalism is to extract today’s risk neutral pdf f0,T from market
data:
1. Today (at time t = 0) observe call option market prices C0,T (S0 , K, θ0 ), all of
them written on the same underlying asset S (with today’s value S0 ) and for
18
maturity T , but for a range of strikes K. As above, θ0 symbolizes the possible
dependence on further stochastic quantities.
2. From these market data extract the empirical risk neutral pdf f0,T (ST |S0 , θ0 )
as valid at time 0, for the event that the underlying asset price at expiry T will
be ST , given that it was S0 at time 0.
There are several ways to infer the risk neutral pdf from market data, be they
expressed in terms of prices or implied volatilities. These approaches can be grouped
as follows [Bahra, 1997], [Jondeau and Rockinger, 2000]: non-parametric methods,
semi-parametric methods, and parametric methods. Non-parametric methods try to
infer the risk free pdf with a minimum of assumptions. One way is to use the formula
[Breeden and Litzenberger, 1978] (proof see Appendix C)
∂ 2 C0,T (S0 , K, θ0 ) = f0,T (ST |S0 , θ0 ) .
∂K 2
K=ST
(3.1)
Besides absence of arbitrage, this approach only assumes that there exist enough strike
prices K to approximate the density numerically. With too few data points, however,
numerical derivatives are known to be unstable. Furthermore, data quality is a problem as well. For instance, errors can be due to data recording errors, asynchronous
data, discreteness of data or liquidity issues (in-the-money options are generally much
less liquid than at-the-money options [Ait-Sahalia, Wang and Yared, 2001]). Resulting small concavities in market values for C0,T lead to negative probabilities in Eq.(3.1)
[Bahra, 1997]. In order to avoid these problems, other non-parametric approaches
assume a minimum of functional structure, like a mixture of log-normal distributions [Bahra, 1997] or a skewed student-t distribution [de Jong and Huisman, 2000].
Another possibility is to use a binomial tree and to adjust its parameters in order
to reproduce option prices [Rubinstein, 1994]. Semi-parametric approaches, on the
other hand, try to capture deviations from log-normality. Examples are Edgeworth
expansions around the log-normal density, or a multiplicative perturbation using Hermite polynomials [Jondeau and Rockinger, 2000]. Finally, parametric methods assume that the risk neutral density is generated by a specific model (with some free
parameters to fit to the market). Examples are jump diffusion or stochastic volatility.
Since for parametric methods the connection between the risk neutral and the real
world distribution is given, they can be applied even in the absence of option prices.
While all these techniques lead to distributions with similar variance, there tend
to be differences in the third and fourth moments [Jondeau and Rockinger, 2000].
19
Another important issue is the robustness of these methods to errors in input data
[Bliss and Panigirtzoglou, 2002]. Finally, one might ask if option markets really do deliver arbitrage-free prices. [Ait-Sahalia, Wang and Yared, 2001] addressed this question by comparing the risk neutral density inferred from option prices on the S&P 500
to the risk neutral density inferred from the S&P 500 returns themselves. The latter
density can be obtained from the real world distribution in combination with the
risk neutral drift rate which is observable from the spot-forward parity relationship.
It turns out that the two distributions are different, as the option-inferred density
has stronger skew and fatter tails. [Ait-Sahalia, Wang and Yared, 2001] propose to
reconcile this disagreement with the assumption that options incorporate a premium
for jump risks in the underlying index that is absent from its recorded time series (as
it has small probability).
Summing up, due to a lack of (quality) data there are technical difficulties in
implying the risk neutral density from market data. Nevertheless, we proceed with the
assumption that we know an expression for today’s risk neutral pdf f0,T (ST |S0 , θ0 ).
3.2
Model
We are going to simulate the future risk neutral pdf for a simplified (,,twin”) variable
X first. This has the advantage that we can use simpler model distributions. In
particular, we can de-couple parameters Xt and θt (see below). In the next subsection,
we will map X onto the underlying asset price S, and we will connect the model
distributions to f0,T , p0,t and ft,T in a way that ensures the no-arbitrage conditions
Eqs.(1.1) and (1.2) for ft,T .
1. First, we need to make the step from time 0 to time t. We postulate that, as of
today, Xt is distributed according to a specific pdf q0,t
q0,t (Xt |X0 , θ0 ) .
(3.2)
Note that this pdf does not depend on θt .
2. Next, we need to make the step from time t to expiry T . Our description consists
of a pdf which is assumed to capture the time persisting structure of the risk
neutral distribution, and which itself depends on stochastic parameters. Thus,
we postulate that, at time t, variable XT has a pdf with a specific form gt,T
gt,T (XT |Xt , θt ) .
20
(3.3)
θt is a parameter of this ,,structural” pdf gt,T (for instance its variance). θt may
also symbolize several parameters (for instance the variance and the skew). In
this case it should be understood as a vector, and dθt will symbolize integration
over all components of θt .
3. Finally, parameter θt is assumed to be stochastic as well, with pdf
k0,t (θt | θ0 ) .
(3.4)
This ,,mixing” density k models the time evolution properties of the risk neutral
distribution. If θt symbolizes several parameters, their distribution is assumed to
1
2
be independent (,,orthogonal shocks”): k0,t (θt | θ0 ) = k0,t
(θt1 | θ01 ) k0,t
(θt2 | θ02 ) . . .
We can choose q0,t , gt,T and k0,t from a wide range of possible distributions. Nevertheless, some conditions have to be met, see Sec. 3.5.
3.3
Connection between market and model
The connection between our twin model variable X and the underlying asset price S
is constructed as follows:
1. The values at time 0, S0 and X0 , are set equal
S0 ≡ X0 .
(3.5)
2. The values at expiry, ST and XT , are mapped onto each other by
−1
ST ≡ F0,T
[M0,t,T (XT ) |S0 , θ0 ] ,
−1
XT = M0,t,T [F0,T (ST |S0 , θ0 )] .
(3.6)
where F0,T is the cumulative density function (cdf) of f0,T . In the following we
will always assume that F0,T is invertible (i.e. that f0,T vanishes nowhere except
at the boundaries). Function M0,t,T is defined to be the following mixture of
distributions
Z Z
M0,t,T (ξ) ≡
dXt dθt Gt,T (ξ |Xt , θt ) q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 ) ,
(3.7)
with the cdf Gt,T of gt,T . This mapping brings the empirical information contained in F0,T into the model, and it ensures Eq.(1.1) (proof see Sec. 4.3.2).
21
3. The values at time t, St and Xt , are mapped onto each other by
St ≡ Lt,T (Xt , θt ) ,
Xt = L−1
t,T (St , θt ) ,
where
(3.8)
Z
Lt,T (ξ, θt ) ≡
dXT ST (XT ) gt,T (XT |ξ, θt ) .
(3.9)
Note that Lt,T depends on the specific realization of θt . This mapping ensures
Eq.(1.2) (proof see Sec. 4.3.2).
Distributions p0,t and ft,T are connected to q0,t , gt,T as follows:
4. The future risk neutral distribution is obtained from the model via
Ft,T (ST |St , θt ) ≡ Gt,T (XT |Xt , θt ) ,
(3.10)
which implies
ft,T (ST |St , θt ) = gt,T (XT |Xt , θt )
dXT
.
dST
(3.11)
5. The joint distribution of St and θt is defined as
p0,t (St , θt |S0 , θ0 ) ≡ q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 )
dXt
(θt ) .
dSt
(3.12)
Notice that, since dXt /dSt is θt -dependent, St and θt will usually not be independent.
3.4
Summary
Summing up, the twin formalism can be represented as shown in Table 3.1.
Expressions L and M in Table 3.1 are defined by
Z
Lt,T (Xt , θt ) ≡
dXT ST gt,T ,
Z Z
M0,t,T (XT ) ≡
dXt dθt Gt,T q0,t k0,t .
The twin formalism guarantees the properties (for a proof see Sec. 4.3.2)
Z
dST ST ft,T = St ,
Z Z
dSt dθt ft,T p0,t = f0 .
22
S-Space
←
S0
St
ST
θ 0 , θt
k0,t (θt | θ0 )
F0,T (ST |S0 , θ0 )
f0,T (ST |S0 , θ0 )
p0,t (St , θt |S0 , θ0 )
Ft,T (ST |St , θt )
ft,T (ST |St , θt )
Connection
S0 ≡ X0
St ≡ Lt,T (Xt , θt )
−1
ST ≡ F0,T
[M0,t,T (XT ) |S0 , θ0 ]
unchanged
unchanged
unchanged
unchanged
t
p0,t ≡ q0,t k0,t dX
dSt
Ft,T ≡ Gt,T
T
ft,T = gt,T dX
dST
→
X-Space
X0
Xt
XT
θ0 , θt
k0,t (θt | θ0 )
F0,T (ST |S0 , θ0 )
f0,T (ST |S0 , θ0 )
q0,t (Xt |S0 , θ0 )
Gt,T (XT |Xt , θt )
gt,T (XT |Xt , θt )
Table 3.1: Overview of the twin formalism: Connection between S and its twin
variable X.
3.5
Conditions for the model distributions
3.5.1
Conditions for the time dependence
All three model distributions q0,t , gt,T and k0,t should be delta-peaks if their time
indices are equal. Furthermore, for consistency we need for t = 0
g0,T (s |s0 , θ0 ) = f0,T (s |s0 , θ0 ) ,
(3.13)
q0,T (s |s0 , θ0 ) = f0,T (s |s0 , θ0 ) .
(3.14)
and for t = T
3.5.2
Conditions for the X-dependence of g
Since we want to ensure that the mapping between XT and ST , as well as between Xt
and St exists, we need to impose two further conditions on the model cdf Gt,T . These
−1
two conditions ensure the existence of M0,t,T
and L−1
t,T (for a proof see Secs. 4.1.1 and
4.2.1):
1. Gt,T (ξ |Xt , θt ) has to increase strictly monotonously with ξ for all Xt , θt .
2a. If µ1 < µ2 then φ12 (ξ) ≡ G−1 [G (ξ |µ2 , θt )| µ1 , θt ] has to be smaller than ξ for
all θt .
2b. Alternatively, it is also sufficient if
gt,T (XT + µ1 |µ1 , θt ) = gt,T (XT + µ2 |µ2 , θt )
for all µ1 , µ2 , θt .
23
Comment to condition 1: This condition implies that Gt,T (ξ) is invertible and,
thus, ensures that M0,t,T (ξ) is invertible as well, see Sec. 4.1.1. It is fulfilled by all
distributions that vanish nowhere except at the boundaries.
Comment to condition 2a: This condition implies that, if we integrate a strictly
monotonously increasing function of x weighted by gt,T (x |µ ), the result is a strictly
monotonously increasing function of µ. This ensures that Lt,T (µ) is invertible, see
Sec. 4.2.1. Condition 2a is for instance fulfilled by the displaced diffusion distribution
(see Appendix A)
1
1
dd x x0 , σ 2 , γ ≡ √
2πσ (1 − γ) x + γ
 2 
(1 − γ) x + γ
2 2
 − log (1 − γ) x + γ + (1 − γ) σ /2 


0
exp 
,
2 2


2 (1 − γ) σ
DD x x0 , σ 2 , γ ≡
Z
x
dξ dd ξ x0 , σ 2 , γ .
For this distribution,
φ12 (ξ) =
1
[[(1 − γ) µ1 + γ] ξ + (µ1 − µ2 ) γ]
(1 − γ) µ2 + γ
is smaller than ξ if µ1 < µ2 . The two special cases of the displaced diffusion distribution, the normal distribution (γ = 1) and the lognormal distribution (γ = 0) fulfill
condition 2a as well.
Comment to condition 2b: This condition has the same effect as condition 2a. It
is fulfilled by all distributions that do not change their form if their mean is shifted
(,,rigid translation”). Examples are the normal distribution, and the student-t distribution (with location parameter x0 , degrees of freedom n, and scale parameter
γ > 0)
1 Γ
stud t [x |x0 , n, γ ] ≡ √
nπ Γ
where
Z
Γ (m) ≡
n+1
2
n
γ
2
(x − x0 )2
1+
nγ 2
∞
dt tm−1 e−t .
0
24
!− n+1
2
,
Chapter 4
Properties
In this chapter we derive general analytical results for the quantities defined in Chapter 3. In particular, we show that these quantities are well-defined and well-behaved.
4.1
4.1.1
Properties of XT
Existence of XT
Condition 1 for G in Sec. 3.5.2 ensures that XT is well-defined. In particular, XT and
ST are strictly monotonously increasing functions of each other.
Proof: Since, according to Eq.(3.6),
−1
XT = M0,t,T
[F0 (ST |S0 , θ0 )] ,
XT is well-defined iff M is invertible. According to condition 1, Gt,T [ξ |Xt , θt ] increases strictly monotonously with ξ. Therefore,
Z Z
M0,t,T (ξ) =
dXt dθt Gt,T (ξ |Xt , θt ) q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 )
increases strictly monotonously as well. Thus, M is invertible. 4.1.2
Case t = 0
If t = 0, then
Z Z
M0,0,T (ξ) =
dX dθ G0,T (ξ |X, θ ) q0,0 (X |S0 , θ0 ) k0,0 (θ| θ0 )
= G0,T (ξ |S0 , θ0 )
= F0,T (ξ |S0 , θ0 ) ,
25
where the last equality follows from Eq.(3.13). Consequently, Eq.(3.6) implies for
t=0
XT = ST .
Furthermore,
Z
dXT ST (XT ) g0,T (XT |ξ, θ0 )
L0,T (ξ, θ0 ) =
Z
dXT XT g0,T (XT |ξ, θ0 )
=
= ξ,
which, when compared with Eq.(3.8), is consistent with X0 = S0 .
4.1.3
Case t = T
If t = T , then
Z
M0,T,T (ξ) =
Z
dX GT,T (ξ |X, θ ) q0,T (X |S0 , θ0 ) k0,T (θ| θ0 )
dθ
Z
=
Z
dθ
Z
=
ξ
dX q0,T (X |S0 , θ0 ) k0,T (θ| θ0 )
ξ
dX q0,T (X |S0 , θ0 )
= F0,T (ξ |S0 , θ0 ) ,
where the last equality follows from Eq.(3.14). Consequently, Eq.(3.6) implies for
t=T
XT = ST .
4.1.4
Derivatives
Let us define the derivative of M0,t,T (XT ) w.r.t. XT
Z Z
m0,t,T (XT ) ≡
dXt dθt gt,T (XT |Xt , θt ) q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 ) .
(4.1)
Since M0,t,T (XT ) increases strictly monotonously with XT (Sec. 4.1.1), m0,t,T (XT ) is
always positive. Furthermore,
dF0,T (ST |S0 , θ0 )
dM0,t,T (XT )
=
= m0,t,T (XT ) ,
dXT
dXT
where we have used Eq.(3.6).
26
(4.2)
The derivative of XT itself has a particular simple form
dXT
dST
dXT
dF0,T (ST |S0 , θ0 )
dF0,T (ST |S0 , θ0 )
dST
−1
dF0,T (ST )
=
f0,T (ST ) ,
dX (ST )
=
which implies, together with Eq.(4.2),
dXT
f0,T (ST |S0 , θ0 )
=
.
dST
m0,t,T (XT )
(4.3)
Eq.(4.3) allows us to transform derivation w.r.t. ST into derivation w.r.t. XT
f0,T (ST |S0 , θ0 ) d
d
=
.
dST
m0,t,T (XT ) dXT
4.2
4.2.1
Properties of Xt
Existence of Xt
Condition 2a for G in Sec. 3.5.2 ensures that Xt is well-defined. In particular, Xt and
St are strictly monotonously increasing functions of each other.
Proof: Since
Xt ≡ L−1
t,T (St , θt ) ,
Xt is well-defined iff L is invertible. Choose µ1 < µ2 , and define φ12 by
G (φ12 (ξ) |µ1 , θt ) = G (ξ |µ2 , θt ) .
This implies
g [φ12 (ξ) |µ1 , θt ]
Thus (with the transformation
Z
Lt,T (µ1 , θt ) =
Z
=
Z
=
Z
<
dφ12 (ξ)
= g (ξ |µ2 , θt ) .
dξ
x ≡ φ12 (ξ)),
dx ST (x) gt,T (x |µ1 , θt )
dφ12 (ξ)
dξ ST [φ12 (ξ)] gt,T [φ12 (ξ) |µ1 , θt ]
dξ
dξ ST [φ12 (ξ)] g (ξ |µ2 , θt )
dξ ST (ξ) g (ξ |µ2 , θt ) = Lt,T (µ2 , θt ) .
27
The inequality holds since ST (ξ) increases strictly monotonously with ξ (see Sec.
4.1.1), and since, by condition 2a, φ12 (ξ) < ξ. Consequently, L (µ, θt ) is invertible
w.r.t. µ. Condition 2b for G in Sec. 3.5.2 ensures that Xt is well-defined. Again, Xt and St
are strictly monotonously increasing functions of each other.
Proof: Since
Xt ≡ L−1
t,T (St , θt ) ,
Xt is well-defined iff L is invertible. Choose µ1 < µ2 . Then (with condition 2b)
Z
Lt,T (µ1 , θt ) =
dx ST (x) gt,T (x |µ1 , θt )
Z
=
dx ST (x + µ1 ) gt,T (x + µ1 |µ1 , θt )
Z
=
dx ST (x + µ1 ) gt,T (x + µ2 |µ2 , θt )
Z
=
dx ST [x − (µ2 − µ1 )] gt,T (x |µ2 , θt )
Z
<
dx ST (x) g (x |µ2 , θt ) = Lt,T (µ2 , θt ) .
The inequality holds since ST (ξ) increases strictly monotonously with ξ (see Sec.
4.1.1). Consequently, L (µ, θt ) is invertible w.r.t. µ. 4.2.2
Derivatives
The derivative of L w.r.t. Xt is
dLt,T (Xt , θt )
=
dXt
Z
dXT ST (XT )
dgt,T (XT |Xt , θt )
.
dXt
(4.4)
Conditions 2a or 2b in Sec. 3.5.2 guarantee that Lt,T (Xt , θt ) increases strictly monotonously with Xt . Therefore, the r.h.s. of this equation is positive.
The derivative of Xt itself is
dXt
dXt
dLt,T (Xt , θt )
=
dSt
dLt,T (Xt , θt )
dSt
−1
dLt,T (Xt , θt )
dSt
=
,
dXt
dSt
where in the last step we have used Eq.(3.8). With Eq.(4.4) we obtain
1
dXt
=R
.
dSt
dXT ST (XT ) dgt,T (XT |Xt , θt ) /dXt
28
(4.5)
4.3
4.3.1
Properties of ft,T
Normalization
Since gt,T is normalized, Eq.(3.11) implies
Z
Z
dXT
dST ft,T (ST |St , θt ) =
dST gt,T (XT |Xt , θt )
dST
Z
=
dXT gt,T (XT |Xt , θt )
= 1.
4.3.2
No arbitrage
The future distribution fulfills Eq.(1.1)
Z Z
dSt dθt ft,T (ST |St , θt ) p0,t (St , θt |S0 , θ0 ) = f0 (ST |S0 , θ0 ) .
Proof:
Z Z
dSt dθt Ft,T (ST |St , θt ) p0,t (St , θt |S0 , θ0 ) =
Z Z
dSt dθt {Gt,T
=
dXt
(XT |Xt , θt )} q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 )
dSt
Z Z
=
dXt dθt Gt,T (XT |Xt , θt ) q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 )
= M0,t,T (XT )
= F0,T (ST |S0 , θ0 )
Derive this equation w.r.t. ST to obtain the result. The future distribution fulfills Eq.(1.2)
Z
dST ST ft,T (ST |St , θt ) = St .
Proof:
Z
Z
ST ft,T (ST |St , θt ) dST =
dXT
ST gt,T (XT |Xt , θt )
dST
dST
Z
=
ST (XT ) gt,T (XT |Xt , θt ) dXT
= Lt,T (Xt , θt )
= St .
29
4.3.3
Explicit form of ft,T
The future implied pdf is given by Eq.(3.11)
ft,T (ST |St , θt ) = gt,T (XT |Xt , θt )
dXT
,
dST
or, if we use Eq.(4.3)
ft,T (ST |St , θt ) = f0,T (ST |S0 , θ0 )
gt,T (XT |Xt , θt )
,
m0,t,T (XT )
(4.6)
where
Z Z
m0,t,T (XT ) ≡
dXt dθt gt,T (XT |Xt , θt ) q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 ) .
In this formulation, mixing property Eq.(1.1) of the future implied distribution is
especially transparent
Z Z
dSt dθt ft,T (ST |St , θt ) p0,t (St , θt |S0 , θ0 ) =
R
= f0,T (ST |S0 , θ0 )
dXt dθt gt,T (XT |Xt , θt ) q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 )
mt,T (XT )
= f0,T (ST |S0 , θ0 ) .
4.3.4
Possible and impossible events
Eq.(4.6) shows that ft,T cannot assign a finite probability to events that were impossible at t = 0: if f0,T vanishes, so does ft,T . On the other hand, it may happen that
eT , ft,T vanishes as well, no
events become impossible: if gt,T vanishes at some point X
matter
if f0,T is finite. However, this cannot be the
case
for all realizations of θt : if
eT |Xt , θt = 0 for all Xt and θt , then m0,t,T X
eT = 0 in contradiction to the
gt,T X
fact that m is always positive (Sec. 4.1.4). Summing up:
• Impossible events remain impossible.
• Possible events may become impossible for certain (but not for all) realizations
of Xt and θt .
This means that, compared to today, the future distributions cannot ,,broaden”
their range. This is a no-arbitrage feature.
30
4.3.5
Tail behaviour
If the conditional values Xt and θt do not influence the tail behaviour of gt,T
gt,T (XT |Xt , θt )
= 1,
ST →∞
m0,t,T (XT )
lim
then Eq.(4.6) implies (assuming that all limits exist) that the tails of ft,T behave like
the tails of f0,T
lim ft,T (ST |St , θt ) = lim f0,T (ST |S0 , θ0 ) .
ST →∞
4.3.6
ST →∞
Time homogeneity
If f0,T and the model distributions are time homogeneous
f0,T (u |ς, θ ) = fτ,T +τ (u |ς, θ ) ,
q0,t (ξ |x, θ ) = qτ,t+τ (ξ |x, θ ) ,
gt,T (ξ |x, θ ) = gt+τ,T +τ (ξ |x, θ ) ,
k0,t (ϑ| θ) = kτ,t+τ (ϑ| θ) ,
then so is ft,T
ft,T (u |s, θ ) = ft+τ,T +τ (u |s, θ ) .
Proof: The time homogeneity of the model distributions implies
Z Z
Mτ,t+τ,T +τ (ξ) =
dx dϑ Gt+τ,T +τ (ξ |x, ϑ ) qτ,t+τ (x |ς, θ ) kτ,t+τ (ϑ| θ)
Z Z
=
dx dϑ Gt,T (ξ |x, ϑ ) q0,t (x |ς, θ ) k0,t (ϑ| θ)
= M0,t,T (ξ) .
Therefore,
−1
XT +τ = Mτ,t+τ,T
+τ [Fτ,T +τ (u |ς, θ )]
−1
= M0,t,T
[F0,T (u |ς, θ )]
= XT .
Notice that the mapping Eq.(3.6) between X and S depends on earlier times as well.
Thus, it would be more precise to write this result like Xτ,t+τ,T +τ (u) = X0,t,T (u).
31
Further implications are
Z
dx ST +τ (x) gt+τ,T +τ (x |ξ, θ )
Lt+τ,T +τ (ξ, θ) =
Z
=
dx ST (x) gt,T (x |ξ, θ )
= Lt,T (ξ, θ) ,
as well as
Xt+τ = Xt .
Therefore,
Ft+τ,T +τ (u |s, θ ) = Gt+τ,T +τ (XT +τ (u) |Xt+τ (s, θ) , θ )
= Gt,T (XT (u) |Xt (s, θ) , θ )
= Ft,T (u |s, θ ) .
32
Chapter 5
Numerical evaluation schemes
In this chapter we describe step-by-step procedures for the numerical implementation of the twin formalism introduced in Chapter 3. The only non-trivial numerical
technique needed is numerical integration. A further advantage is that the mapping
ST ↔ XT has to be established only once and can then be stored and used for further
calculations.
5.1
Evaluation of the mapping S ↔ X
−1
1. Choose a function f0,T to fit the market data. Ideally, F0,T
should be known
analytically.
2. Decide on the form of distributions q0,t , gt,T and k0,t .
3. Choose parameter values for t, T, θ0 and S0 .
4. Establish the mapping ST ↔ XT by
• scanning through XT
• calculating (numerical integration)
Z Z
M0,t,T (XT ) =
dXt dθt Gt,T (XT |Xt , θt ) q0,t (Xt |S0 , θ0 )
k0,t (θt | θ0 )
• and solving
−1
ST = F0,T
[M0,t,T (XT ) |S0 , θ0 ]
5. Pick a realization of θt .
33
6. Given θt , establish the mapping St ↔ Xt by
• scanning through Xt
• and calculating (numerical integration)
Z
St = ST (XT ) gt,T (XT |Xt , θt ) dXT
5.2
Evaluation of the future distribution
We assume that we have carried out the scheme described in Sec. 5.1. Thus, we have
chosen f0,T , q0,t , gt,T and k0,t , as well as values for t, T, θ0 and S0 . Furthermore, we
know the mapping ST ↔ XT , we have chosen a value for θt , and we have established
the corresponding mapping St ↔ Xt . Now:
1. Pick a value for St .
2. Scan through XT and
• calculate (numerical integration)
Z Z
m0,t,T (XT ) =
dXt dθt gt,T (XT |Xt , θt ) q0,t (Xt |S0 , θ0 )
k0,t (θt | θ0 )
• infer ST and St from XT and Xt .
• calculate
ft,T (ST |St , θt ) = f0,T (ST |S0 , θ0 )
5.3
gt,T (XT |Xt , θt )
m0,t,T (XT )
Valuation of a call at time t
The value of a plain vanilla European call with strike K and expiry T is at time t
Z
Ct,T (St , K, θt ) = dST {ST − K}+ ft,T (ST |St , θt ) ,
given that spot St and parameter value θt obtain. If we transform this equation into
our model variables, we get
Z
Ct,T (St , K, θt ) =
dST {ST − K}+ ft,T (ST |St , θt )
Z
dXT
=
dST {ST − K}+ gt,T (XT |Xt , θt )
,
dST
34
or
Z
dXT {ST (XT ) − K}+ gt,T (XT |Xt , θt ) .
Ct,T (St , K, θt ) =
(5.1)
Usually, this integration needs to be carried out numerically, using the mappings
ST ↔ XT and St ↔ Xt established in Sec. 5.1.
5.4
Valuation of a forward start call
If we transform Eq.(1.9) for the value of a forward start call into our model variables,
we obtain
Z Z
dSt dθt Ct,T [St , φ (St ) , θt ] p0,t (St , θt |S0 , θ0 )
V0,t,T (S0 , θ0 ) =
Z Z
dSt dθt Ct,T [St , φ (St ) , θt ] q0,t (Xt |S0 , θ0 )
=
k0,t (θt | θ0 )
dXt
,
dSt
or
Z Z
V0,t,T (S0 , θ0 ) =
dXt dθt Ct,T [St (Xt , θt ) , φ (St (Xt , θt )) , θt ]
q0,t (Xt |S0 , θ0 ) k0,t (θt | θ0 ) ,
(5.2)
where Ct,T is given by Eq.(5.1). We will have to carry out the three integrations in
Eqs.(5.2) and (5.1) numerically.
35
Chapter 6
Applications
In this chapter we show how the twin formalism developed in earlier chapters can be
used to implement different models. Although we are going to draw some conclusions
and to compare some results with market data, the emphasis does not lie on the
usefulness of the models themselves − they only serve as case studies. The goal of
this chapter is rather to illustrate how the theoretical concepts developed earlier work
in practice.
We start in Sec. 6.1 with the plain Black-Scholes case which obtains naturally if
all distributions are chosen to be lognormal with constant volatility. This exemplifies
the case of no dependence on additional parameters θ. Next (in Sec. 6.2), we make
volatility stochastic in a minimal way by allowing it to jump to a different value at
one given point in time. This jump could model market reaction to some external
information that will become available at this specific time. Again, however, we do
not stress how useful or realistic this model actually is. It is only used as an especially
simple example for a model with a stochastic parameter θ (i.e. volatility). Finally, Sec.
6.3 shows the application to a displaced diffusion distribution with a skew parameter
γ that has a deterministic time dependence. This third example could be counted as
a model which depends on a deterministic parameter θ (i.e. the skew parameter γ).
All numerical results in this chapter have been obtained according to the evaluation schemes described in Chapter 5. They have been implemented as Visual Basic
scripts. The corresponding MS-Excel sheets can be obtained from the author upon
request.
36
6.1
Black-Scholes model
6.1.1
Distributions
The Black-Scholes special case is obtained if we set
BS
f0,T
(ST |S0 ) = logN ST S0 , σ02 (T − 0) ,
BS
q0,t
(Xt |S0 ) = logN Xt S0 , σ02 (t − 0) ,
BS
gt,T
(XT |Xt ) = logN XT Xt , σ02 (T − t) ,
where there is no additional θ dependence. The Black-Scholes model is time-homogeneous (cf. Sec. 4.3.6). Eq.(B.2) implies
M0,t,T (XT ) = LogN XT S0 , σ02 T .
Thus, due to the sum properties of the lognormal distribution, the result of the mixing
integral M is again lognormal. Therefore,
−1
XT = M0,t,T
[F0,T (ST |S0 , θ0 )]
= LogN−1 LogN ST S0 , σ02 T S0 , σ02 T
= ST .
This leads to
Z
Lt,T (ξ) =
Z
=
BS
dXT ST (XT ) gt,T
(XT |ξ )
BS
dXT XT gt,T
(XT |ξ )
= ξ,
and thus
St = Lt,T (Xt ) = Xt .
Summing up
BS
Ft,T
(ST |St ) = GBS
t,T (XT |Xt )
= GBS
t,T (ST |St )
= LogN ST St , σ02 (T − t) .
BS
Hence, the future pdf ft,T
(ST |St ) will be lognormal again, as it should be in the
BS
Black-Scholes model. The time evolution of ft,T
is shown in Fig. 6.1. Notice how
weight around the value St = 0.25 increases as time to maturity decreases.
37
BS
ft,T (ST)
t=0.00
t=0.75
t=0.95
0.00
0.13
0.25
ST
0.38
0.50
Figure 6.1: Time evolution à la Black-Scholes: The lognormal risk neutral pdf
BS
ft,T
(ST ) at three different times t. Parameter values are: σ0 = 40%, t0 = 0, T = 1,
and St = 0.25 for all three values t = 0.00, t = 0.75, and t = 0.95.
6.1.2
Call at time t
The value of a call in the Black-Scholes model at time t is
Z
BS
Ct,T (St , K) =
dST {ST − K}+ logN ST St , σ02 (T − t)
= St N d0+ |0, 1 − K N d0− |0, 1 ,
where
d0± ≡
6.1.3
log [St /K] ± σ02 (T − t) /2
p
.
σ0 (T − t)
(6.1)
(6.2)
Forward start call
The value of a forward start call in the Black-Scholes model is, according to Eq.(5.2),
Z
BS
BS
V0,t,T (S0 ) = dSt Ct,T
[St , φ (St )] logN St S0 , σ02 t .
38
BS
Ct,T
is the value of a call with strike φ (St ), as given by Eq.(6.1). If φ (St ) = aSt , with
some constant a, quantities d0± in Eq.(6.2) become independent of St
d0± =
− log [a] ± σ02 (T − t) /2
p
.
σ0 (T − t)
In this case, the integral over St can be calculated explicitly as well
Z
BS
V0,t,T (S0 ) =
dSt St N d0+ |0, 1 − aSt N d0− |0, 1 logN St S0 , σ02 t
Z
0
0
dSt St logN St S0 , σ02 t
= N d+ |0, 1 − aN d− |0, 1
= N d0+ |0, 1 − aN d0− |0, 1 S0 .
This is the value of a call with spot S0 , strike aS0 and time to maturity T − t
BS
BS
V0,t,T
(S0 ) = Ct,T
(S0 , aS0 ) .
6.2
6.2.1
(6.3)
Black-Scholes model with a volatility jump
Distributions
We again assume that today’s distribution f0,T , as well as q0,t and gt,T are lognormal.
Now, however, there is an additional stochastic parameter (i.e. model parameter θ in
the twin formalism): volatility. Thus,
J
f0,T
(ST |S0 , σ0 ) = logN ST S0 , σ02 (T − 0) ,
J
q0,t
(Xt |S0 , σ0 ) = logN Xt S0 , σ02 (t − 0) ,
J
gt,T
(XT |Xt , σt ) = logN XT Xt , σt2 (T − t) .
We assume that volatility σt at time t can either remain equal to σ0 or jump to a
different value σ1 with probability
k0 ≡ 1 − αt/T for σt = σ0
k0,t (σt | σ0 ) =
,
k1 ≡ αt/T
for σt = σ1
with some constant 0 ≤ α ≤ 1. Like the Black-Scholes model, this model is timehomogeneous (cf. Sec. 4.3.6). In case α = 0 the Black-Scholes model is recovered. The
volatility jump could for instance model the reaction to external information (like a
company announcement, an election result, a rate cut etc.) that becomes available
at time t. The time evolution of the risk neutral pdf without and with a jump (from
40% to 60% volatility) is shown in Fig. 6.2. Notice that the jump to higher volatility
39
ft,TJ(ST)
t=0.0
t=0.5 (no jump)
t=0.5 (jump)
0
50
100
150
200
ST
J
Figure 6.2: Time evolution with a volatility jump: The risk neutral pdf ft,T
(ST ) at
two different times: t = 0.0 and t = 0.5. For the latter time, the pdf without and with
a jump in volatility is shown. The other parameter values are: σ0 = 40%, σ1 = 60%,
α = 1, t0 = 0, T = 1, and S0 = St = 100.
J
almost completely compensates the effect of the passed time to maturity (i.e. ft,T
J
).
after the jump is almost equal to f0,T
Eq.(B.2) implies
M0,t,T (XT ) = LogN XT S0 , σ02 T k0 + LogN XT S0 , σ02 t + σ12 (T − t) k1 .
Therefore, according to Eq.(3.6), ST and XT are connected via
LogN ST S0 , σ02 T = LogN XT S0 , σ02 T k0
+LogN XT S0 , σ02 t + σ12 (T − t) k1 .
(6.4)
Numerical calculations show that ST (XT ), as given by Eq.(6.4), is (for reasonable
parameter values) predominantly linear
ST (XT ) ≈ bXT ,
40
(6.5)
with a factor b close to 1, see Fig. 6.3. Together with Eq.(3.9) this implies St (Xt ) ≈
bXt . We will use this approximation below for qualitative discussions.
200
ST
150
100
50
0
0
50
100
150
200
XT
Figure 6.3: Almost linear: Mapping between ST and its twin variable XT for the
Black-Scholes model with a volatility jump (dotted line), compared to ST = XT
(straight line). The parameter values are: σ0 = 40%, σ1 = 60%, α = 1, t0 = 0,
t = 0.5, T = 1, and S0 = St = 100.
6.2.2
Call at time t
In the present case, Eq.(5.1) has the form
Z
J
Ct,T (St , K, σt ) = dXT {ST (XT ) − K}+ logN XT Xt , σt2 (T − t) .
41
(6.6)
How do implied volatilities derived from this equation look like? We can give a crude
answer by using the approximation ST (XT ) ≈ bXT
Z
J
Ct,T (St , K, σt ) ≈
dXT {bXT − K}+ logN XT St /b, σt2 (T − t)
Z
= b dXT {XT − K/b}+ logN XT St /b, σt2 (T − t)
BS
= bCt,T
(St /b, K/b, σt )
BS
(St , K, σt ) .
= Ct,T
Thus, implied volatilities will essentially remain flat. Numerically, the following corrections to this result are found: implied volatilities are shifted away from σt . Furthermore, there is a slight frown, see Fig. 6.4.
60%
implied volatility
55%
50%
t=0.0
t=0.5
45%
40%
35%
30%
60
80
100
120
140
K
Figure 6.4: No smile: Implied volatilities vs. strike K calculated in the Black-Scholes
model with a volatility jump. The parameter values are: σ0 = 40%, σ1 = 60%, α = 1,
t0 = 0, t = 0.5, T = 1, and S0 = St = 100.
42
6.2.3
Forward start call
The value of a forward start call in the Black-Scholes model with a volatility jump is
Z
J
V0,t,T (S0 , σ0 ) =
dXt [k0 Ct,T [St (Xt , σ0 ) , φ (St (Xt , σ0 )) , σ0 ]
+k1 Ct,T [St (Xt , σ1 ) , φ (St (Xt , σ1 )) , σ1 ]]
logN Xt S0 , σ02 t .
(6.7)
If we set φ (St ) = St , and use approximation (6.5), we obtain the following expression
J
V0,t,T
(S0 , σ0 ) ≈
Z
≈
dXt [k0 Ct,T [bXt , bXt , σ0 ] + k1 Ct,T [bXt , bXt , σ1 ]] logN Xt S0 , σ02 t
Z
BS
BS
dXt k0 bCt,T
[Xt , Xt , σ0 ] + k1 bCt,T
[Xt , Xt , σ1 ] logN Xt S0 , σ02 t
= b k0 N d0+ |0, 1 − N d0− |0, 1 + k1 N d1+ |0, 1 − N d1− |0, 1
Z
dXt Xt logN Xt S0 , σ02 t
= b S0 k0 N d0+ |0, 1 − N d0− |0, 1 + k1 N d1+ |0, 1 − N d1− |0, 1
,
≈
where d1± has the same form as d0± , but with volatility σ1 . Consequently,
J
BS
BS
V0,t,T
(S0 , σ0 ) ≈ b k0 Ct,T
(S0 , S0 , σ0 ) + k1 Ct,T
(S0 , S0 , σ1 )
approx
≡ V0,t,T
(S0 , σ0 ) .
(6.8)
Thus, we expect that the value of a forward start call in our model is a weighted
mean of the Black-Scholes values for a forward start call with volatility σ0 and σ1 .
Table 6.1 shows numerical results for (forward start) call values. b is the parameter
from Eq.(6.5), obtained from a linear fit to the ST (XT ) values between XT = 30 and
BS
XT = 200. Ct,T
is the value of an at-the-money call with spot S0 in the Black-Scholes
approx
model, and V0,t,T
is the value of the forward start call approximated by Eq.(6.8).
43
σ1 = 20%
b
1.09
BS
Ct,T (σ0 ) 11.3
BS
Ct,T
(σ1 ) 5.6
approx
V0,t,T
9.2
J
V0,t,T
9.4
σ1 = 30%
1.02
11.3
8.5
10.1
10.4
σ1 = 50%
0.91
11.3
14.0
11.5
11.9
σ1 = 60%
0.89
11.3
16.8
12.5
12.3
Table 6.1: Numerical results in the Black-Scholes model with a volatility jump for
several values of σ1 , with S0 = 100, t = 0.5, T = 1, σ0 = 40%, and α = 1. For details
see text.
Summing up, we can draw the following conclusions:
J
• V0,t,T
increases with increasing σ1 .
J
• V0,t,T
lies between the Black-Scholes values for σ0 and σ1 .
approx
J
• V0,t,T
.
gives a reasonable approximation for V0,t,T
• Implied volatilities at time t remain essentially flat (there is only a slight frown),
and are shifted away from σt .
6.3
6.3.1
Displaced diffusion model
Distributions
In this model we assume that today’s distribution f0,T , as well as q0,t and gt,T are of
displaced diffusion form
D
f0,T
(ST |S0 , γT −0 ) = dd ST S0 , σ02 (T − 0) , γT −0 ,
D
q0,t
(Xt |S0 , γt−0 ) = dd Xt S0 , σ02 (t − 0) , γt−0 ,
D
gt,T
(XT |Xt , γT −t ) = dd XT Xt , σ02 (T − t) , γT −t ,
cf. Appendix A. The skew parameter γ is assumed to grow deterministically with
decreasing time τ to expiry
T −0
,
γτ ≡ γmax − (γmax − γmin ) exp κ 1 −
τ
(6.9)
where γmin and γmax are constants with 0 ≤ γmin ≤ γmax ≤ 1, and κ is a positive
constant. Note that γT = γmin .
44
1.2
1.0
γτ
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
τ
Figure 6.5: Turning on the skew: Skew parameter γτ as a function of time to maturity
τ calculated from Eq.(6.9). The parameter values are T = 1, γmin = 0 and γmax =
κ = 1.
Fig. 6.5 shows γτ as a function of τ for some specific parameter values. There is
no special motivation for the choice of a time dependence in the form of Eq.(6.9),
it is only intended to serve as an example. The displaced diffusion model is timehomogeneous (cf. Sec. 4.3.6). M is in this model
Z
M0,t,T (ξ) ≡ dXt DD ξ Xt , σ02 (T − t) , γT −t dd Xt S0 , σ02 t, γt .
This integral has to be solved numerically, except for the special case γt = γT −t ≡ γ
(i.e. either γmin = γmax or t = T − t). In this case, Eq.(B.3) implies
M0,t,T (ξ) = DD ξ S0 , σ02 T, γ .
45
D
ft,T (ST)
t=0.00
t=0.75
t=0.95
0.00
0.13
0.25
ST
0.38
0.50
Figure 6.6: Time evolution in the displaced diffusion model: The risk neutral pdf
D
ft,T
(ST ) at three different times t. Parameter values are: σ0 = 40%, t0 = 0, T = 1,
and St = 0.25 for all three values t = 0.00, t = 0.75, and t = 0.95. Cf. Fig. 6.1.
D
Figure 6.6 shows the time evolution of ft,T
. The parameter values are the same
as in Fig. 6.1. Notice that, as time to maturity decreases, the displaced diffusion pdf
(dashed line in Fig. 6.6) shrinks far slower than its lognormal counterpart (dashed
line in Fig. 6.1). This is due to the growing skew parameter γ.
6.3.2
Call at time t
The value of a call at time t in the displaced diffusion model is
Z
D
Ct,T (St , K) = dXT {ST (XT ) − K}+ dd XT Xt , σ02 (T − t) , γT −t .
(6.10)
Judging from our discussion in Sec. 2.3.3, the displaced diffusion model should reproduce some sort of implied volatility skew that grows as time to maturity decreases.
Figure 6.7 shows that this is indeed the case. For certain parameter values the model
generates even more structure in the implied volatility (not shown).
46
27.5%
27.0%
implied volatility
26.5%
26.0%
t=0.0
t=0.7
t=0.9
25.5%
25.0%
24.5%
24.0%
23.5%
23.0%
0.8
0.9
1.0
1.1
1.2
K
Figure 6.7: Increasing skew: Implied volatilities vs. strike K calculated in the displaced diffusion model at three different times t. Parameter values are: σ0 = 25%,
t0 = 0, T = 1, γmin = 0, γmax = κ = 1 and St = 0.25 for all three values t = 0.0,
t = 0.7, and t = 0.9.
47
6.3.3
Forward start call
The value of a forward start call in the displaced diffusion model is
Z
D
V0,t,T (S0 ) = dXt Ct,T [St (Xt ) , φ (St (Xt ))] dd Xt S0 , σ02 t, γt .
6.3.4
(6.11)
Negative values of ST
As mentioned in Sec. 2.3.3, one drawback of displaced diffusion is that it allows for
negative values of the price ST . This feature can be avoided in the present formulation,
D
if negative values of ST are impossible in today’s risk neutral pdf f0,T
. For instance
D
this holds true if γmin = 0, i.e. if f0,T is purely lognormal. In this case, Eq.(4.6)
D
ensures that negative values of ST are excluded in future distributions ft,T
as well,
D
f0,T (ST), gt,T (XT), ft,T (ST)
D
even if gt,T
attaches finite weight to negative values of XT .
D
D
today
model
future
-0.25
0.00
0.25
0.50
0.75
ST, XT
Figure 6.8: No negative ST allowed: Comparison between today’s risk neutral pdf
D
D
D
f0,T
(ST ), the model pdf gt,T
(XT ), and the future pdf ft,T
(ST ) which is connected to
D
the former two via Eq.(4.6). Notice that, although gt,T attaches a sizeable probability
D
to negative XT values, ft,T
still allows only for positive values of ST . Parameter values
are: σ0 = 40%, t0 = 0, t = 0.75, T = 1, γmin = 0, γmax = κ = 1, and S0 = St = 0.25.
48
0.8
0.6
ST
0.4
0.2
0.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
XT
Figure 6.9: From X-space to S-space: Mapping between ST and its twin variable
XT for the displaced diffusion model. Note that ST is always positive, cf. Fig. 6.8.
Parameter values are: σ0 = 40%, t0 = 0, t = 0.75, T = 1, γmin = 0, γmax = κ = 1,
and S0 = St = 0.25.
This effect is illustrated in Fig. 6.8. It is just an example for the principle that, in
the twin formalism, ,,impossible events remain impossible” (see Sec. 4.3.4). Another
way to look at this matter is to consider the mapping between XT and ST . As shown
in Fig. 6.9, all values of XT are mapped onto positive values of ST . This elimination
of negative ST has no particular influence on the volatility skews which result from
the displaced diffusion model, but it clearly makes the model more plausible.
49
6.3.5
Comparison to market data
We conclude our discussion of the displaced diffusion model with a comparison to
equity market data. In doing so we leave aside the question if displaced diffusion really
is, from the financial point of view, a good model for equity data [Rebonato, 2002].
The point of this comparison is rather to show that our comparatively simple model
is already flexible enough to allow for a market data fit that is at least qualitatively
correct.
As our empirical target we choose the FTSE data from Fig. 2.1. Our model
parameters are σ0 = 26%, γmin = 0.17, γmax = 1.00, and κ = 0.05. Furthermore:
t0 = 0, T = 7, and t equal to either 0 (i.e. 7 years time to maturity), 2 (5y), 6
(1y), 6.66 (i.e. 4 months time to maturity), or 6.92 (1m). Notice that in comparing
these data with Fig. 2.1 we are assuming time homogeneity: the value today of a
call with five years to maturity is assumed to be equal to the value in two years of
a call with today seven years to maturity, cf. Sec. 4.3.6. Finally, for γ 6= 0 the scale
of ST matters (as it does not cancel with the scale of St in the displaced diffusion
distribution). Therefore, we map the values ϕ of the FTSE onto our model variable
S using the linear transformation
S=
ϕ 7
− 1,
5648 4
where 5648 is the spot value of the FTSE and corresponds to S = 0.75 in our model.
The values 7/4 and −1 have been chosen so as to obtain a good fit to the data.
This mapping is also desirable as it makes the model dependent on the spot, i.e. it
generates a ,,floating” smile, see Sec. 2.2.
50
50%
45%
40%
0.45-0.5
0.4-0.45
0.35-0.4
0.3-0.35
0.25-0.3
35%
30%
1m
4m
25%
1y
5y
4523 5088
5371 5654
7y
5936 6219
6785
Figure 6.10: Theoretical fit: Smile surface for calls from the displaced diffusion model,
cf. Fig. 2.1. The x-axis displays the strike K in index points (with spot value 5648),
the y-axis displays time to maturity T (in years y or months m), and the z-axis
displays the corresponding implied volatilities σimp . For details see the text.
The results are shown in Fig. 6.10. As can be seen in comparison with Fig. 2.1,
a good qualitative fit of the whole smile surface is obtained. For the 7 years to
maturity curve the fit is even quantitatively correct. The shortcomings of the model
are, however, clear as well: for short times to maturity, the skew is not steep enough.
Still, a good qualitative fit of 35 data points using a model with 6 free parameters (σ0 ,
γmin , γmax , κ, and the two parameters from the linear mapping of S) is a reasonable
result.
51
Chapter 7
Summary and outlook
7.1
Summary
In this thesis we have presented a formalism, the twin formalism, that allows to
simulate how the risk neutral probability density function (pdf) evolves with time.
The risk neutral pdf is seen as a mixture of possible future distributions which may
depend on further stochastic variables θ. The twin formalism provides a general
framework in which a wide variety of models can be evaluated. We have shown some
examples in Chapter 6. By design, the twin formalism ensures no-arbitrage conditions
Eqs.(1.1) and (1.2) between the future and today’s risk neutral pdf. The modeling
framework is formulated in terms of probability distributions, rather than processes.
This means that one does not have to specify the dynamics (i.e. stochastic differential
equations) of the underlying processes. It suffices to determine the distributions only
at those points of time that are relevant for the option one wants to price (i.e. times at
which fixings, exercises or payoffs may happen). The twin formalism is flexible enough
to allow for a perfect fit of today’s risk neutral pdf (and, hence, today’s corresponding
market prices).
This thesis builds on the foundation laid by [Skiadopoulos and Hodges, 2002]. We
improve and deepen these author’s approach in several important ways:
• Firstly, we take account of the conditional dependencies of all distributions
(Chapter 3).
• Secondly, we give expressions for pdf’s at earlier times, i.e. q0,t and p0,t (Chapter
3).
• Thirdly, the new transformation L (Sec. 3.3) ensures martingale property Eq.(1.2).
52
• Fourthly, we specify all conditions the model distributions have to fulfill (Sec.
3.5).
• Finally, we list and prove a number of analytical results (Chapter 4).
The basic idea of the twin formalism is to simulate the time development of the
risk neutral pdf in a simplified (,,twin”) model space which is characterized by a
stochastic variable X. This X-space is a convenient environment for the design of
models. The connection to today’s risk neutral pdf, as well as the implementation of
no arbitrage conditions is ensured by transformations that lead from twin space X
to the space of the actual underlying S. These transformations M and L have the
form of integrals over the model distributions (Sec. 3.3). For this reason, the twin
formalism can readily be implemented numerically − the only non-trivial technique
needed is numerical integration (Chapter 5). On the other hand, there are some
limitations: The twin formalism is only applicable to options with a limited amount
of relevant points in time (i.e. times at which fixings, exercises or payoffs may happen).
Thus, American-style options cannot be evaluated. Furthermore, complete analytical
solutions are possible only in exceptional cases.
As an illustration of the power of the twin formalism we have implemented several
models, especially in connection with the so-called smile problem (Chapter 2). In
particular, we have discussed the plain Black-Scholes case (Sec. 6.1), we have added
a volatility jump (Sec. 6.2), and we have made calculations in a displaced diffusion
model (Sec. 6.3). We have shown how the volatility jump influences implied volatilities
and forward start call prices, and we have used the displaced diffusion model to fit
equity market data. Finally, we note that our framework enables us to avoid negative
prices due to displaced diffusion distributions in a natural way (Sec. 6.3.4).
7.2
Outlook
There are two ways in which one could build upon the work presented in this thesis: practical implementation and theoretical development. On the practical side
it may be worthwhile to apply the twin formalism to other markets (e.g. interest
rates) or to more realistic models. For instance, one could add stochastic volatility
(as parameter θ in the twin formalism) to the displaced diffusion model. Using these
more realistic models, one could try to predict the future risk neutral pdf for real
markets, and back test these predictions afterward. One interesting question that
could be addressed in this way is how time homogeneous market prices are. From
53
the theoretical point of view, it would be interesting to investigate in more detail
how the no-arbitrage conditions imposed by the twin formalism influence implied
volatility. For instance, one could investigate additional structures found in implied
volatility (as mentioned in Secs. 6.2.2 and 6.3.2). More generally, as suggested by
[Skiadopoulos and Hodges, 2002] it could be worthwhile to clarify the relation between mixtures (as used in the twin formalism) and the corresponding stochastic
processes. Finally, as a possible extension of the twin formalism one may try to allow
for several correlated parameters θ, or to try to generalize the twin formalism for the
application to American options.
54
Appendix A
The displaced diffusion distribution
Displaced diffusion is characterized by the distribution function
1
1
dd x x0 , σ 2 , γ ≡ √
2πσ (1 − γ) x + γ
 2 
(1 − γ) x + γ
2 2
 − log (1 − γ) x + γ + (1 − γ) σ /2 


0
exp 
.
2 2


2 (1 − γ) σ
Notice that
dd x x0 , σ 2 , γ = (1 − γ) logN (1 − γ) x + γ (1 − γ) x0 + γ, (1 − γ)2 σ 2 ,
(A.1)
and, consequently,
DD x x0 , σ 2 , γ = LogN (1 − γ) x + γ (1 − γ) x0 + γ, (1 − γ)2 σ 2 .
(A.2)
For γ = 0 this distribution is purely lognormal
"
#
2
2
1
1
−
[log
(x/x
)
+
σ
/2]
0
dd x x0 , σ 2 , 0 = √
exp
,
2σ 2
2πσ x
(A.3)
whereas it is normal for γ → 1 (cf. Fig. A.1)
"
#
2
1
−
(x
−
x
)
0
dd x x0 , σ 2 , 1 = √
exp
.
2
2σ
2πσ
(A.4)
Proof : Set
m (γ) ≡ log [(1 − γ) x0 + γ] − (1 − γ)2 σ 2 /2,
then
"
#
2
1
1
−
[log
[(1
−
γ)
x
+
γ]
−
m
(γ)]
dd x x0 , σ 2 , γ ≡ √
exp
.
2πσ (1 − γ) x + γ
2 (1 − γ)2 σ 2
55
dd(x\γ)
γ=0.0
γ=0.1
γ=0.5
γ=1.0
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
x
Figure A.1: From lognormal to normal: Displaced diffusion pdf dd(x|γ) for different
values of γ. The other parameter values are σ = 40% and x0 = 0.25.
By de L’Hôpital
− [log [(1 − γ) x + γ] − m (γ)]2
→
γ→1
(1 − γ)2
lim
[log [(1 − γ) x + γ] − m (γ)]
1−x
0
→
− m (γ)
(1 − γ)
(1 − γ) x + γ
[log [(1 − γ) x + γ] − m (γ)] (1 − x) [log [(1 − γ) x + γ] − m (γ)] m0 (γ)
→
−
1−γ
(1 − γ)2 x + γ − γ 2
(1 − x)
1−x
0
0
→
+ m (γ)
− m (γ)
−2 (1 − γ) x + 1 − 2γ
(1 − γ) x + γ
2
→ − [x − (1 − m0 (1))] .
Since
m0 (γ) =
1 − x0
+ (1 − γ) σ 2 ,
(1 − γ) x0 + γ
56
we obtain m0 (1) = 1 − x0 . Therefore,
"
#
2
1
−
(x
−
x
)
0
lim dd x x0 , σ 2 , γ = √
exp
.
γ→1
2σ 2
2πσ
dd[x |x0 , σ 2 , γ ] has mean x0 .
Proof: For γ = 1 this follows from Eq.(A.4). For γ < 1 use Eq.(A.1) and set
x
e ≡ (1 − γ) x + γ
Z
dx x dd x x0 , σ 2 , γ =
Z
= (1 − γ) dx x logN (1 − γ) x + γ (1 − γ) x0 + γ, (1 − γ)2 σ 2
Z
x
e−γ
=
dx
e
logN x
e (1 − γ) x0 + γ, (1 − γ)2 σ 2
1−γ
[(1 − γ) x0 + γ] − γ
=
1−γ
= x0 .
The value of a call for a displaced diffusion distribution is for γ < 1
h
h
i i
γ
γ
dd
2
e
e
C
x0 , K, σ , γ = x0 +
N d+ |0, 1 − K +
N d− |0, 1 , (A.5)
1−γ
1−γ
where
(1 − γ) x0 + γ
log
± (1 − γ)2 σ 2 /2
(1 − γ) K + γ
de± ≡
.
(1 − γ) σ
(A.6)
Proof: Use Eq.(A.1) with x
e ≡ (1 − γ) x+γ, and the Black-Scholes result Eq.(6.1)
to obtain
C
dd
2
x0 , K, σ , γ
Z
dx {x − K}+ dd x x0 , σ 2 , γ
+
Z
x
e−γ
=
de
x
−K
1−γ
logN x
e (1 − γ) x0 + γ, (1 − γ)2 σ 2
h
h
i i
γ
γ
e
e
=
x0 +
N d+ |0, 1 − K +
N d− |0, 1 .
1−γ
1−γ
=
57
Finally, we note without proof that the value of a call for a displaced diffusion
distribution is in the normal case (γ = 1)
C dd
#
"
2
x
−
K
σ
(x
−
K)
0
0
x0 , K, σ 2 , 1 = (x0 − K) N
|0, 1 + √ exp −
. (A.7)
σ
2σ 2
2π
58
Appendix B
Analytical solutions for M
If q and g are normal
q0,t (Xt |S0 , σq ) = n Xt S0 , σq2 ,
gt,T (XT |Xt , σt ) = n XT Xt , σt2 ,
then
Z
M0,t,T (ξ) =
dσt N ξ S0 , σt2 + σq2 k0,t (σt | σ0 ) .
(B.1)
If q and g are lognormal
q0,t (Xt |S0 , σq ) = logN Xt S0 , σq2 ,
gt,T (XT |Xt , σt ) = logN XT Xt , σt2 ,
then
Z
M0,t,T (ξ) =
dσt LogN ξ S0 , σt2 + σq2 k0,t (σt | σ0 ) .
(B.2)
If q and g are of displaced diffusion form with the same parameter γ
q0,t (Xt |S0 , σq ) = dd Xt S0 , σq2 , γ ,
gt,T (XT |Xt , σt ) = dd XT Xt , σt2 , γ ,
then
Z
M0,t,T (ξ) =
dσt DD ξ S0 , σt2 + σq2 , γ k0,t (σt | σ0 ) .
59
(B.3)
Proof: We start
Z
M0,t,T (ξ) ≡
Z
=
Z
=
by proving Eq.(B.3).
Z
dσt dXt Gt,T (ξ |Xt , σt ) q0,t (Xt |S0 , σq ) k0,t (σt | σ0 )
Z
dσt dXt DD ξ Xt , σt2 , γ dd Xt S0 , σq2 , γ k0,t (σt | σ0 )
Z
dσt dXt LogN (1 − γ) ξ + γ (1 − γ) Xt + γ, (1 − γ)2 σt2
(1 − γ) logN (1 − γ) Xt + γ (1 − γ) S0 + γ, (1 − γ)2 σq2
k0,t (σt | σ0 )
Z
Z
h i
h i
et LogN ξeX
et , σ
et Se0 , σ
=
dσt dX
et2 logN X
eq2 k0,t (σt | σ0 ) ,
where
et ≡ (1 − γ) Xt + γ
X
,
σ
et2 ≡ (1 − γ)2 σt2
σ
eq2 ≡ (1 − γ)2 σq2 ,
,
ξe ≡ (1 − γ) ξ + γ
,
Se0 ≡ (1 − γ) S0 + γ
and where we have used Eqs.(A.1) and (A.2). Thus,
M0,t,T
ξe
Z
1 1 1
et 1
√
(ξ) =
dσt
dx √
k0,t (σt | σ0 ) dX
et
2πe
σt x 2πe
σq
X
 h i2 
i2 h 2
2
e
e
e
eq /2 
et /2
log Xt /S0 + σ
 − log x/Xt + σ
−
exp 
.
2e
σt2
2e
σq2
Z
Z
We set
i2 h i2
h i2
h et + σ
et /Se0 + σ
et − B
et2 /2
eq2 /2
log x/X
log X
log X
D
−
−
=−
− .
2
2
2e
σt
2e
σq
2C
2
et we obtain
By expanding in orders of log X
h h ii
1
2
2
2
2
e0 − σ
σ
e
log
(x)
+
σ
e
/2
+
σ
e
log
S
e
/2
,
t
t
q
σ
et2 + σ
eq2 q
1
σ
e2 σ
e2 ,
C =
2
σ
et + σ
eq2 t q
h i2
1
2
2
e0 + σ
.
D =
log
x/
S
e
+
σ
e
q /2
t
σ
et2 + σ
eq2
B =
60
Therefore,
1 1 1
D √
√
(ξ) =
dσt
dx √
k0,t (σt | σ0 ) exp −
2πC
2
2πσt x 2πσq
Z
1
1
=
dσt k0,t (σt | σ0 ) √ q
2π σ
et2 + σ
eq2
 h i2 
2
2
Z ξe
e
et + σ
eq /2 
1
 − log x/S0 + σ
dx exp 

x
2 σ
et2 + σ
eq2
Z
h i
=
dσt LogN ξeSe0 , σ
et2 + σ
eq2 k0,t (σt | σ0 ) .
Z
M0,t,T
Z
ξe
Finally, we transform our variables back and use Eq.(A.2) again
Z
M0,t,T (ξ) =
dσt LogN (1 − γ) ξ + γ (1 − γ) S0 + γ, (1 − γ)2 σt2 + σq2
k0,t (σt | σ0 )
Z
=
dσt DD ξ S0 , σt2 + σq2 , γ k0,t (σt | σ0 ) ,
which proves Eq.(B.3).
For the proof of Eqs.(B.1) and (B.2) we set γ = 1 and γ = 0 in Eq.(B.3), respectively. 61
Appendix C
Proof of the Breeden Litzenberger
formula
If we reduce the strike K of a call we enhance the probability that the underlying
asset price S will be in the money at expiry T . Therefore, the higher the increase
in the probability, the higher should also be the increase in the call’s price C0,T . For
that reason we expect a relationship between the change of call prices as a function
of strike on the one side, and, on the other side, the probability that the underlying
asset price will hit this strike at expiry. This relationship is given by the Breeden
Litzenberger formula [Banz and Miller, 1978], [Breeden and Litzenberger, 1978]
∂ 2 C0,T (S0 , K, θ0 ) = f0,T (ST |S0 , θ0 ) .
∂K 2
K=ST
As always, θ0 symbolizes a possible dependence on further parameters.
Proof: Consider the value of a call
Z
C0,T (S0 , K, θ0 ) = ds {s − K}+ f0,T (s |S0 , θ0 ) .
Since
∂
{ST − K}+ = −H (ST − K) ,
∂K
∂2
{ST − K}+ = δ (ST − K) ,
∂K 2
with the Heaviside function H and the delta function δ, we obtain
Z
∂ 2 C0,T (S0 , K, θ0 ) =
ds δ (s − ST ) f0,T (s |S0 , θ0 )
∂K 2
K=ST
= f0,T (ST |S0 , θ0 ) .
Note that we did not need to make any assumptions about the underlying asset
price dynamics.
62
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