Zhu J. Applications of Fourier Transform to Smile Modeling: Theory and Implementation

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2.2 Understanding Characteristic Functions 37

(2). The Heston Model (1993). This model is characterized by the following two

processes:

dS(t)

S(t)

= r(t)dt +



V(t)dW

(t)

or (2.49)

dx(t)=[r(t) −

V(t)]dt +



V(t)dW

(t),

dV(t)=[

κθ

−(

)V(t)]dt +



V(t)dW

(t)

with

(t)dW

(t)=

dt.

Here the squared volatilities follow a mean-reverting square-root process which was

used in ﬁnance for the ﬁrst time by CIR (1985b) to specify stochastic interest rates.

Hence, we have

(T)=exp(−rT +x(T ) −x

), g

(T)=1.

The detailed derivation of this pricing formula for options will be given in the next

chapter.

(3). The model of Bakshi, Cao and Chen (1997). This model includes stochastic

volatility, stochastic interest rates and random jumps where the ﬁrst two factors are

speciﬁed by a mean-reverting square root process. Random jumps are expressed by

Poisson process with lognormally distributed jump sizes.

dS(t)

S(t)

=[r(t) −

λμ

]dt +



V(t)dW

(t)+J(t)dY(t)

dx(t)=[r(t) −

λμ

−

V(t)]dt +



V(t)dW

(t)+ln[1 + J(t)]dY(t),

(2.50)

dV(t)=

[

−V(t)]dt +



V(t)dW

(t),

dr(t)=

[

−r(t)]dt +



r(t)dW

(t)

with

(t)dW

(t)=

dt,

ln[1+J(t)] ∼ N(ln[1 +

] −

dY(t) ∼

dt +

(1−

)dt,

where

is an indicator function for the value n. Here the stock price process in-

cludes a jump component J(t)dq(t). Since the jump has no impact on the It

o process,

after taking the market price of the jump risk

λμ

into account, we can construct the

38 2 Characteristic Functions in Option Pricing

same g

(t), j = 1, 2, as the ones without jumps. So we have

(T)=exp



−



r(t)dt + x(T) −x



(T)=exp



−



r(t)dt



/B(0,T).

We will show that it is very convenient to derive the option pricing formula in this

complicated model by applying the idea of modular pricing.

Scott (1997) devel-

oped a similar model incorporating stochastic interest rates and jumps.

(4). The Bates (1996) model for currency option. This model differs only slightly

from the above one due to a different process for the underlying asset. Let S(t)

denote the exchange rate,

dS(t)

S(t)

=[r −r

∗

−

λμ

]dt +



V(t)dW

(t)+J(t)dY(t)

dx(t)=[r −r

∗

−

λμ

−

V(t)]dt +



V(t)dW

(t)+ln[1 + J(t)]dY(t),

(2.51)

dV(t)=[

−

V(t)]dt +



V(t)dW

(t),

with

(t)dW

(t)=

dt,

ln[1+J(t)] ∼ N(ln[1 +

] −

dY(t) ∼

dt +

(1−

)dt.

Thus it follows

(T)=exp(−(r −r

∗

)T + x(T) −x

), g

(T)=1.

The above models are some representatives in the vast option pricing literature.

All of these models (except for the Black-Scholes model) have some common prop-

erties: First of all, stock prices (underlying assets) are characterized only by a Brow-

nian motion with a correlated process for variance. Squared volatility (variance) fol-

lows a square-root process. Secondly, the process of short rates is one-dimensional

and also follows a square root process. As shown later, a square root process allows

for a nice derivation of characteristic functions, but, obviously, it is not the only

way to describe volatilities and interest rates. The Ornstein-Uhlenbeck process, for

example, may be an appealing alternative (Vasicek (1977), Stein and Stein (1991),

Sch

obel and Zhu (1999)). Furthermore, we can extend the one-dimensional stock

The original derivation of the option pricing formula given by Bakshi, Cao and Chen is based on

the PDEs, and rather complicated.

2.2 Understanding Characteristic Functions 39

price process to a multi-dimensional case. We will face such multi-dimensional

problems in pricing basket options and correlation options.

2.2.4 Relationship between CF to PDE

In this section, we will show that there is a relationship between the Fourier trans-

form approach and the partial differential equation (PDE) approach to asset pricing.

As shown in Chapter 1, a PDE called the Black-Scholes equation is established to

derive option pricing formula, and can be applied to value any derivative within the

Black-Scholes framework if the corresponding boundary and terminal conditions

are imposed. In fact, the original works of Heston (1993), Bates (1996), BCC (1997)

are based on a multi-dimensional PDE-approach with which particular option pric-

ing formula may be given. However, a closed-form solution to a multi-dimensional

PDE is generally not trivial, and we can not directly utilize some useful results of

stochastic calculus if a PDE is applied to value options. Now we take stochastic

volatility models, particularly the Heston model and the the BCC model, as ex-

amples to demonstrate the relationship between the CF and the PDE by applying

Kolmogorov’s backward equation.

Given y as a n-dimensional It

o process and a function h ∈ C(R

), then the Kol-

mogorov’s backward equation states:

∂

∂τ

= K u for u = E

[h(

,y)],

 0, y ∈ R

with the boundary condition

u(0,a)=h(0,a), y(0)=a ∈ R

and an operator K deﬁned by

K =

∑

drift

∂

∑

i, j

cov

∂

Now denote

= T for the notation consistence. We have already known that CFs

(

)=E

[exp(i

x(T))] and f

(

)=E

[exp(i

x(T))] are the CFs associated

with the probability measures Q

and Q

respectively, and both twice differentiable,

therefore belong to the function class C(R

). Furthermore, assume f

(

) and f

(

)

have the following functional form

(

)= f

= lnS

,T;

), a =[x

]

and

(

)= f

= lnS

,T;

), a =[x

K is the generator of an It

o diffusion.

40 2 Characteristic Functions in Option Pricing

Applying the Girsanov theorem given in Section 2.1, x(t) in the Heston model under

the measures Q

follows a process

dx(t)=



r +

V(t)



dt +



V(t)dW

and the stochastic variance V(t) follows a mean-reverting square root process

dV(t)=



[

−V(t)] +

ρσ

V(t)



dt +



V(t)dW

In this model setup, we have then y = {x(t),V(t)}. Hence, a direct application of

the operator K yields

K f

∂

ρσ

∂



r +



∂

(

−V)+

ρσ

∂

As a result, we obtain the corresponding Kolmogorov’s backward equation under

the measure Q

∂

ρσ

∂



r +



∂

(

−V)+

ρσ

∂

, (2.52)

with which the CF f

(

) may be derived for initial condition.

If the risk-free interest rate r is constant as in the Heston model, we have shown

that Q

is identical to Q, this means that x(t) and V(t) in the Heston model under

the measures Q

are governed by the original risk-neutral processes respectively

dx(t)=



r −

V(t)



dt +



V(t)dW

and

dV(t)=

[

−V(t)]dt +



V(t)dW

It follows the corresponding Kolmogorov’s backward equation under the mea-

sure Q

∂

ρσ

∂



r −



∂

(

−V)

∂

. (2.53)

This is the PDE to solve the CF f

(

Of course, if the instantaneous variance V(t) is constant, the Heston model re-

duces to the Black-Scholes model, the CFs f

(

), j = 1,2, will be simpliﬁed to

2.2 Understanding Characteristic Functions 41

(

)= f

,T;

), a =[x

It is not hard to obtain the similar Black-Scholes equations for f

(

In the above mentioned BCC model where the instantaneous interest rate r(t) is

speciﬁed to follow a mean-reverting square root process, we shall extend f

(

) by

(

)= f

), a =[x

]

and

(

)= f

,T;

), a =[x

Since r(t) is uncorrelated with x(t) in the BCC model, the change of measure Q to

does not affect the process of r(t), namely

dr(t)=

[

−r(t)]dt +



r(t)dW

(t). (2.54)

However, in this case, the change of measure Q to Q

will change also the drift term

of the process of r(t). More precisely, under the forward T-measure Q

,wehavean

adjusted process for r(t),

dr(t)=[

(

−r(t)) +

b(t, T )]dt +



r(t)dW

(t) (2.55)

with an exponential form for a zero-coupon price B(t, T ),

B(t, T )=e

a(t,T)+b(t,T )r(t)

Based on these two different processes under two different measures, we can estab-

lish the corresponding PDEs for f

(

) in the BCC model as follows:

∂

ρσ

∂



r +



∂

(

−V)+

ρσ

∂

(

−r)

∂

(2.56)

and

∂

ρσ

∂



r +



∂

(

−V)

∂

(

−r)+

b(t, T )]

∂

. (2.57)

Therefore in the sense of Kolmogorov’s backward equation, an equivalent re-

lationship between the Fourier transform approach and the PDE approach can be

42 2 Characteristic Functions in Option Pricing

established. If a process under a particular measure is given, then the PDE for its CF

is just the corresponding Kolmogorov’s backward equation.

2.2.5 Advantages of CF and Modular Pricing

In general, when we compare the Fourier transform approach with the traditional

pricing method, a natural question arises: what are the advantages of the Fourier

transform approach by using the CF to value European-style options? The answer

to this question is at least fourfold:

1. The ﬁrst gain is the tractability and easy manipulation of CFs which allows us

to arrive at closed-form solutions in terms of CFs. In contrast, by applying the

PDE approach it is difﬁcult to make variable transformations to get closed-form

pricing solutions. As shown above, due to the equivalence between the Fourier

transform approach and the PDE approach, PDEs that are relevant to option pric-

ing can be solved, at least numerically. However, if our focus is to derive the

closed-form pricing solution for options, numerical solutions are no longer use-

ful. In many cases, it is not easy to get a “right” guess for an analytical solution

for CFs by using the PDE approach as in the Heston model. For example, with

the volatility speciﬁed as a mean-reverting Ornstein-Uhlenbeck process, we will

encounter problems to ﬁnd analytical solutions if the PDE method is directly

applied. In contrast, if we ﬁrst calculate and manipulate the CFs to reduce di-

mensionality, then apply the PDEs as done in Scott (1997), Sch

obel and Zhu

(1999), we can indeed arrive at closed-form solutions for the CFs. As shown in

the following chapters, a direct calculation of the CF may be applied broadly

to almost all option pricing models with few restrictions, even in the present of

jump components. Therefore, one can sometimes only solve a PDE numerically

although a closed-form solution for the CF does exist.

2. Secondly, as demonstrated by the Heston model, the Fourier transform method

provides us with a powerful tool to give a new form of option pricing formula.

In fact, Fourier transform is often the only elegant and compact way to deal with

other stochastic volatility models, as shown in Lewis (2000) and Zhu (2000).

In a similar manner, stochastic interest rates can be incorporated into an option

pricing model conveniently via CFs, as shown in BCC (1997) and, Dai and Sin-

gleton (2000). As shown later, the pricing formulas both for stochastic volatility

models and for stochastic interest rate models share a common pricing kernel as

long as they are speciﬁed to follow the same process. This point becomes espe-

cially more obvious by using Fourier analysis and was discussed in Madan and

Bakshi (2000), Goldstein (1997), as well as Carr and Wu (2004). Consequently,

we could incorporate the dynamics of interest rates into an option pricing model

more ﬂexibly and efﬁciently in the framework of Fourier transform. In fact, al-

most new researches on option pricing since 2000 are based on the technique

of Fourier transform. The broad applications of Fourier transform revolutionized

the valuation theory and practice.

2.2 Understanding Characteristic Functions 43

3. Thirdly, CFs are always continuous in their domain even for a discontinuous dis-

tribution. This property is of great advantage when we consider to model jumps

in an asset price process. The classical approach to dealing with discontinuous

phonomania in stock price movements is to describe jump frequency with Pois-

son process and to describe jump size with some independent distributions. Ad-

ditionally, more general jumps characterized by L

evy measures, for example, the

jumps driven by a variance-gamma process (Madan, Carr and Chang (1999)),

and a normal inverse Gaussian process (Barndorff-Nielsen (1998)), are applied

to capture the dynamics of stock prices. We will show that CF is also a con-

venient and elegant way to arrive at closed-form option pricing formulas in all

theses jumps models. Indeed, in these so-called L

evy jump models, there exists

no general PDE for the relevant CF.

4. Finally, Fourier transform allows us to build up a ﬂexible option pricing frame-

work wherein each stochastic factor can be considered as a module. The idea of

this building-block approach is referred to as the modular pricing of options in

this book. In the light of the property of CF given by (2.27), the CFs in a com-

prehensive option pricing model are simply composed of the CFs of independent

stochastic factors. Roughly and intuitively, we have

f (

)= f

(

) f

(

) f

Jump

(

where SV , SI and Jump represent the stochastic volatility, the stochastic inter-

est rate and the jumps respectively, which again could be speciﬁed by various

stochastic processes. Hence, the Fourier transform enables us to develop a num-

ber of option pricing models by simply combining different stochastic factors.

In this sense, stochastic volatilities, stochastic interest rates and random jumps

work as modules, and can be withdrawn from or inserted into a pricing formula

by assembling the corresponding CFs. However, all pricing formulas for options

have a common shell: the Fourier inversion given in (2.18).

From the point of view of ﬁnancial engineering, the immediate gain of this mod-

ular concept is its convenient implementation. For example, we can deﬁne three

basic classes for stochastic volatilities, stochastic interest rates and jumps. The

particular speciﬁcation of these stochastic factor may be implemented in the de-

rived classes. Therefore, the implementation of a special new pricing model is

just a simple assemble of three concrete classes where the corresponding CFs are

implemented.

Chapter 3

Stochastic Volatility Models

In this chapter we discuss stochastic volatility models that play a crucial role in

smile modeling not only on the theoretical side, but also on the practical side.

Stochastic volatility models are extensively examined by ﬁnancial researchers, and

widely used in investment banks and ﬁnancial institutions. In many trading ﬂoors,

this type of option pricing model has replaced the Black-Scholes model to be the

standard pricing engine, especially for exotic derivatives. Intuitively, it is a natural

way to capture the volatility smile by assuming that volatility follows a stochas-

tic process. Simple observations propose that the stochastic process for volatility

should be stationary with some possible features such as mean-reverting, correla-

tion with stock dynamics. In this sense, a mean-reverting square root process and

a mean-reverting Ornstein-Uhlenbeck process adopted from the interest rate mod-

eling are two ideal candidate processes for stochastic volatilities. Heston (1993)

speciﬁed stochastic variances with a mean-reverting square root process and de-

rived a pioneering pricing formula for options by using CFs. Stochastic volatility

model with a mean-reverting Ornstein-Uhlenbeck process is examined by many re-

searchers. Sch

obel and Zhu (1999) extended Stein and Stein’s (1991) solution for

a zero-correlation case to a general non-zero correlation case. In this chapter, we

will expound the basic skills to search for a closed-form formula for options in a

stochastic volatility model. Generally, we have two approaches available: the PDE

approach and the expectation approach, both are linked via the Feynman-Kac theo-

rem. However, as demonstrated in following, the expectation approach utilizes the

techniques of stochastic calculus and is then an easier and more effective way. To

provide an alternative stochastic volatility model, we also consider a model with

stochastic variances speciﬁed as a mean-reverting double square root process.

3.1 Introduction

Since the Black-Scholes formula was derived, a number of empirical studies have

concluded that the assumption of constant volatility is inadequate to describe stock

J. Zhu, Applications of Fourier Transform to Smile Modeling, Springer Finance,

DOI 10.1007/978-3-642-01808-4

 Springer-Verlag Berlin Heidelberg 2000, 2010

46 3 Stochastic Volatility Models

returns, based on two ﬁndings: (1) volatilities of stock returns vary over time,

but persist in a certain level (mean-reversion property). This ﬁnding can be traced

back to the empirical works of Mandelbrot (1963), Fama (1965) and Blattberg and

Gonedes (1974) with the results that the distributions of stock returns are more lep-

tokurtic than normal; (2) volatilities are correlated with stock returns, and more

precisely, they are usually inversely correlated. Furthermore, volatility smile pro-

vides a direct evident for inconsistent volatility pattern with moneyness in the Black-

Scholes model. Black (1976a), Schmalensee and Trippi (1978), Beckers (1980) in-

vestigated the time-series property between stock returns and volatilities, and found

an imperfect negative correlation. Bakshi, Cao and Chen (BCC, 1997) and Nandi

(1998) also reported a negative correlation between the implied volatilities and stock

returns. Moreover, Beckers (1983), Pozerba and Summers (1984) gave evidence that

shocks to volatility persist and have a great impact on option prices, but tend to

decay over time. These uncovered properties associated with volatility such as lep-

tokurtic distribution, correlation, mean-reversion and persistence of shocks, should

be considered in a suitable option pricing model.

In order to model the variability of volatility and to capture the volatility smile,

several approaches have been suggested. One approach was the so-called constant

elasticity of variance diffusion model developed by Cox (1975), and may be re-

garded as a representative of a more general model class labeled to local volatility

model. Cox assumed that volatility is a function of the stock price with the following

form:

v(S(t)) = aS(t)

−1

, with a > 0, 0 

 1. (3.1)

Since v(S(t)) is a decreasing function of S(t), volatilities are inversely correlated

with stock returns. However, this deterministic function can not describe other de-

sired features of volatility. Derman and Kani (1994), Dupire (1994) and Rubinstein

(1994) hypothesized that volatility is a deterministic function of the stock price and

time, and developed a deterministic volatility function (DVF), with which they at-

tempt to ﬁt the observed cross-section of option prices exactly. This approach, as

reported by Dumas, Fleming and Whaley (1998), does not perform better than an

ad hoc procedure that merely smooths implied volatilities across strike prices and

times to maturity.

A more general approach is to model volatility by a diffusion process and has

been examined Johnson and Shanno (1987), Wiggins (1987), Scott (1987), Hull

and White (1987), Stein and Stein (1991), Heston (1993), Sch

obel and Zhu (1999)

and Lewis (2000). The models following this approach are the so-called stochastic

volatility models. Table (3.1) gives an overview of some representatives of stochas-

tic volatility models.

There are no known closed-form option pricing formulas in the case of non-zero

correlation between volatilities and stock returns for all models listed in Table 3.1

except for model (5) [Sch

obel and Zhu, 1999], model (6) [Heston, 1993] as well as

model (7). Models (1) and (3) perform no mean-reversion property and therefore can

not capture the effects of shocks to volatility in the valuation of options. Models (1),

(2), (3) (4) and (8) are not stationary processes and violate the feature of stationarity

3.1 Introduction 47

(1): dv(t)=

vdt +

vdW(t) Johnson/Shanno (1987)

(2): d lnv(t)=

(

−lnv)dt +

dW(t) Wiggins (1987)

(3): dv(t)

dt +

dW(t) Hull/White (1987)

(4): dv(t)

(

−v)dt +

dW(t) Hull/White (1987)

(5): dv(t)=

(

−v)dt +

dW(t) Stein and Stein (1991),

Sch

obel/Zhu (1999)

(6): dV(t)=

(

−V (t))dt +



V(t)dW(t) Heston (1993)

(7): dV(t)=

(

−



V(t) −

V(t))dt +



V(t)dW(t) suggested in this book

(8): dV(t)=

(

−V

(t))dt +

V(t)

3/2

dW(t) Lewis (2000)

Table 3.1 Overview of one-factor stochastic volatility models.

of volatility or variance. The so-called 3/2 model of Lewis (2000) is unbounded,

and as a result, option prices under the 3/2 model are not martingales, but only local

martingales, therefore not appropriate for practical applications. Consequently, only

models (5) and (6) are reasonable and worthwhile to study in details.

As shown in Hull and White (1987), Ball and Roma (1994), if stock returns are

uncorrelated with volatilities, the option price can be approximated very accurately

by replacing the variance in the Black-Scholes formula by the expected average

variance which is deﬁned by

AV = E





v(t)



. (3.2)

The mathematical background for this replacement is that the conditional distribu-

tion of terminal stock prices, given the expected average variance, is still lognormal.

However, this argument fails in the presence of the correlation between stock re-

turns and volatilities. Heston (1993) showed that the correlation between volatility

and spot returns is necessary to create negative skewness and excess kurtosis in a

leptokurtic distribution of spot returns. BCC (1997), Heston and Nandi (1997), and

Nandi (1998) demonstrated that non-zero correlation is important in eliminating the

smile effect of implied volatilities and leads to signiﬁcant improvements both in

pricing and in hedging options.

Some econometric models also attract attention when studying stochastic volatil-

ity. In fact, each model from (1) to (7) in Table (3.1) may correspond to an economet-

ric counterpart by discretizating them on time points, and some could be referred to

as the so-called autoregressive random variance models (ARV). Nelson and Foster

(1994), Duan (1996) showed that some existing stochastic volatility models can be

considered as the week limits of some generalized autoregressive conditional het-

eroscedasticity (GARCH) models.

The discrete-time versions of stochastic mod-

els play also an important role in empirical tests. For instances, Heston and Nandi

(1997) suggested a GARCH option pricing model and derived a closed-form so-

The crucial difference between ARV models and GRACH models is that ARV models include

two innovations to generate data while GRACH models have only one innovation with conditional

variances dependent upon the past information.