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2.1 Constructing Characteristic Functions (CFs) 27

framework to derive the pricing formulas for European-style options with a minimal

dependence upon the speciﬁcations of factors.

Summarizing the above results, we have the following principle for constructing

characteristic functions for option pricing using the unique risk-neutral measure Q.

Theorem 2.1.1 The Principle for Constructing Characteristic Functions 1: If the

risk-neutral processes of stock price and interest short rate are of form in (2.4)

and (2.3), respectively, and the drift and diffusion components of these processes

satisfy all technical conditions so that the solutions to (2.4) and (1.21) exist, then

the characteristic functions of the exercise probabilities F

and F

for a European-

style call option

C(S

,T;K)=S

−KB(0, T )F

have the expression of (2.14) and (2.15) respectively. This is independent of the

speciﬁcations of the stock price process and the short-rate process.

An enhanced principle for constructing CFs will be given in the following sec-

tion. In fact, one can extend the one-factor version of the short-rate process to a

multi-factor version without having any effect on the above results. This princi-

ple allow us to construct much more complicated and interesting variants of option

pricing formulas. Note that there are no particular constraints on the processes of

S(t), v(t) and r(t). Therefore, these factors can be speciﬁed rather arbitrarily. The

following chapters will show the effectiveness of this method in developing new

closed-form solutions.

In this manuscript, if not explicitly noted, expectations are always taken under the

measure Q. Obviously, the principle of constructing CFs is also valid for European-

style put options,

= KB(0,T)F

−S

(2.19)

with

−



∞



(

)

exp(−i

lnK)



, j = 1,2. (2.20)

Expressing a probability via its characteristic function is equivalent to expressing

a probability via a density function from many points of view. Firstly, both alterna-

tives involve a single integral (for the one-dimensional case), and hence need a sim-

ilar procedure to calculate the integral. Additionally, the one-to-one correspondence

between a characteristic function and its distribution guarantees a unique form of

the option pricing formula.

2.1.3 Special Case: FX Options

As argued by Garman and Kohlhagen (1983) in the Black-Scholes framework for

European-style FX options, since no arbitrage in two-country economics is ensured

by the interest rate parity, one needs a modiﬁcation for the exchange rate risk-neutral

process and therefore also for g

(t). We recall the risk-neutral process of exchange

28 2 Characteristic Functions in Option Pricing

rate S(t) given in (2.5),

dS(t)

S(t)

=[r(t) −r

∗

(t)]dt + v(t)dW

(t) (2.21)

with r(t) and r

∗

(t) as the domestic and foreign interest rates respectively. The pro-

cess of r(t) has been given before in (2.3). We may propose a similar process for

∗

(t) and assume that the solution for a foreign zero-coupon bond price

∗

(t, T )=E



exp(−



∗

(u)du)



exists. As indicated by the process for exchange rate in (2.21), the discounted risk-

neutral exchange rate S(t) by the money market account H(t) is no longer a martin-

gale. Instead by interest rate parity, S(t) is a martingale only if it is discounted by

the difference between domestic and foreign money market account, namely



S(t)exp(−



[r(u) −r

∗

(u)]du)



= E



S(t)

∗

(t)

H(t)



= 1

with

∗

(t)=exp





∗

(u)du



This gives us a hint to deﬁne a Radon-Nikodym derivative as follows:

∗

(t)=

S(t)H

∗

(t)

S(0)H(t)

. (2.22)

The FX options can then be valued similarly as the asset options,

C = E



exp



−



r(t)dt



(S(T) −K) ·1

(S(T)>K)



= S

∗



exp



−



∗

(t)dt



(x(T)>lnK)



−B(0,T )KE



(x(T)>lnK)



where in the ﬁrst term there is still the stochastic foreign discount factor. Therefore

we can not consider the ﬁrst term as a probability under measure Q

∗

, and face the

same situation as computing F

, this means, we need an additional foreign forward

measure to eliminate the stochastic foreign discount factor. To this end, we deﬁne

the following Radon-Nikodym derivative,

∗

(t)=

∗

(t, T )H

∗

(0)

∗

(0,T)H

∗

(t)

∗

(t, T )

∗

(0,T)H

∗

(t)

. (2.23)

Particularly,

∗

(T)=exp



−



∗

(t)dt



∗

(0,T)

2.2 Understanding Characteristic Functions 29

Applying this Radon-Nikodym derivative yields immediately

C = S

∗

(0,T)E



(x(T)>lnK)



−B(0,T)KE



(x(T)>lnK)



, (2.24)

which is exactly the formula we want to obtain, in order to construct two CFs un-

der appropriate measures. The ﬁnal change of the risk-neutral measure Q to the

new equivalent martingale measure Q

is already demonstrated by above two-step

change and given by

(t)=

∗

(t)=

S(t)B

∗

(t, T )

S(0)B

∗

(0,T)H(t)

. (2.25)

Its form at time T is a bit simpler

(T)=g

(T)=exp



−



r(u)du



S(T)

∗

(0,T)

, (2.26)

which can be used to compute CF f

(

) and the corresponding ﬁrst exercise prob-

ability F

for FX options. With F

keeping unchanged, the pricing formula for FX

call options can be generally expressed as

= S

∗

(0,T)F

−KB(0, T )F

that is an extension of the Garman and Kohlhagen formula in a general model setup.

2.2 Understanding Characteristic Functions

2.2.1 Properties of Characteristic Functions

The characteristic function of a random variable is essentially a Fourier transform

mapping the real random variable into a frequency domain. Therefore, CFs f

(

)

derived above share any property of Fourier transform, and also any property of

integral operator. We now brieﬂy examine some special features of CF with respect

to probability.

The ﬁrst useful property of CF lies in dealing with independent random vari-

ables. Let X

,··· ,X

, be some independent, but not necessarily identical random

variables, and Y

∑

j=1

is the sum of these variables, then the CF of Y

is just

the product of the CF of each X

, namely

f (

∏

j=1

f (

). (2.27)

Since X

, j = 1, 2, ··· ,n, are not necessarily identically distributed, it is convenient

to examine the joint distribution of X

using the joint CF f (

). As shown later,

30 2 Characteristic Functions in Option Pricing

if stock returns are modeled by a Brownian motion and a jump process whose in-

crements are governed by Poisson law, we can calculate the distribution of stock

returns by the joint CF which consists of the CF of Gaussian distribution and the CF

of Poisson distribution. This allows us to deal with each stochastic factor in an asset

process as a building block. From the point of view of engineering, each stochastic

factor can be regarded as module in a whole stochastic process. The idea of modular

pricing presented later in this book then utilizes this property of CF.

Now let y denote the inverse Fourier transform of CF f of a random variable x.

As mentioned above, y is then the corresponding density function, and equal to

y(x)=



f (

−i

There is a Fourier transform relation between y(x) and f(

). We denote this relation

with

f (

) ⇐⇒ y(x).

Some properties of the CF f (

) and its inverse transform y(x) can be given as fol-

lows:

• Linearity:

(

)+bf

(

) ⇐⇒ ay

(x)+by

(x).

• Multiplication:

(

) f

(

) ⇐⇒

∗y

)(x).

• Convolution:

( f

∗ f

)(

) ⇐⇒ y

(x)y

(x).

• Conjugation:

f (

) ⇐⇒ y(−x).

• Scaling:

f (a

) ⇐⇒

|a|

y(x/a), a ∈ R, a = 0.

• Time reversal:

f (−

) ⇐⇒ y(−x).

This feature can be considered as a special case of the scaling feature.

• Time shift:

f (

) ⇐⇒ y(−x).

Note that f

and f

here are two arbitrary CFs, not same as the ﬁrst and second CFs

given above in the context of option pricing.

The moments of a random variable can also be derived from its characteristic

function. Thus, some important statistical parameters such as variance, skewness

and kurtosis are automatically available if the characteristic function is known.

Let m

∗

(x(T)) denote the n-th order non-central moment of x(T), we can express

∗

(x(T)) with the CF f (

;x(T)) by the following equation:

2.2 Understanding Characteristic Functions 31

∗

(x(T)) = (−i)



∂

f (

;x(T))

∂φ



. (2.28)

It must be emphasized that the CF f (

;x(T)) for calculating the moments should

be derived from the original measure Q since we need the moments under actual

measure,

that is

f (

;x(T)) ≡ E

[exp(i

x(T))] . (2.29)

Hence f(

;x(T)) is generally not identical to f

(

) as given above in (2.13). In fact,

the moment-generating function of x(T) is deﬁned by

mg f (

)=E

[exp(

x(T))] (2.30)

for a positive real number

> 0. Obviously, a moment-generating function is

very similar to a CF, and there is an interchange between them by setting

= −i

= i

f (−i

)=mg f (

), mg f(i

)= f (

In this sense, the CF f (

) is equivalent to the moment-generating function, and

therefore can be applied to calculate the moments of X(T ).

To compute skewness and kurtosis, the ﬁrst four central moments m

(x(T))

should be known and can be given by a recursive scheme (Stuard and Ord, 1994):

= m

∗

−(m

∗

)

= m

∗

−3m

∗

+ 2(m

∗

)

, (2.31)

= m

∗

−4m

∗

+ 6m

∗

)

−3(m

∗

)

The skewness and kurtosis of x(T ) are then given by

skewness ≡

)

3/2

, kurtosis ≡

, (2.32)

respectively.

As seen in the Black-Scholes model, x(t) is Gaussian and therefore has a skew-

ness of zero and a kurtosis of three. It is empirically evident and also supported by

smile effect of implied volatilities that the actual distribution of stock returns x(t)

exhibits a negative skewness and an excess kurtosis. Therefore, in order to capture a

leptokurtic distribution and to recover smile effect, a reasonable stock price process

should imply

)

3/2

< 0,

> 3.

These two statistics may be checked in advance by employing CF under the risk-

neutral measure or the original statistical measure.

Strictly speaking, the moments are calculated under the risk-neutral measure Q. To obtain the

actual probability measure of stock returns, we must change the risk-neutral parameters to their

empirical counterparts.

32 2 Characteristic Functions in Option Pricing

2.2.2 Economic Interpretation of CFs

At ﬁrst glance, expressing the probabilities F

and F

by the Fourier inversion of CFs

(

) and f

(

) respectively is more technical than economic. In fact, an economic

interpretation of CFs could be exploited by its implicit spanning of state space. This

aspect gains signiﬁcance especially with respect to market completeness and Arrow-

Debreu securities in a state-space approach. Ross (1976) has proved that contingent

claims (options) enhances market efﬁciency in a state-space framework since creat-

ing options can complete (or span) markets in an uncertain world. Bakshi and Madan

(2000) demonstrated that spanning via options and spanning via CFs are completely

interchangeable. To see this, we note that the CF f (

;x(T)) ≡E

[exp(i

x(T))] un-

der the risk-neutral measure Q can be rewritten as follows:

f (

;x(T)) ≡ E

[exp(i

x(T))]

= E

[cos(

x(T)) + isin(

x(T))] . (2.33)

Now let us imagine two “contingent claims” of x(T) with payoff at time T :

cos(

x(T)) and sin(

x(T)).

Thus, from the point of view of valuation, f(x(T );

)

is a “security” consisting of two trigonometric assets of x(T ) while the stock price

is only an exponential asset of x(T ). To show the spanning equivalence via options

and via CFs, the following two equations hold:

cos(

x(T)) = 1 −



∞

cos(

lnK)max(0,x(T) −lnK)d lnK,

sin(

x(T)) = x(T) −



∞

sin(

lnK)max(0,x(T) −lnK)d lnK.

These two equations state that trigonometric functions can be expressed by options.

Hence, it follows that the spanned security space of options and their underlying is

the same one spanned by cos(

x(T)) and sin(

x(T)). We can consider two proba-

bilities F

and F

as Arrow-Debreu prices in the space spanned by options and the

underlying primitive assets; Analogously, CFs f

(

) and f

(

) may be interpreted

as the Arrow-Debreu prices in the transformed space.

Starting from this equivalence, we can price more general contingent claims on

x(T) by using CFs. As a useful feature of CFs, differentiation and translation of

CFs can simplify considerably the construction of primitive and derivative assets on

x(T). To expound this interesting feature of CF in the context of option pricing, we

deﬁne a discounted CF (also called the CF of the remaining uncertainty):

The contingent claim with payoff cos(

) or sin(

) is an “asset” with unlimited liability

since cos(

) and sin(

) can become negative.

It is well-known that normal Arrow-Debreu prices should (must) be positive and smaller than

one. Since the transformed spanned space is the complex plane, we can impose the usual L

-norm

on f

and f

. It follows immediately that

≤ 1and

≤ 1. Thus, f

and f

are well-deﬁned.

See Bakshi and Madan (2000).

2.2 Understanding Characteristic Functions 33

∗

(

;x(T)) = E



exp



−



r(t)dt



exp(i

x(T))



. (2.34)

Setting

= 0, we have

∗

(0;x(T)) = E



exp



−



r(t)dt



= B(0,T ). (2.35)

Thus the CF f

(

) in (2.13) can be expressed by

(

∗

(

;x(T))

∗

(0;x(T))

B(0,T)

∗

(

;x(T)). (2.36)

Note that

∗

(

−i;x(T )) = E



exp



−



r(t)dt



exp((1 + i

)x(T))



∗

(−i;x(T)) = E



exp



−



r(t)dt



exp(x(T))



= S

we immediately obtain an alternative expression for f

(

∗

(

−i;x(T ))

∗

(−i;x(T))

∗

(

−i;x(T ))

, (2.37)

where x(T )=lnS(T ). As indicated in (2.28), we can construct the moments of the

random variable x(T) by partially differentiating the CF with respect to

.Inthe

same manner, partially differentiating the discounted CF with respect to

yields

the present value of the moments of x(T). Generally, we have the following result

on the relationship between the n-th order polynomial of x(T) and the n-th order

partial differential of f

∗

(

;x(T)),

(−i)



∂

f (

;x(T))

∂φ



= E



exp



−



r(t)dt



(T) exp(i

x(T))



= E



exp



−



r(t)dt



(T)



=(−i)

∗

(0;x(T)). (2.38)

It follows

x(T)

= E



exp



−



r(t)dt



x(T)



∗

(0;x(T)), (2.39)

where H

x(T)

is the present value of x(T). Note that a real payoff function L(x(T)) ∈

∞

can be sufﬁciently approximated by a (Taylor-) polynomial series, and hence

the expected discounted price of L(x(T)) can be approximated by a series of

∗

(0;x(T)). This approach states that f

∗

(0;x(T)) forms a polynomial basis for

34 2 Characteristic Functions in Option Pricing

the market of contingent claims on x(T). Without loss of generality, we denote the

spot price or the forward price of the contingent claim L(x(T)) by L

and expand

L(x(T)) around zero,

G(T,

)=E



exp



−



r(t)dt



L(x(T)) exp(i

x(T))



= f

∗

(

;x(T))L(0)+

∞

∑

k=1

∗

(

;x(T))

(i)

∂

(0) (2.40)

and

G(T,0)=L

= E



exp



−



r(t)dt



L(x(T))



= f

∗

(0;x(T))L(0)+

∞

∑

k=1

∗

(0;x(T))

(i)

∂

(0). (2.41)

In many cases, for example, in the case of stock options, the spot price L

, i.e.,

G(T,0) is directly observable and does not have to be derived from the CFs. G(T,0)

plays an important role in valuation only for the case where no spot prices or for-

ward prices of the contingent claim L(x(T)) are quoted in the markets. Using L

numeraire, we can construct the ﬁrst CF,

(

)=E



exp



−



r(t)dt



L(x(T))

exp(i

x(T))



G(T,

)

G(T,0)

. (2.42)

To guarantee f

(

) to be a well-deﬁned CF, a sufﬁcient and necessary condition

must be satisﬁed



exp



−



r(t)dt



L(x(T))



= 1. (2.43)

Equation (2.43) implies that L(x) is an arbitrage-free process and admits no free-

lunch. The construction of CF f

(

) for pricing general contingent claims follows

a similar way. Therefore, we are able to derive closed-form valuation formulas for

a call on particular contingent claims in an uniﬁed framework. A simple example

can be an option on interest rates if we specify x(t) as r(t). Summing up, we have a

more general principle for constructing CFs:

Theorem 2.2.1 The Principle for Constructing Characteristic Functions 2: Assume

a (primitive) asset return process x(t) with x(0)=0 follows an It

o process and satis-

ﬁes the usual necessary technical conditions. In the market there is a basic security

L(x) ∈ C

∞

on this return, which is strictly positive and invertible. Furthermore, the

arbitrage-free condition (2.43) is satisﬁed. Then a European-style call option C on

L(x) with a strike price K and a maturity time of T can be valued by

This principle is essentially identical to the case 3 in Bakshi and Madan (2000).

2.2 Understanding Characteristic Functions 35

C(x

,T;K)=L

−KB(0, T )F

with

−



∞



(

)

exp(−i

−1

(K))



, j = 1,2,

where

= G(T, 0), B(0,T)= f

∗

(0;x(T)),

(

G(T,

)

G(T,0)

, f

(

∗

(

;x(T))

∗

(0;x(T))

G(T,

)= f

∗

(

;x(T))L(0)+

∞

∑

k=1

∗

(

;x(T))

(i)

∂

(0).

This generalized principle for constructing CFs is useful in deriving a tractable

valuation formula for options on securities with an unconventional payoff function

L(x) and provides us with a structural and amenable tool to analyze quantitative

effects of uncertainty in economics and ﬁnance.

2.2.3 Examination of Existing Option Models

Since Black and Scholes (1973) derived their celebrated option pricing formula, an

enormous number of variants of option pricing models have been developed. Mer-

ton (1973) examined an option valuation problem with stochastic interest rates using

the assumption that the zero-coupon bond price is lognormally distributed. Geske

(1979) gave a pricing formula for compound options. Merton (1976), Cox and Ross

(1976) studied issues of option valuation where the underlying asset price follows

a jump process and introduced for the ﬁrst time a discontinuous price process in

option pricing theory. Since the middle of 1980s, valuation with stochastic volatil-

ities has drawn more and more the attention of ﬁnancial economists. The models

of Stein and Stein (1990), Heston (1993), BCC (1997), as well as Sch

obel and Zhu

(1999) are the most inﬂuential works on modeling stochastic volatility. Recently,

pure jumps characterized by L

evy process are introduced to describe leptokurtic

features of stock returns in the hope of recovering the smile of implied volatility.

The representatives of option pricing models with L

evy process are the variance-

Gamma model of Madan, Carr and Chang (1998), and the normal inverse Gaussian

model of Barndorff-Nielsen (1998). Here, we give a brief overview of some impor-

tant closed-form pricing solutions for options under the risk-neutral measure using

the Fourier transform techniques. The detailed discussions for these models will be

given in the following chapters.

(1). The Black-Scholes Model (1973). As mentioned in Chapter 1, it includes

only one stochastic process

36 2 Characteristic Functions in Option Pricing

dS(t)=rS(t)dt +

S(t)dW(t).

Let x(t)=lnS(t),

dx(t)=(r −

)dt +

dW(t). (2.44)

Hence,

(T)=exp(−rT +x(T ) −x

), g

(T)=1.

The two CFs can be calculated as follows:

(

)=E



(T)e

x(T)



= E

[exp((1+i

)x(T) −rT −x

]

= exp



(rT + x

) −

1 +i

T +

(1+i

)



= exp





rT + x



−



= exp(i

+ rT)) f

(

) (2.45)

with

(

)=exp







−



. (2.46)

and

(

)=E



(T)e

x(T)



= E

[expi

x(T)]

= exp





rT + x

−



−



= exp(i

+ rT)) f

(

) (2.47)

with

= exp





−



−



. (2.48)

It is not hard to verify that

N(d



∞



(

)

exp(−i

lnK)



, j = 1,2.

N(d

) are the usual probability terms in the Black-Scholes formula.

Thus, we have

shown that the Black-Scholes formula can be easily derived by using the Fourier

inversion.

Since the CF of a normal density function n(

) is f (

)=exp



φμ

−



, two proba-

bilities N(d

), j = 1, 2, in the Black-Scholes formula are Gaussian with cumulative distribution

functions N(rT + x

T) respectively.