E–cient Options Pricing Using the Fast Fourier Transformmaykwok/piblications/FFT_paperx.pdf ·...

Efficient Options Pricing Using the FastFourier Transform

Yue Kuen Kwok1, Kwai Sun Leung2, and Hoi Ying Wong3

1 Hong Kong University of Science and Technology, Hong Kong [email protected] The Chinese University of Hong Kong, Hong Kong [email protected] The Chinese University of Hong Kong, Hong Kong [email protected]

1 Introduction

The earliest option pricing models originated by Black and Scholes (1973) andMerton (1973) use the Geometric Brownian process to model the underlyingasset price process. However, it is well known among market practitioners thatthe lognormal assumption of asset price returns suffers from serious deficien-cies that give rise to inconsistencies as exhibited by smiles (skewness) and termstructures in observed implied volatilities. The earlier remedy to resolve thesedeficiencies is the assumption of state and time dependence of the volatility ofthe asset price process (see Derman and Kani (1998) and Dupire (1994)). Onthe other hand, some researchers recognize the volatility of asset returns as ahidden stochastic process, which may also undergo regime change. Examplesof these pioneering works on stochastic volatility models are reported by Steinand Stein (1991), Heston (1993), and Naik (2000). Starting from the seminarpaper by Merton (1973), jumps are introduced into the asset price processesin option pricing. More recently, researchers focus on option pricing modelswhose underlying asset price processes are the Levy processes (see Cont andTankov (2004) and Jackson et al. (2008)).

In general, the nice analytic tractability in option pricing as exhibitedby Black-Scholes-Merton’s Geometric Brownian process assumption cannotbe carried over to pricing models that assume stochastic volatility and Levyprocesses for the asset returns. Stein and Stein (1991) and Heston (1993)manage to obtain an analytic representation of the European option pricefunction in the Fourier domain. Duffie et al. (2000) propose transform methodsfor pricing European options under the affine jump-diffusion processes. Fouriertransform methods are shown to be an effective approach to pricing an optionwhose underlying asset price process is a Levy process. Instead of applying thedirect discounted expectation approach of computing the expectation integralthat involves the product of the terminal payoff and the density function ofthe Levy process, it may be easier to compute the integral of their Fouriertransform since the characteristic function (Fourier transform of the density

2 Yue Kuen Kwok, Kwai Sun Leung, and Hoi Ying Wong

function) of the Levy process is easier to be handled than the density functionitself. Actually, one may choose a Levy process by specifying the characteristicfunction since the Levy-Khinchine formula allows a Levy process to be fullydescribed by the characteristic function.

In this chapter, we demonstrate the effective use of the Fourier transformapproach as an effective tool in pricing options. Together with the Fast Fouriertransform (FFT) algorithms, real time option pricing can be delivered. Theunderlying asset price process as modeled by a Levy process can allow for moregeneral realistic structure of asset returns, say, excess kurtosis and stochasticvolatility. With the characteristic function of the risk neutral density beingknown analytically, the analytic expression for the Fourier transform of theoption value can be derived. Option prices across the whole spectrum of strikescan be obtained by performing Fourier inversion transform via the efficientFFT algorithms.

This chapter is organized as follows. In the next section, the mathemati-cal formulations for building the bridge that links the Fourier methods withoption pricing are discussed. We first provide a brief discussion on Fouriertransform and FFT algorithms. Some of the important properties of Fouriertransform, like the Parseval relation, are presented. We also present the defi-nition of a Levy process and the statement of the Levy-Khintchine formula. InSec. 3, we derive the Fourier representation of the European call option pricefunction. The Fourier inversion integrals in the option price formula can beassociated with cumulative distribution functions, similar to the Black-Scholestype representation. However, due to the presence of a singularity arising fromnon-differentiability in the option payoff function, the Fourier inversion inte-grals cannot be evaluated by applying the FFT algorithms. We then presentvarious modifications of the Fourier integral representation of the option priceusing the damped option price method and time value method (see Carr andMadan (1999)). Details of the FFT implementation of performing the Fourierinversion in option valuation are illustrated. In Sec. 4, we consider the exten-sion of the FFT techniques for pricing multi-asset options. Unlike the finitedifference approach or the lattice tree methods, the FFT approach does notsuffer from the curse of dimensionality of the option models with regard to anincrease in the number of risk factors in defining the asset return distribution(see Dempster and Hong (2000) and Hurd and Zhou (2009)). In Sec. 5, weshow how one can price Bermudan style options under Levy processes usingthe FFT techniques by reformulating the risk neutral valuation formulationas a convolution. We show how the property of the Fourier transform of a con-volution product can be effectively applied in pricing a Bermudan option (seeLord et al. (2008)). In Sec. 6, we illustrate an innovative FFT-based networkapproach for pricing options under Levy processes by extending the finitestate Markov chain approach in option pricing. Similar to the forward shoot-ing grid technique in the usual lattice tree algorithms, the approach can beadapted to valuation of options with exotic path dependence (see Wong andGuan (2009)). In Sec. 7, we derive the partial integral-differential equation

Efficient Options Pricing Using the Fast Fourier Transform 3

formulation that governs option prices under the Levy process assumption ofasset returns. We then show how to apply the Fourier space time steppingtechniques that solve the partial differential-integral equation for option pric-ing under Levy processes. This versatile approach can handle various forms ofpath dependence of the asset price process and features / constraints in theoption models (see Jackson et al. (2008)). We present summary and conclusiveremarks in the last section.

2 Mathematical Preliminaries on Fourier TransformMethods and Levy Processes

Fourier transform methods have been widely used to solve problems in mathe-matics and physical sciences. In recent years, we have witnessed the continualinterests in developing the FFT techniques as one of the vital tools in optionpricing. In fact, the Fourier transform methods become the natural math-ematical tools when we consider option pricing under Levy models. This isbecause a Levy process Xt can be fully described by its characteristic functionφX(u), which is defined as the Fourier transform of the density function ofXt.

2.1 Fourier Transform and its Properties

First, we present the definition of the Fourier transform of a function andreview some of its properties. Let f(x) be a piecewise continuous real functionover (−∞,∞) which satisfies the integrability condition:

∫ ∞

−∞|f(x)| dx < ∞.

The Fourier transform of f(x) is defined by

Ff (u) =∫ ∞

−∞eiuyf(y) dy. (1)

Given Ff (u), the function f can be recovered by the following Fourier inver-sion formula:

f(x) =12π

∫ ∞

−∞e−iuxFf (u) du. (2)

The validity of the above inversion formula can be established easily via thefollowing integral representation of the Dirac function δ(y − x), where

δ(y − x) =12π

∫ ∞

−∞eiu(y−x) du.

Applying the defining property of the Dirac function


f(x) =∫ ∞

−∞f(y)δ(y − x) dy

and using the above integral representation of δ(y − x), we obtain

f(x) =∫ ∞

−∞f(y)

12π

∫ ∞

−∞eiu(y−x) dudy

=12π

∫ ∞

−∞e−iux

(∫ ∞

−∞f(y)eiuy dy

)du.

This gives the Fourier inversion formula (2).Sometimes it may be necessary to take u to be complex, with Im u 6=

0. In this case, Ff (u) is called the generalized Fourier transform of f . Thecorresponding Fourier inversion formula becomes

f(x) =12π

∫ iImu+∞

iImu−∞e−iuxFf (u) du.

Suppose the stochastic process Xt has the density function p, then theFourier transform of p

Fp(u) =∫ ∞

−∞eiuxp(x) dx = E

[eiuX

](3)

is called the characteristic function of Xt.The following mathematical properties of Ff are useful in our later dis-

cussion.

1. DifferentiationFf ′ (u) = −iuFf (u).

2. ModulationFeλxf (u) = Ff (u− iλ), λ is real.

3. ConvolutionDefine the convolution between two integrable functions f(x) and g(x) by

h(x) = f ∗ g(x) =∫ ∞

−∞f(y)g(x− y) dy,

thenFh = FfFg.

4. Parseval relationDefine the inner product of two complex-valued square-integrable func-tions f and g by

< f, g > =∫ ∞

−∞f(x)g(x) dx,

then< f, g > =

12π

< Ff (u),Fg(u) > .


We would like to illustrate an application of the Parseval relation in optionpricing. Following the usual discounted expectation approach, we formallywrite the option price V with terminal payoff VT (x) and risk neutral densityfunction p(x) as

V = e−rT

∫ ∞

−∞VT (x)p(x) dx = e−rT < VT (x), p(x) > .

By the Parseval relation, we obtain

V =e−rT

2π< Fp(u),FVT (u) > . (4)

The option price can be expressed in terms of the inner product of the char-acteristic function of the underlying process Fp(u) and the Fourier transformof the terminal payoff FVT

(u). More applications of the Parseval relation inderiving the Fourier inversion formulas in option pricing and insurance canbe found in Dufresne et al. (2009).

2.2 Discrete Fourier Transform

Given a sequence xk, k = 0, 1, · · · , N − 1, the discrete Fourier transform ofxk is another sequence yj, j = 0, 1, · · · , N − 1, as defined by

yj =N−1∑

k=0

e2πijk

N xk, j = 0, 1, · · · , N − 1. (5)

If we write the N -dimensional vectors

x = (x0 x1 · · · xN−1)T and y = (y0 y1 · · · yN−1)T,

and define a N ×N matrix FN whose (j, k)th entry is

FNj,k = e

2πijkN , 1 ≤ j, k ≤ N,

then x and y are related byy = FNx. (6)

The computation to find y requires N2 steps.However, if N is chosen to be some power of 2, say, N = 2L, the compu-

tation using the FFT techniques would require only 12NL = N

2 log2 N steps.The idea behind the FFT algorithm is to take advantage of the periodicityproperty of the N th root of unity. Let M = N

2 , and we split vector x into twohalf-sized vectors as defined by

x′ = (x0 x2 · · · xN−2)T and x′′ = (x1 x3 · · · xN−1)T.

We form the M -dimensional vectors


y′ = FMx′ and y′′ = FMx′′,

where the (j, k)th entry in the M ×M matrix FM is

FMj,k = e

2πijkM , 1 ≤ j, k ≤ M.

It can be shown that the first M and the last M components of y are givenby

yj = y′j + e2πij

N y′′j , j = 0, 1, · · · ,M − 1,

yj+M = y′j − e2πij

N y′′j , j = 0, 1, · · · ,M − 1. (7)

Instead of performing the matrix-vector multiplication FNx, we now reducethe number of operations by two matrix-vector multiplications FMx′ andFMx′′. The number of operations is reduced from N2 to 2

(N2

)2= N2

2 . Thesame procedure of reducing the length of the sequence by half can be ap-plied repeatedly. Using this FFT algorithm, the total number of operations isreduced from O(N2) to O(N log2 N).

2.3 Levy Processes

An adapted real-valued stochastic process Xt, with X0 = 0, is called a Levyprocess if it observes the following properties:

1. Independent incrementsFor every increasing sequence of times t0, t1, · · · , tn, the random variablesXt0 , Xt1 −Xt0 , · · · , Xtn −Xtn−1 are independent.

2. Time-homogeneousThe distribution of Xt+s −Xs; t ≥ 0 does not depend on s.

3. Stochastically continuousFor any ε > 0, P [|Xt+h −Xt| ≥ ε] → 0 as h → 0.

4. Cadlag processIt is right continuous with left limits as a function of t.

Levy processes are a combination of a linear drift, a Brownian process,and a jump process. When the Levy process Xt jumps, its jump magnitude isnon-zero. The Levy measure w of Xt defined on R\0 dictates how the jumpoccurs. In the finite-activity models, we have

∫R w(dx) < ∞. In the infinite-

activity models, we observe∫R w(dx) = ∞ and the Poisson intensity cannot

be defined. Loosely speaking, the Levy measure w(dx) gives the arrival rateof jumps of size (x, x + dx). The characteristic function of a Levy process canbe described by the Levy-Khinchine representation

φX(u) = E[eiuXt ]

= exp

(aitu− σ2

2tu2 + t

∫

R\0

(eiux − 1− iux1|x|≤1

)w(dx)

)

= exp(tψX(u)), (8)


where∫Rmin(1, x2)w(dx) < ∞, a ∈ R, σ2 ≥ 0. We identify a as the drift

rate and σ as the volatility of the diffusion process. Here, ψX(u) is called thecharacteristic exponent of Xt. Actually, Xt

d= tX1. All moments of Xt canbe derived from the characteristic function since it generalizes the moment-generating function to the complex domain. Indeed, a Levy process Xt isfully specified by its characteristic function φX . In Table 1, we present a listof Levy processes commonly used in finance applications together with theircharacteristic functions.

Levy process Xt Characteristic function φX(u)

Finite-activity modelsGeometric Brownian motion exp

(iuµt− 1

2σ2tu2

)

Lognormal jump diffusion exp(iuµt− 1

2σ2tu2 + λt(eiuµJ− 1

2 σ2J u2 − 1)

)

Double exponential jumpdiffusion

exp

(iuµt− 1

2σ2tu2 + λt

(1− η2

1 + u2η2eiuκ − 1

))

Infinite-activity models

Variance gamma exp(iuµt)(1− iuνθ + 12σ2νu2)

tν

Normal inverse Gaussian exp(iuµt + δt

√α2 − β2 −

√α2 − (β + iu)2

)

Generalized hyperbolic exp(iuµt)

(α2 − β2

α2 − (β + iu)2

) λt2

(Kλ

(δ√

α2−(β+iu)2)

Kλ

(δ√

α2−β2)

)t

where Kλ(z) =π

2

Iν(z)− I−ν(z)

sin(νπ),

Iν(z) =(z

2

)ν∞∑

k=0

(z2/4)k

k!Γ (ν + k + 1)

Finite-moment stable exp(iuµt− t(iuσ)α sec πα

2

)CGMY exp(CΓ (−Y ))[(M − iu)Y −MY + (G + iu)Y −GY ]

where C, G, M > 0 and Y > 2

Table 1. Characteristic functions of some parametric Levy processes

3 FFT Algorithms for Pricing European Vanilla Options

The renowned discounted expectation approach of evaluating a European op-tion requires the knowledge of the density function of the asset returns underthe risk neutral measure. Since the analytic representation of the character-istic function rather than the density function is more readily available forLevy processes, we prefer to express the expectation integrals in terms of thecharacteristic function. First, we derive the formal analytic representation of


a European option price as cumulative distribution functions, like the Black-Scholes type price formula. We then examine the inherent difficulties in thedirect numerical evaluation of the Fourier integrals in the price formula.

Under the risk neutral measure Q, suppose the underlying asset priceprocess assumes the form

St = S0 exp(rt + Xt), t > 0,

where Xt is a Levy process and r is the riskless interest rate. We write Y =log S0 + rT and let FVT

denote the Fourier transform of the terminal payofffunction VT (x), where x = log ST . By applying the discounted expectationvaluation formula and the Fourier inversion formula (2), the European optionvalue can be expressed as (see Lewis (2001))

V (S0, T ) = e−rT EQ[VT (x)]

=e−rT

2πEQ

[∫ iµ+∞

iµ−∞e−izxFVT

(z) dz

]

=e−rT

2π

∫ iµ+∞

iµ−∞e−izY φXT

(−z)FVT(z) dz,

where µ = Im z and ΦXT(z) is the characteristic function of XT . The above

formula is consistent with (4) derived using the Parseval relation.In our subsequent discussion, we set the current time to be zero and write

the current stock price as S. For the T -maturity European call option withterminal payoff (ST −K)+, its value is given by (see Lewis (2001))

C(S, T ; K) =−Ke−rT

2π

∫ iµ+∞

iµ−∞

e−izκφXT (−z)z2 − iz

dz

=−Ke−rT

2π

[∫ iµ+∞

iµ−∞e−izκφXT (−z)

i

zdz

−∫ iµ+∞

iµ−∞e−izκφXT (−z)

i

z − idz

]

= S

[12

+1π

∫ ∞

0

Re(

eiuκφXT(u− i)

iuφXT(−i)

)du

]

−Ke−rT

[12

+1π

∫ ∞

0

Re(

eiuκφXT(u)

iu

)du

], (9)

where κ = log SK + rT . This representation of the call price resembles the

Black-Scholes type price formula. However, due to the presence of the sin-gularity at u = 0 in the integrand function, we cannot apply the FFT toevaluate the integrals. If we expand the integrals as Taylor series in u, theleading term in the expansion for both integral is O

(1u

). This is the source

of the divergence, which arises from the discontinuity of the payoff function


at ST = K. As a consequence, the Fourier transform of the payoff functionhas large high frequency terms. Carr and Madan (1999) propose to dampenthe high frequency terms by multiplying the payoff by an exponential decayfunction.

3.1 Carr-Madan Formulation

As an alternative formulation of European option pricing that takes advan-tage of the analytic expression of the characteristic function of the underlyingasset price process, Carr and Madan (1999) consider the Fourier transform ofthe European call price (considered as a function of log strike) and computethe corresponding Fourier inversion to recover the call price using the FFT.Let k = log K, the Fourier transform of the call price C(k) does not exist sinceC(k) is not square integrable. This is because C(k) tends to S as k tends to−∞.

Modified Call Price MethodTo obtain a square-integrable function, Carr and Madan (1999) propose toconsider the Fourier transform of the damped call price c(k), where

c(k) = eαkC(k),

for α > 0. Positive values of α are seen to improve the integrability of themodified call value over the negative k-axis. Carr and Madan (1999) showthat a sufficient condition for square-integrability of c(k) is given by

EQ

[Sα+1

T

]< ∞.

We write ψT (u) as the Fourier transform of c(k), pT (s) as the densityfunction of the underlying asset price process, where s = log ST , and φT (u)as the characteristic function (Fourier transform) of pT (s). We obtain

ψT (u) =∫ ∞

−∞eiukc(k) dk

=∫ ∞

−∞e−rT pT (s)

∫ s

−∞

[es+αk − e(1+α)k

]eiuk dkds

=e−rT φT (u− (α + 1) i)

α2 + α− u2 + i(2α + 1)u. (10)

The call price C(k) can be recovered by taking the Fourier inversion trans-form, where

C(k) =e−αk

2π

∫ ∞

−∞e−iukψT (u) du

=e−αk

π

∫ ∞

0

e−iukψT (u) du, (11)


by virtue of the properties that ψT (u) is odd in its imaginary part and evenin its real part [since C(k) is real]. The above integral can be computed usingFFT, the details of which will be discussed next. From previous numericalexperience, usually α = 3 works well for most models of asset price dynamics.It is important to observe that α has to be chosen such that the denominatorhas only imaginary roots in u since integration is performed along real valueof u.

FFT ImplementationThe integral in (11) with a semi-infinite integration interval is evaluated bynumerical approximation using the trapezoidal rule and FFT. We start withthe choice on the number of intervals N and the stepwidth ∆u. A numericalapproximation for C(k) is given by

C(k) ≈ e−αk

π

N∑

j=1

e−iujkψT (uj)∆u, (12)

where uj = (j − 1)∆u, j = 1, · · · , N . The semi-infinite integration domain[0,∞) in the integral in (11) is approximated by a finite integration domain,where the upper limit for u in the numerical integration is N∆u. The errorintroduced is called the truncation error. Also, the Fourier variable u is nowsampled at discrete points instead of continuous sampling. The associatederror is called the sampling error. Discussion on the controls on various formsof errors in the numerical approximation procedures can be found in Lee(2004).

Recall that the FFT is an efficient numerical algorithm that computes thesum

y(k) =N∑

j=1

e−i 2πN (j−1)(k−1)x(j), k = 1, 2, · · · , N. (13)

In the current context, we would like to compute around-the-money call optionprices with k taking discrete values: km = −b + (m− 1)∆k, m = 1, 2, · · · , N .From one set of the FFT calculations, we are able to obtain call option pricesfor a range of strike prices. This facilitates the market practitioners to capturethe price sensitivity of a European call with varying values of strike prices. Toeffect the FFT calculations, we note from (13) that it is necessary to choose∆u and ∆k such that

∆u∆k =2π

N. (14)

A compromise between the choices of ∆u and ∆k in the FFT calculationsis called for here. For fixed N , the choice of a finer grid ∆u in numericalintegration leads to a larger spacing ∆k on the log strike.

The call price multiplied by an appropriate damping exponential factor be-comes integrable and the Fourier transform of the modified call price becomesan analytic function of the characteristic function of the log price. However,


at short maturities, the call value tends to the non-differentiable terminalcall option payoff causing the integrand in the Fourier inversion to becomehighly oscillatory. As shown in the numerical experiments performed by Carrand Madan (1999), this causes significant numerical errors. To circumventthe potential numerical pricing difficulties when dealing with short-maturityoptions, an alternative approach that considers the time value of a Europeanoption is shown to exhibit smaller pricing errors for all range of strike pricesand maturities.

Modified Time Value MethodFor notational convenience, we set the current stock price S to be unity anddefine

zT (k) = e−rT

∫ ∞

−∞

[(ek − es)1s<k,k<0 + (es − ek)1s>k,k<0

]pT (s) ds,

(15)which is seen to be equal to the T -maturity call price when K > S and theT -maturity put price when K < S. Therefore, once zT (k) is known, we canobtain the price of the call or put that is currently out-of-money while thecall-put parity relation can be used to obtain the price of the other optionthat is in-the-money.

The Fourier transform ζT (u) of zT (k) is found to be

ζT (u) =∫ ∞

−∞eiukzT (k) dk

= e−rT

[1

1 + iu− erT

iu− φT (u− i)

u2 − iu

]. (16)

The time value function zT (k) tends to a Dirac function at small maturityand around-the-money, so the Fourier transform ζT (u) may become highlyoscillatory. Here, a similar damping technique is employed by considering theFourier transform of sinh(αk)zT (k) (note that sinh αk vanishes at k = 0).Now, we consider

γT (u) =∫ ∞

−∞eiuk sinh(αk)zT (k) dk

=ζT (u− iα)− ζT (u + iα)

2,

and the time value can be recovered by applying the Fourier inversion trans-form:

zT (k) =1

sinh(αk)12π

∫ ∞

−∞e−iukγT (u) du. (17)

Analogous FFT calculations can be performed to compute the numerical ap-proximation for zT (km), where


zT (km) ≈ 1π sinh(αkm)

∑Nj=1 e−i 2π

N (j−1)(m−1)eibuj γT (uj)∆u, (18)

m = 1, 2, · · · , N, and km = −b + (m− 1)∆k.

4 Pricing of European Multi-Asset Options

Apparently, the extension of the Carr-Madan formulation to pricing Euro-pean multi-asset options would be quite straightforward. However, dependingon the nature of the terminal payoff function of the multi-asset option, the im-plementation of the FFT algorithm may require some special considerations.

The most direct extension of the Carr-Madan formulation to the multi-asset models can be exemplified through pricing of the correlation option, theterminal payoff of which is defined by

V (S1, S2, T ) = (S1(T )−K1)+(S2(T )−K2)+. (19)

We define si = log Si, ki = log Ki, i = 1, 2, and write pT (s1, s2) as the jointdensity of s1(T ) and s2(T ) under the risk neutral measure Q. The character-istic function of this joint density is defined by the following two-dimensionalFourier transform:

φ(u1, u2) =∫ ∞

−∞

∫ ∞

−∞ei(u1s1+u2s2)pT (s1, s2) ds1ds2. (20)

Following the Carr-Madan formulation, we consider the Fourier transformψT (u1, u2) of the damped option price eα1k1+α2k2VT (k1, k2) with respect tothe log strike prices k1, k2, where α1 > 0 and α2 > 0 are chosen such thatthe damped option price is square-integrable for negative values of k1 and k2.The Fourier transform ψT (u1, u2) is related to φ(u1, u2) as follows:

ψT (u1, u2) =e−rT φ(u1 − (α1 + 1)i, u2 − (α2 + 1)i)

(α1 + iu1)(α1 + 1 + iu1)(α2 + iu2)(α2 + 1 + iu2). (21)

To recover CT (k1, k2), we apply the Fourier inversion on ψT (u1, u2). Followinganalogous procedures as in the single-asset European option, we approximatethe two-dimensional Fourier inversion integral by

CT (k1, k2) ≈ e−α1k1−α2k2

(2π)2

N−1∑m=0

N−1∑n=0

e−i(u1mk1+u2

nk2)ψT (u1m, u2

n)∆1∆2, (22)

where u1m =

(m− N

2

)∆1 and u2

n =(n− N

2

)∆2. Here, ∆1 and ∆2 are the

stepwidths, and N is the number of intervals. In the two-dimensional form ofthe FFT algorithm, we define

k1p =

(p− N

2

)∆1 and k1

q =(

q − N

2

)∆2,


where λ1 and λ2 observe

λ1∆1 = λ2∆2 =2π

N.

Dempster and Hong (2000) show that the numerical approximation to theoption price at different log strike values is given by

CT (k1p, k2

q) ≈ e−α1k1p−α2k2

q

(2π)2Γ (k1

p, k2q)∆1∆2, 0 ≤ p, q ≤ N, (23)

where

Γ (k1p, k2

q) = (−1)p+qN−1∑m=0

N−1∑n=0

e−2πiN (mp+nq)

[(−1)m+nψT (u1

m, u2n)

].

The nice tractability in deriving the FFT pricing algorithm for the corre-lation option stems from the rectangular shape of the exercise region Ω of theoption. Provided that the boundaries of Ω are made up of straight edges, theabove procedure of deriving the FFT pricing algorithm still works. This is be-cause one can always take an affine change of variables in the Fourier integralsto effect the numerical evaluation. What would be the classes of option payofffunctions that allow the application of the above approach? Lee (2004) lists4 types of terminal payoff functions that admit analytic representation of theFourier transform of the damped option price. Another class of multi-assetoptions that possess similar analytic tractability are options whose payoff de-pends on taking the maximum or minimum value among the terminal valuesof a basket of stocks (see Eberlein et al. (2009)). However, the exercise regionof the spread option with terminal payoff

VT (S1, S2) = (S1(T )− S2(T )−K)+ (24)

is shown to consist of a non-linear edge. To derive the FFT algorithm of similarnature, it is necessary to approximate the exercise region by a combinationof rectangular strips. The details of the derivation of the corresponding FFTpricing algorithm are presented by Dempster and Hong (2000).

Hurd and Zhou (2009) propose an alternative approach to pricing theEuropean spread option under Levy model. Their method relies on an elegantformula of the Fourier transform of the spread option payoff function. LetP (s1, s2) denote the terminal spread option payoff with unit strike, where

P (s1, s2) = (es1 − es2 − 1)+.

For any real numbers ε1 and ε2 with ε2 > 0 and ε1+ε2 < −1, they establish thefollowing Fourier representation of the terminal spread option payoff function:

P (s1, s2) =1

(2π)2

∫ ∞+iε2

−∞+iε2

∫ ∞+iε1

−∞+iε1

ei(u1s1+u2s2)P (u1, u2) du1du2, (25)


where

P (u1, u2) =Γ (i(u1 + u2)− 1)Γ (−iu2)

Γ (iu1 + 1).

Here, Γ (z) is the complex gamma function defined for Re(z) > 0, where

Γ (z) =∫ ∞

0

e−ttz−1 dt.

To establish the Fourier representation in (25), we consider

P (u1, u2) =∫ ∞

−∞

∫ ∞

−∞e−i(u1s1+u2s2)P (s1, s2) ds2ds1.

By restricting to s1 > 0 and es2 < es1 − 1, we have

P (u1, u2) =∫ ∞

0

e−iu1s1

∫ log(es1−1)

−∞e−iu2s2(es1 − es2 − 1) ds2ds1

=∫ ∞

0

e−iu1s1(es1 − 1)1−iu2

(1

−iu2− 1

1− iu2

)ds1

=1

(1− iu2)(−iu2)

∫ 1

0

ziu1

(1− z

z

)1−iu2 dz

z,

where z = e−s1 . The last integral can be identified with the beta function:

β(a, b) =Γ (a)Γ (b)Γ (a + b)

=∫ 1

0

za−1(1− z)b−1 dz,

so we obtain the result in (25). Once the Fourier representation of the terminalpayoff is known, by virtue of the Parseval relation, the option price can beexpressed as a two-dimensional Fourier inversion integral with integrand thatinvolves the product of P (u1, u2) and the characteristic function of the jointprocess of s1 and s2. The evaluation of the Fourier inversion integral can beaffected by the usual FFT calculations (see Hurd and Zhou (2009)). Thisapproach does not require the analytic approximation of the two-dimensionalexercise region of the spread option with a non-linear edge, so it is consideredto be more computationally efficient.

The pricing of European multi-asset options using the FFT approach re-quires availability of the analytic representation of the characteristic functionof the joint price process of the basket of underlying assets. One may incor-porate a wide range of stochastic structures in the volatility and correlation.Once the analytic forms in the integrand of the multi-dimensional Fourierinversion integral are known, the numerical evaluation involves nested sum-mations in the FFT calculations whose dimension is the same as the numberof underlying assets in the multi-asset option. This contrasts with the usualfinite difference / lattice tree methods where the dimension of the scheme in-creases with the number of risk factors in the prescription of the joint process


of the underlying assets. This is a desirable property over other numericalmethods since the FFT pricing of the multi-asset options is not subject tothis curse of dimensionality with regard to the number of risk factors in thedynamics of the asset returns.

5 Convolution Approach and Pricing of Bermudan StyleOptions

We consider the extension of the FFT technique to pricing of options thatallow early exercise prior to the maturity date T . Recall that a Bermudanoption can only be exercised at a pre-specified set of time points, say T =t1, t2, · · · , tM, where tM = T . On the other hand, an American option canbe exercised at any time prior to T . By taking the number of time pointsof early exercise to be infinite, we can extrapolate a Bermudan option tobecome an American option. In this section, we would like to illustrate howthe convolution property of Fourier transform can be used to price a Bermudanoption effectively (see Lord et al. (2008)).

Let F (S(tm), tm) denote the exercise payoff of a Bermudan option attime tm, m = 1, 2, · · · ,M . Let V (S(tm), tm) denote the time-tm value of theBermudan option with exercise point set T ; and we write ∆tm = tm+1 − tm,m = 1, 2, · · · , M−1. The Bermudan option can be evaluated via the followingbackward induction procedure:

terminal payoff: V (S(tM ), tM ) = F (S(tM ), tM )For m = M − 1,M − 2, · · · , 1, compute

C(S(tm), tm) = e−r∆tm

∫ ∞

−∞V (y, tm+1)p(y|S(tm)) dy

V (S(tm), tm) = maxC(S(tm), tm), F (S(tm), tm).

Here, p(y|S(tm)) represents the probability density that relates the transitionfrom the price level S(tm) at tm to the new price level y at tm+1. By virtue ofthe early exercise right, the Bermudan option value at tm is obtained by takingthe maximum value between the time-tm continuation value C(S(tm), tm) andthe time-tm exercise payoff F (S(tm), tm).

The evaluation of C(S(tm), tm) is equivalent to the computation of thetime-tm value of a tm+1-maturity European option. Suppose the asset priceprocess is a monotone function of a Levy process (which observes the indepen-dent increments property), then the transition density p(y|x) has the followingproperty:

p(y|x) = p(y − x). (26)

If we write z = y − x, then the continuation value can be expressed as aconvolution integral as follows:


C(x, tm) = e−r∆tm

∫ ∞

−∞V (x + z, tm+1)p(z) dz. (27)

Following a similar damping procedure as proposed by Carr and Madan(1999), we define

c(x, tm) = eαx+r∆tmC(x, tm)

to be the damped continuation value with the damping factor α > 0. Applyingthe property of the Fourier transform of a convolution integral, we obtain

Fxc(x, tm)(u) = Fyv(y, tm+1)(u)φ(−(u− iα)), (28)

and φ(u) is the characteristic function of the random variable z.Lord et al. (2008) propose an effective FFT algorithm to calculate the

following convolution:

c(x, tm) =12π

∫ ∞

−∞e−iuxv(u)φ(−(u− iα)) du, (29)

where v(u) = Fv(y, tm). The FFT calculations start with the prescriptionof uniform grids for u, x and y:

uj = u0 + j∆u, xj = x0 + j∆x, yj = y0 + j∆y, j = 0, 1, · · · , N − 1.

The mesh sizes ∆x and ∆y are taken to be equal, and ∆u and ∆y are chosento satisfy the Nyquist condition:

∆u∆y =2π

N.

The convolution integral is discretized as follows:

c(xp) ≈ e−iu0(x0+p∆y)

2π∆u

N−1∑

j=0

e−ijp 2πN eij(y0−x0)∆uφ(−(uj − iα))v(uj), (30)

where

v(uj) ≈ eiu0y0∆y

N−1∑n=0

eijn2π/Neinu0∆ywnv(yn),

w0 = wN−1 =12, wn = 1 for n = 1, 2, · · · , N − 2.

For a sequence xp, p = 0, 1, · · · , N − 1, its discrete Fourier transform and thecorresponding inverse are given by

Djxn =N−1∑n=0

eijn2π/Nxn, D−1n xj =

1N

N−1∑

j=0

e−ijn2π/Nxj .


By setting u0 = −N2 ∆u so that einu0∆y = (−1)n, we obtain

c(xp) ≈ eiu0(y0−x0)(−1)pD−1p eij(y0−x0)∆uφ(−(uj − iα))Dj(−1)nwnv(yn).

(31)In summary, by virtue of the convolution property of Fourier transform,

we compute the discrete Fourier inversion of the product of the discrete char-acteristic function of the asset returns φ(−(uj − iα)) and the discrete Fouriertransform of option prices Dj(−1)nwnv(yn). It is seen to be more efficientwhen compared to the direct approach of recovering the density function bytaking the Fourier inversion of the characteristic function and finding the op-tion prices by discounted expectation calculations (see Zhylyevsky (2009)).

6 FFT-Based Network Method

As an extension to the usual lattice tree method, an FFT-based networkapproach to option pricing under Levy models has been proposed by Wongand Guan (2009). The network method somewhat resembles Duan-Simonato’sMarkov chain approximation method (Duan and Simonato (2001)). This newapproach is developed for option pricing for which the characteristic functionof the log-asset value is known. Like the lattice tree method, the networkmethod can be generalized to valuation of path dependent options by adoptingthe forward shooting grid technique (see Kwok (2009)).

First, we start with the construction of the network. We perform the space-time discretization by constructing a pre-specified system of grids of time andstate: t0 < t1 < · · · < tM , where tM is the maturity date of the option, andx0 < x1 < · · · < xN , where X = xj |j = 0, 1, · · · , N represents the set ofall possible values of log-asset prices. For simplicity, we assume uniform gridsizes, where ∆x = xj+1 − xj for all j and ∆t = ti+1 − ti for all i. Unlikethe binomial tree where the number of states increases with the number oftime steps, the number of states is fixed in advance and remains unchangedat all time points. In this sense, the network resembles the finite differencegrid layout. The network approach approximates the Levy process by a finitestate Markov chain, like that proposed by Duan and Simonato (2001). Weallow for a finite probability that the log-asset value moves from one stateto any possible state in the next time step. This contrasts with the usualfinite difference schemes where the linkage of nodal points between successivetime steps is limited to either one state up, one state down or remains at thesame state. The Markov chain model allows greater flexibility to approximatethe asset price dynamics that exhibits finite jumps under Levy model withenhanced accuracy. A schematic diagram of a network with 7 states and 3time steps is illustrated in Fig. 1.

After the construction of the network, the next step is to compute thetransition probabilities that the asset price goes from one state xi to anotherstate xj under the Markov chain model, 0 ≤ i, j ≤ N . The corresponding


x4

x3

x2

x1

x4

x3

x2

x1

x4

x3

x2

x1

x5

x5 x

5

x4

x3

x2

x1

x5

x6

x6

x6

x6

x0

x0

x0

x0

t0

t1

t2

t3

Fig. 1. A network model with 3 time steps and 7 states.

transition probability is defined as follows:

pij = P [Xt+∆t = xj |Xt = xi], (32)

which is independent of t due to the time homogeneity of the underlying Levyprocess. We define the corresponding characteristic function by

φi(u) =∫ ∞

−∞eiuzfi(z|xi) dz,

where fi(z|xi) is the probability density function of the increment Xt+∆t−Xt

conditional on Xt = xi. The conditional probability density function can berecovered by Fourier inversion:

fi(xj |xi) = F−1u φi(u)(xj). (33)

If we take the number of Markov chain states to be N + 1 = 2L for someinteger L, then the above Fourier inversion can be carried out using the FFTtechniques. The FFT calculations produce approximate values for fi(xj |xi)for all i and j. We write these approximate conditional probability valuesobtained from the FFT calculations as fi(xj |xi). The transition probabilitiesamong the Markov chain states are then approximated by

pij ≈ fi(xj |xi)∑Ni=0 fi(xj |xi)

, 0 ≤ i, j ≤ N. (34)


Once the transition probabilities are known, we can perform option valuationusing the usual discounted expectation approach. The incorporation of variouspath dependent features can be performed as in usual lattice tree calculations.Wong and Guan (2009) illustrate how to compute the Asian and lookback op-tion prices under Levy models using the FFT-based network approach. Theirnumerical schemes are augmented with the forward shooting grid technique(see Kwok (2009)) for capturing the asset price dependency associated withthe Asian and lookback features.

7 Fourier Space Time Stepping Method

When we consider option pricing under Levy models, the option price func-tion is governed by a partial integral-differential equation (PIDE) where theintegral terms in the equation arise from the jump components in the under-lying Levy process. In this section, we present the Fourier space time stepping(FST) method that is based on the solution in the Fourier domain of the gov-erning PIDE (see Jackson et al. (2008)). This is in contrast with the usualfinite difference schemes which solve the PIDE in the real domain. We discussthe robustness of the FST method with regard to its symmetric treatment ofthe jump terms and diffusion terms in the PIDE and the ease of incorporationof various forms of path dependence in the option models. Unlike the usualfinite difference schemes, the FST method does not require time stepping cal-culations between successive monitoring dates in pricing Bermudan optionsand discretely monitored barrier options. In the numerical implementationprocedures, the FST method does not require the analytic expression for theFourier transform of the terminal payoff of the option so it can deal easierwith more exotic forms of the payoff functions. The FST method can be eas-ily extended to multi-asset option models with exotic payoff structures andpricing models that allow regime switching in the underlying asset returns.

First, we follow the approach by Jackson et al. (2008) to derive the gov-erning PIDE of option pricing under Levy models and consider the Fouriertransform of the PIDE. We consider the model formulation under the generalmulti-asset setting. Let S(t) denote a d-dimensional price index vector of theunderlying assets in a multi-asset option model whose T -maturity payoff is de-noted by VT (S(T )). Suppose the underlying price index follows an exponentialLevy process, where

S(t) = S(0)eX(t),

and X(t) is a Levy process. Let the characteristic component of X(t) be thetriplet (µ,M, ν), where µ is the non-adjusted drift vector, M is the covariancematrix of the diffusion components, and ν is the d-dimensional Levy density.The Levy process X(t) can be decomposed into its diffusion and jump com-ponents as follows:

X(t) = µ(t) + MW(t) + Jl(t) + limε→0

Jε(t), (35)


where the large and small components are

Jl(t) =∫ t

0

∫

|y|≥1

y m(dy × ds)

Jε(t) =∫ t

0

∫

ε≤|y|<1

y [m(dy × ds)− ν(dy × ds)],

respectively. Here, W(t) is the vector of standard Brownian processes, m(dy×ds) is a Poisson random measure counting the number of jumps of size yoccurring at time s, and ν(dy × ds) is the corresponding compensator. Oncethe volatility and Levy density are specified, the risk neutral drift can bedetermined by enforcing the risk neutral condition:

E0[eX(1)] = er,

where r is the riskfree interest rate. The governing partial integral-differentialequation (PIDE) of the option price function V (X(t), t) is given by

∂V

∂t+ LV = 0 (36)

with terminal condition: V (X(T ), T ) = VT (S(0), eX(T )), where L is the in-finitesimal generator of the Levy process operating on a twice differentiablefunction f(x) as follows:

Lf(x) =

(µT ∂

∂x+

∂

∂x

T

M∂

∂x

)f(x)

+∫

Rn\0[f(x + y)− f(x)]− yT ∂

∂xf(x)1|y|<1 ν(dy). (37)

By the Levy-Khintchine formula, the characteristic component of the Levyprocess is given by

ψX(u) = iµTu− 12uTMu

+∫

Rn

(eiuTy − 1− iuTy1|y|<1

)ν(dy). (38)

Several numerical schemes have been proposed in the literature that solvethe PIDE (36) in the real domain. Jackson et al. (2008) propose to solve thePIDE directly in the Fourier domain so as to avoid the numerical difficultiesin association with the valuation of the integral terms and diffusion terms.An account on the deficiencies in earlier numerical schemes in treating thediscretization of the integral terms can be found in Jackson et al. (2008).

By taking the Fourier transform on both sides of the PIDE, the PIDE isreduced to a system of ordinary differential equations parametrized by the


d-dimensional frequency vector u. When we apply the Fourier transform tothe infinitesimal generator L of the process X(t), the Fourier transform canbe visualized as a linear operator that maps spatial differentiation into multi-plication by the factor iu. We define the multi-dimensional Fourier transformas follows (a slip of sign in the exponent of the Fourier kernel is adopted herefor notational convenience):

F [f ](u) =∫ ∞

−∞f(x)e−iuTx dx

so thatF−1[Ff ](u) =

12π

∫ ∞

−∞FfeiuTx du.

We observe

F[

∂

∂xf

]= iuF [f ] and F

[∂2

∂x2f

]= iuF [f ] iuT

so thatF [LV ](u, t) = ψX(u)F [V ](u, t). (39)

The Fourier transform of LV is elegantly given by multiplying the Fouriertransform of V by the characteristic component ψX(u) of the Levy processX(t). In the Fourier domain, F [V ] is governed by the following system ofordinary differential equations:

∂

∂tF [V ](u, t) + ψX(u)F [V ](u, t) = 0 (40)

with terminal condition: F [V ](u, T ) = FVT(u, T ).

If there is no embedded optionality feature like the knock-out feature orearly exercise feature between t and T , then the above differential equationcan be integrated in a single time step. By solving the PIDE in the Fourierdomain and performing Fourier inversion afterwards, the price function of aEuropean vanilla option with terminal payoff VT can be formally representedby

V (x, t) = F−1F [VT ](u, T )eψX(u)(T−t)

(x, t). (41)

In the numerical implementation procedure, the continuous Fourier transformand inversion are approximated by some appropriate discrete Fourier trans-form and inversion, which are then effected by FFT calculations. Let vT andvt denote the d-dimensional vector of option values at maturity T and timet, respectively, that are sampled at discrete spatial points in the real domain.The numerical evaluation of vt via the discrete Fourier transform and inver-sion can be formally represented by

vt = FFT −1[FFT [vT ]eψX(T−t)], (42)


where FFT denotes the multi-dimensional FFT transform. In this numericalFFT implementation of finding European option values, it is not necessary toknow the analytic representation of the Fourier transform of the terminal pay-off function. This new formulation provides a straightforward implementationof numerical pricing of European spread options without resort to elaboratedesign of FFT algorithms as proposed by Dempster and Hong (2000) andHurd and Zhou (2009) (see Section 4).

Suppose we specify a set of preset discrete time points X = t1, t2, · · · , tN,where the option may be knocked out (barrier feature) or early exercised(Bermudan feature) prior to maturity T (take tN+1 = T for notational con-venience). At these time points, we either impose constraints or perform op-timization based on the contractual specification of the option. Consider thepricing of a discretely monitored barrier option where the knock-out featureis activated at the set of discrete time points X . Between times tn and tn+1,n = 1, 2, · · · , N , the barrier option behaves like a European vanilla option sothat the single step integration can be performed from tn to tn+1. At timetn, we impose the contractual specification of the knock-out feature. Say, theoption is knocked out when S stays above the up-and-out barrier B. Let Rdenote the rebate paid upon the occurrence of knock-out, and vn be the vec-tor of option values at discrete spatial points. The time stepping algorithmcan be succinctly represented by

vn = HB(FFT −1[FFT [vn+1]eψX(tn+1−tn)]),

where the knock-out feature is imposed by defining HB to be (Jackson et al.(2008))

HB(v) = v1x<log BS(0) + R1x≥log B

S(0).No time stepping is required between two successive monitoring dates.

8 Summary and Conclusions

The Fourier transform methods provide the valuable and indispensable toolsfor option pricing under Levy processes since the analytic representation of thecharacteristic function of the underlying asset return is more readily availablethan that of the density function itself. When used together with the FFTalgorithms, real time pricing of a wide range of option models under Levyprocesses can be delivered with high accuracy, efficiency and reliability. Inparticular, option prices across the whole spectrum of strikes can be obtainedin one set of FFT calculations.

In this chapter, we review the most commonly used option pricing algo-rithms via FFT calculations. When the European option price function isexpressed in terms of Fourier inversion integrals, option pricing can be deliv-ered by finding the numerical approximation of the Fourier integrals via FFTtechniques. Several modifications of the European option pricing formulation


in the Fourier domain, like the damped option price method and time valuemethod, have been developed so as to avoid the singularity associated withnon-differentiability of the terminal payoff function. Alternatively, the pricingformulation in the form of a convolution product is used to price Bermudanoptions where early exercise is allowed at discrete time points. Depending onthe structures of the payoff functions, the extension of FFT pricing to multi-asset models may require some ingeneous formulation of the correspondingoption model. The order of complexity in the FFT calculations for pricingmulti-asset options generally increases with the number of underlying assetsrather than the total number of risk factors in the joint dynamics of the un-derlying asset returns. When one considers pricing of path dependent optionswhose analytic form of the option price function in terms of Fourier integralsis not readily available, it becomes natural to explore various extensions ofthe lattice tree schemes and finite difference approach. The FFT-based net-work method and the Fourier space time stepping techniques are numericalapproaches that allow greater flexibility in the construction of the numericalalgorithms to handle various form of path dependence of the underlying as-set price processes through the incorporation of the auxiliary conditions thatarise from modeling the embedded optionality features. The larger numberof branches in the FFT-based network approach can provide better accuracyto approximate the Levy process with jumps when compared to the usualtrinomial tree approach. The Fourier space time stepping method solves thegoverning partial integral-differential equation of option pricing under Levymodel in the Fourier domain. Unlike usual finite difference schemes, no timestepping procedures are required between successive monitoring instants inoption models with discretely monitored features.

In summary, a rich set of numerical algorithms via FFT calculations havebeen developed in the literature to perform pricing of most types of optionmodels under Levy processes. For future research topics, one may considerthe pricing of volatility derivatives under Levy models where the payoff func-tion depends on the the realized variance or volatility of the underlying priceprocess. Also, more theoretical works should be directed to error estimationmethods and controls with regard to sampling errors and truncation errorsin the approximation of the Fourier integrals and other numerical Fouriertransform calculations.

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