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Pricing Asian Options under a Hyper-Exponential Jump

Diffusion Model

Ning Cai1and S. G. Kou2

1Room 5521, Department of IELM, HKUST

2313 Mudd Building, Department of IEOR, Columbia University

ningcai@ust.hk and sk75@columbia.edu

June 2011

Abstract

We obtain a closed-form solution for the double-Laplace transform of Asian options

under the hyper-exponential jump diffusion model (HEM). Similar results are only available

previously in the special case of the Black-Scholes model (BSM). Even in the case of the

BSM, our approach is simpler as we essentially use only Itˆ o’s formula and do not need

more advanced results such as those of Bessel processes and Lamperti’s representation. As

a by-product we also show that a well-known recursion relating to Asian options has a

unique solution in a probabilistic sense. The double-Laplace transform can be inverted

numerically via a two-sided Euler inversion algorithm. Numerical results indicate that our

pricing method is fast, stable, and accurate, and performs well even in the case of low

volatilities.

Subject classifications: Finance: asset pricing. Probability: stochastic model applica-

tions.

Area of review: Financial engineering.

1 Introduction

Asian options (or average options), whose payoffs depend on the average of the underlying asset

price over a pre-specified time period, are among the most popular path-dependent options

traded in both exchanges and over-the-counter markets. A main difficulty in pricing Asian

options is that the distribution of the average price may not be available analytically.

There is a large body of literature on Asian options under the Black-Scholes model (BSM).

For example, approaches based on partial differential equations were given in Ingersoll [26],

Rogers and Shi [38], Lewis [31], Dubois and Leli` evre [18], Zhang [51, 52]; Monte Carlo simulation

techniques were discussed in Broadie and Glasserman [7], Glasserman [24] and Lapeyre and

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Temam [29]; analytical approximations were derived in Turnbull and Wakeman [46], Milevsky

and Posner [35] and Ju [27]; lower and upper bounds were given in Curran [16], Henderson et

al. [25], and Thompson [45]. Previous results that are related to ours are: (i) Linetsky [32]

derived an elegant series expansion for Asian options via a one-dimensional affine diffusion.

(ii) Vecer [47] obtained a one-dimensional partial differential equation (PDE) for Asian options

which can be solved numerically in stable ways. (iii) Based on Bessel processes and Lamperti’s

representation, in a celebrated paper Geman and Yor [23] provided an analytical solution of a

single-Laplace transform of the Asian option price with respect to the maturity; see also Yor

[50], Matsumoto and Yor [33, 34], Carr and Schr¨ oder [13] and Schr¨ oder [40]. Significant progress

has been made for the inversion of the single-Laplace transform in Shaw [41, 43]. Dewynne and

Shaw [17] gave a simple derivation of the single-Laplace transform, and provided a matched

asymptotic expansion, which performs well for extremely low volatilities. (iv) Dufresne [19, 20]

obtained many interesting results including a Laguerre series expansion for both Asian and

reciprocal Asian options. (v) Double-Laplace and Fourier-Laplace transforms were proposed in

Fu et al. [21] and Fusai [22], respectively. For the differences between their methods and ours,

see Section 2 and the online supplement (Section 3).

All the papers discussed above are within the Black-Scholes framework. There are only few

papers for alternative models with jumps. For example, Albrecher [2], Albrecher and Predota

[4] and Albrecher et al. [3] derived bounds and approximations for Asian options under certain

exponential L´ evy models; Carmona et al. [11] derived some theoretical representations for Asian

options under some special L´ evy processes; Vecer and Xu [49] gave some representations for

Asian options under semi-martingale models via partial integro-differential equations; Bayraktar

and Xing [6] proposed a numerical approach to Asian options for jump diffusions by constructing

a sequence of converging functions.

In this paper we study the pricing of Asian options under the hyper-exponential jump

diffusion model (HEM) where the jump sizes have a hyper-exponential distribution, i.e., a

mixture of a finite number of exponential distributions. For background on the HEM, see

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Levendorski˘ ı [30] and Cai and Kou [9]. The contribution of the current paper is three-fold:

(1) Even in the special case of the BSM, our approach is simpler as we essentially use only

Itˆ o’s formula and do not need more advanced results such as those of Bessel processes and

Lamperti’s representation. See Section 3. (2) Our approach is more general as it applies to

the HEM (see Section 4). As a by-product we also show that under the HEM a well-known

recursion relating to Asian options has a unique solution in a probabilistic sense, and the

integral of the underlying asset price process at the exponential time has the same distribution

as a combination of a sequence of independent gamma and beta random variables; see Section

4.1. (3) The double-Laplace transform can be inverted numerically via a latest two-sided Euler

inversion algorithm along with a scaling factor proposed in Petrella [37].

We analyze the algorithm’s accuracy, stability, and low-volatility performance by conducting

a detailed comparison with other existing methods. For example our pricing method is highly

accurate compared with the benchmarks from the three existing pricing methods under the

BSM: (i) Linetsky’s method, (ii) Vecer’s method, and (iii) Geman and Yor’s single-Laplace

method via Shaw’s elegant Mathematica implementation. Moreover, our method performs well

even for low volatilities, e.g., 0.05. See Section 5.

The rest of the paper is organized as follows. Section 2 contains a general formulation of

the double-Laplace transform of Asian option prices. Section 3 concentrates on pricing Asian

options under the BSM. In Section 4, we extend the results in Section 3 to the more general

HEM. Section 5 is devoted to the implementation of our pricing algorithm via the latest two-

sided, two-dimensional Euler inversion algorithm with a scaling factor. Some proofs and some

numerical issues are presented in the appendices and the online supplement.

2 A Double-Laplace Transform

For simplicity, we shall focus on Asian call options, as Asian put options can be treated similarly.

The payoff of a continuous Asian call option with a mature time t and a fixed strike K is

?S0At

t

− K?+, where At:=?t

0eX(s)ds, S(t) is the underlying asset price process with S(0) ≡

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S0, and X(t) := log(S(t)/S(0)) is the return process. Standard finance theory says that the

Asian option price at time zero can be expressed as P(t,k) := e−rtE?S0At

is expectation under a pricing probability measure P. Under the BSM the measure P is the

t

− K?+, where E

unique risk neutral measure, whereas under more general models P may be obtained in other

ways, such as using utility functions or mean variance hedging arguments. For more details,

see, e.g., Shreve [44].

A key component of our double-Laplace inversion method is a scaling factor X > S0. More

precisely, with k := ln(X

Kt) we can rewrite the option price P(t,k) =e−rtX

t

E?S0

XAt− e−k?+.

Note that k can be either positive or negative, so the Laplace transform w.r.t. k will be two-

sided. The scaling factor X introduced by Petrella [37] is primarily to control the associated

discretization errors and to let the inversion occur at a reasonable point k; see also Cai et

al. [10], where they introduced a shift parameter for the two-sided Euler inversion algorithm.

Thanks to the scaling factor, the resulting inversion algorithm appears to be accurate, fast, and

stable even in the case of low volatility, e.g. σ = 0.05; see Section 5.

The following result presents an analytical representation for the double-Laplace transform

of f(t,k) := XE(S0

XAt−e−k)+w.r.t. t and k. Note that Theorem 2.1 holds not only under the

BSM but also under other stochastic models. The result reduces the problem of pricing Asian

options to the study of real moments of the exponentially-stopped average E[Aν+1

Tµ].

Theorem 2.1. Let L(µ,ν) be the double-Laplace transform of f(t,k) w.r.t. t and k, respectively.

More precisely, L(µ,ν) =?∞

L(µ,ν) =

µν(ν + 1)

X

where ATµ=?Tµ

{X(t) : t ≥ 0}. Here µ > 0 and ν > 0 should satisfy E[Aν+1

0

?∞

−∞e−µte−νkXE(S0

XE[Aν+1

XAt− e−k)+dkdt. Then we have that

?ν+1

Tµ]

?S0

, µ > 0, ν > 0,

(1)

0

eX(s)ds and Tµis an exponential random variable with rate µ independent of

Tµ] < +∞.

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Proof : Applying Fubini’s theorem yields

L(µ,ν)=

X

?∞

?∞

?∞

?∞

0

e−µtE

??∞

?

?

−ln(S0At/X)

S0

XAt

e−νk

?S0

XAt− e−k

?

dk

?

dt

=

X

0

e−µtE

?∞

−ln(S0At/X)

e−νkdk −

?∞

?

−ln(S0At/X)

e−(ν+1)kdk

?

dt

=

X

0

e−µtE

(S0At/X)ν+1

ν

?S0

−(S0At/X)ν+1

ν + 1

?ν+1

dt

=

X

0

e−µtE[Aν+1

ν(ν + 1)

t

]

X

dt =

X

µν(ν + 1)

?S0

X

?ν+1

· E[Aν+1

Tµ],

from which the proof is completed. ?

The idea of taking Laplace transform w.r.t. the log-strike ln(K) dates back to the work by

Carr and Madan [12]. Here we use the scaled log-strike ln(X/(Kt)) instead, as suggested in

Petrella [37]. A different double-Laplace transform was given in Fu et al. [21] under the BSM,

where the transform is taken w.r.t. t and K.

3 Pricing Asian Options under the BSM

In this section we study Asian option pricing under the BSM via the double-Laplace transforms.

More precisely, we investigate the distribution of ATµso that we can compute E[Aν+1

Tµ] explicitly

and hence obtain analytical solutions for the double-Laplace transforms, thanks to Theorem

2.1.

3.1 Distribution of ATµunder the BSM

The classical BSM postulates that under the risk-neutral measure P, the return process {X(t) =

log(S(t)/S(0)) : t ≥ 0} is given by X(t) =

free rate, σ the volatility, and {W(t) : t ≥ 0} the standard Brownian motion. The infinitesimal

generator of {S(t)} is

Lf(s) =σ2

2s2f??(s) + rsf?(s)

?

r −σ2

2

?

t + σW(t), X(0) = 0, where r is the risk-

(2)

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