Page 1

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

1

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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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for any twice continuously differentiable function f(·), and the L´ evy exponent of {X(t)} is

G(x) :=E?exX(t)?

t

=σ2

2x2+

?

r −σ2

2

?

x.

(3)

Let α1and α2be the two roots of the equation G(x) = µ(> 0) under the BSM. Then,

?

α1=−v +

v2+ 2µ

2

> 0,α2=−v −

?

v2+ 2µ

2

< 0,

(4)

where µ =4µ

σ2and v =2r

σ2− 1.

Consider the following nonhomogeneous ordinary differential equation (ODE)

Ly(s) = (s + µ)y(s) − µ,

for s ≥ 0.

(5)

Note that the equation (5) has two singularities, a regular singularity at 0 and an irregular

singularity at +∞. Due to the singularity, the above equation has infinitely many solutions.

However, if we impose an additional condition that the solution must be bounded, then the

solution is unique.

Theorem 3.1. (Uniqueness of the solution of the ODE (5) via a stochastic represen-

tation) A bounded solution to the ODE (5), if exists, must be unique. More precisely, suppose

a(s) solves the ODE (5) and sups∈[0,∞)|a(s)| ≤ C < ∞ for some constant C > 0. Then we

must have

a(s) = E?exp?−sATµ

??

for any s ≥ 0.

?] as

−

0

(6)

Proof: In terms of S(t), we can rewrite E[exp?−sATµ

E[exp?−sATµ

where the notation Esmeans that the process {S(t)} starts from s, i.e. S(0) = s. By Itˆ o’s

formula, we have that

?] =Es

??∞

0

µexp

?

?t

[µ + S(u)]du

?

dt

?

,

(7)

Mt:= a(S(t))exp

?

−

?t

0

[µ + S(u)]du

?

+

?t

0

µexp

?

−

?v

0

[µ + S(u)]du

?

dv

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is a local martingale. Indeed, since

da(S(t))=

a?(S(t))[rS(t)dt + σS(t)dW(t)] +1

2a??(S(t))σ2S2(t)dt

= [(S(t) + µ)a(S(t)) − µ]dt + a?(S(t))σS(t)dW(t),

where the last equality follows from the fact that a(s) solves the ODE (5), and

dexp

?

−

?t

0

[µ + S(u)]du

?

= exp

?

−

?t

0

[µ + S(u)]du

?

· {−[µ + S(t)]}dt ,

we obtain by some algebra that

dMt

= exp

?

−

?t

0

[µ + S(u)]du

?

· a?(S(t))σS(t)dW(t),

which implies that {Mt} is a local martingale. Actually, {Mt} is a true martingale as Mtis

uniformly bounded, supt≥0|Mt| ≤ supt≥0

S(u) ≥ 0. Thus, a(s) = a(S(0)) = Es[M0] = Es[Mt]. Letting t → +∞, the first term in Mt

goes to zero almost surely because a(·) is bounded, and therefore

?∞

almost surely. Accordingly, by the dominated convergence theorem,

?

C +?t

0µe−µvdv

?

= C + 1 < ∞, because µ > 0 and

Mt→

0

µexp

?

−

?v

0

{µ + S(u)}du

?

dv,

a(s) = Es[ lim

t→∞Mt] = Es

??∞

0

µexp

?

−

?v

0

{µ + S(u)}du

?

dv

?

= E[exp?−sATµ

?],

where the last equality holds due to (7). ?

Theorem 3.1 implies that if we can find a particular bounded solution to the ODE (5),

it must have the stochastic representation in (6). To find such a one, consider a difference

equation (or a recursion) for a function H(ν) defined on (−1,α1)

h(ν)H(ν) = νH(ν − 1) for any ν ∈ (0,α1),and

H(0) = 1,

(8)

h(ν) ≡ µ − G(ν) = −σ2

2ν2−

?

r −σ2

2

?

ν + µ = −σ2

2(ν − α1)(ν − α2).

(9)

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In general, if the above difference equation (8) has one solution H1(ν), then there exist an

infinite number of solutions to (8). In fact, any function in the following class

?

also solves (8). This also partly explains why very few people investigated Asian option pricing

H(ν) = θ(ν)H1(ν) : for some periodic function θ(ν) s.t. θ(ν) = θ(ν − 1) for any ν ∈ (0,α1)

?

based on this recursion. However, we shall show next that the difference equation (8) has a

unique solution if we restrict our attention to random variables.

Theorem 3.2. (A particular bounded solution to the ODE (5)) If there exists a non-

negative random variable X such that H(ν) := E[Xν] satisfies the difference equation (8), then

the Laplace transform of X, i.e. E[e−sX], solves the nonhomogeneous ODE (5).

Proof: Denote the Laplace transform of X by y(s) = E[e−sX], for s ≥ 0. Note that for any

a ∈ (0,min(α1,1)), we have

?+∞

where the second equality holds due to integration by parts. Taking expectations on both sides

0

s−ae−sXds = Γ(1 − a)Xa−1

and

?+∞

0

s−a−1?e−sX− 1?ds = −Γ(1 − a)

a

Xa,

of the two equations above and applying Fubini’s theorem yields

E[Xa−1] =

1

Γ(1 − a)

?∞

0

s−ay(s)ds

and

E[Xa] = −

a

Γ(1 − a)

?∞

0

s−a−1(y(s) − 1)ds.

Thus, by the difference equation (8), we have

−

ah(a)

Γ(1 − a)

?∞

0

s−a−1(y(s) − 1)ds =

a

Γ(1 − a)

?∞

0

s−ay(s)ds,

i.e.

0 =

?∞

0

s−a−1[sy(s) + h(a)(y(s) − 1)]ds,

where h(a) is given by (9). Setting s = e−xand z(x) = y(s) − 1, we have

?∞

0 =

−∞

eax?e−x(z(x) + 1) + h(a)z(x)?dx,

for any a ∈ (0,min(α1,1)).

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For simplicity of notations, rewrite h(a) as h(a) = h0a2+h1a+h2, with h0= −σ2

and h2= µ. Note that integration by parts yields

2, h1= −r+σ2

2,

?∞

−∞

eaxaz(x)dx = −

???

0=

?∞

−∞

eaxz?(x)dx

and

?∞

−∞

eaxa2z(x)dx =

?∞

−∞

eaxz??(x)dx

because [z(x)eax]

+∞

x=−∞= 0 and [z?(x)eax]

?∞

=

−∞

???

∞

x=−∞= 0. Then for any a ∈ (0,min(α1,1)),

eax?e−x(z(x) + 1) +?h0a2+ h1a + h2

eax?e−x(z(x) + 1) + h0z??(x) − h1z?(x) + h2z(x)?dx.

By the uniqueness of the moment generating function, we have an ODE

−∞

?∞

?z(x)?dx

h0z??(x) − h1z?(x) + h2z(x) + e−x(z(x) + 1) = 0.

Now transferring the ODE for z(x) back to that for y(s), with s = e−xwe have z(x) = y(s)−1,

z?(x) = −sy?(s), and z??(x) = sy?(s) + s2y??(s). Then the ODE becomes

h0s2y??(s) + (h1+ h0)sy?(s) + (h2+ s)y(s) = h2.

Substituting h0, h1, and h2into the above equation, we have the nonhomogeneous ODE (5). ?

Theorem 3.3. Under the BSM, we have

ATµ=d

2

σ2

Z(1,−α2)

Z(α1)

(10)

and therefore

E[Aν

Tµ] =

?2

σ2

?νΓ(ν + 1)Γ(α1− ν)Γ(1 − α2)

Γ(α1)Γ(−α2+ ν + 1)

,

for any ν ∈ (−1,α1).

(11)

Here Z(a,b) denotes a beta random variable with parameters a and b, Z(a) a gamma random

variable with scale parameter 1 and shape parameter a, and Γ(·) the gamma function. Moreover,

Z(1,−α2) and Z(α1) are independent with α1and α2given by (4).

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Proof: Consider a random variable χ such that χ =d

?2

and furthermore, it can be easily verified that E[χν] solves the difference equation (8). By

2

σ2Z(1,−α2)/Z(α1). Then

E[χν] =

σ2

?νΓ(ν + 1)Γ(α1− ν)Γ(1 − α2)

Γ(α1)Γ(−α2+ ν + 1)

,

for any ν ∈ (−1,α1),

Theorem 3.2, we conclude that a∗(s) := E[e−sχ] for any s ≥ 0 is a particular bounded solution

to the ODE (5). As a result, it follows from Theorem 3.1 that ATµ=dχ =d

2

σ2Z(1,−α2)/Z(α1),

which gives the distribution of ATµ. ?

Remarks: 1. There are various ways to show that E[Aν

Tµ] satisfies recursions similar to (8)

for general processes. For example, Dufresne [20] used time reversal and Itˆ o’s formula to derive

the recursion (8) for the BSM, and Carmona et al. [11] obtained similar recursions for general

L´ evy-type processes. In Appendix C we shall give a new proof for a recursion similar to (8)

under the HEM, although that proof is not needed to study Asian options.

2. The result (10) coincides with that in Yor [50] and Matsumoto and Yor [33]. However,

compared with the existing proofs involving Bessel processes and Lamperti’s representation,

our approach is simpler and more elementary. Furthermore, we illustrate in Section 4 that our

approach is more general, because it can be extended to the case of the HEM.

3.2 Pricing Formulae and Hedging Parameters under the BSM

Theorem 3.4. Under the BSM, for every µ and ν such that µ > 0 and ν ∈ (0,α1− 1), the

double-Laplace transform of XE(S0

XAt− e−k)+w.r.t. t and k is given by:

?2S0

L(µ,ν) =

X

µν(ν + 1)

Xσ2

?ν+1Γ(ν + 2)Γ(α1− ν − 1)Γ(1 − α2)

Γ(α1)Γ(ν − α2+ 2)

.

(12)

Therefore, the Asian option price is equal to:

P(t,k) =e−rt

t

L−1(L(µ,ν))

???k=ln(X/Kt),

where L−1, a function of t and k, denotes the Laplace inversion of L. Furthermore, for any

maturity t and strike K, two common greeks delta ∆(P(t,k)) and gamma Γ(P(t,k)) can be

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calculated as follows

∆(P(t,k)) =

∂

∂S0P(t,k) =e−rt

t

L−1

?

?

XSν

µν

0

?

2

Xσ2

?ν+1Γ(ν + 2)Γ(α1− ν − 1)Γ(1 − α2)

?ν+1Γ(ν + 2)Γ(α1− ν − 1)Γ(1 − α2)

Γ(α1)Γ(ν − α2+ 2)

????k=ln(X/Kt)

????k=ln(X/Kt).

Γ(P(t,k)) =

∂2

∂S2

0

P(t,k) =e−rt

t

L−1

XSν−1

µ

0

?

2

Xσ2

Γ(α1)Γ(ν − α2+ 2)

Proof. Combining (11) with (1) yields (12). The two greeks can be obtained by interchanging

derivatives and integrals based on Theorem A. 12 on pp. 203-204 in Schiff [39]. ?

Theorem 3.4 requires (0,α1− 1) to be nonempty, i.e. α1> 1, which means some Laplace

parameter µ > 0 may be disqualified. Nonetheless, a broad range of µ meets the requirement.

For example, it is sufficient to have µ > r, because G(1) = r, α1solves the equation G(x) = µ

and G(x) − µ is a continuous function. Furthermore, this restriction µ > r does not present

any difficulty in term of numerical Laplace inversion.

4Pricing Asian Options under the HEM

In the HEM, the asset return process {X(t) : t ≥ 0} under a risk-neutral measure P is given by

?

where r is the risk-free rate, σ the volatility, ζ := E(eY1) − 1 =?m

{W(t) : t ≥ 0} the standard Brownian motion, {N(t) : t ≥ 0} a Poisson process with rate λ,

and {Yi: i ∈ N} i.i.d. hyper-exponentially distributed random variables with the probability

density function (pdf)

X(t) =

r −1

2σ2− λζ

?

t + σW(t) +

N(t)

?

i=1

Yi,X(0) = 0,

i=1

piηi

ηi−1+?n

j=1

qjθj

θj+1− 1,

fY(y) =

m

?

i=1

piηie−ηiyI{y≥0}+

n

?

j=1

qjθjeθjyI{y<0},

(13)

with pi> 0, ηi> 1, for i = 1,··· ,m, qj> 0, ηj> 0, for j = 1,··· ,n, and?m

Due to the jumps, the risk-neutral measure is not unique. Here we assume the risk-neutral

i=1pi+?n

j=1qj= 1.

measure P is chosen within a rational expectations equilibrium setting such that the equilibrium

price of an option is given by the expectation under P of the discounted option payoff. For

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details, refer to Kou [28]. It is worth mentioning that when m = n = 0 the HEM is reduced to

the BSM, and when m = n = 1 the HEM is reduced to the double exponential jump diffusion

model. The L´ evy exponent of {Xt} is given by

G(x) :=E?exX(t)?

t

=1

2σ2x2+

?

r −1

2σ2− λζ

?

x + λ

m

?

i=1

piηi

ηi− x+

n

?

j=1

qjθj

θj+ x− 1

(14)

for any x ∈ (−θ1,η1). It can be shown (see Cai [8]) that for any µ > 0, the equation G(x) = µ

has exactly (m + n + 2) real roots β1,µ, ···, βm+1,µ, γ1,µ, ···, γn+1,µsatisfying

−∞ < γn+1,µ< −θn< γn,µ< ··· < −θ1< γ1,µ< 0 < β1,µ< η1< ··· < βm,µ< ηm< βm+1,µ< ∞.

(15)

Additionally, the infinitesimal generator of {S(t) = S(0)eX(t): t ≥ 0} is given by

Lf(s) =σ2

2s2f??(s) + (r − λζ)sf?(s) + λ

?+∞

−∞

[f(seu) − f(s)]fY(u)du,

(16)

for any twice continuously differentiable function f(·).

4.1 Distribution of ATµunder the HEM

Consider the following nonhomogeneous ordinary integro-differential equation (OIDE)

Ly(s) = (s + µ)y(s) − µ,

(17)

where L is given by (16). Similarly as in the BSM, we have the following theorem.

Theorem 4.1. (Uniqueness of the solution of the OIDE (17) via a stochastic repre-

sentation) There is at most one bounded solution to the OIDE (17). More precisely, suppose

a(s) solves (17) and sups∈[0,∞)|a(s)| ≤ C < ∞ for some constant C > 0. Then we must have

a(s) = E?exp?−sATµ

Proof : See Appendix A. ?

??

for any s ≥ 0.

(18)

Since the proofs for Theorem 3.1 and 4.1 involve only Itˆ o’s formula (see Section 1.2 and

1.3 of Øksendal and Sulem [36] or Applebaum [5] for more general Itˆ o formulae), the result

12

Page 13

holds for more general underlying process S(t) such as exponential L´ evy processes and L´ evy

diffusions. Next, we look for a particular bounded solution to the OIDE (17), which has the

stochastic representation in (18). Consider a difference equation (or a recursion) for a function

H(ν) defined on (−1,β1)

h(ν)H(ν) = νH(ν − 1)for any ν ∈ (0,β1),and

H(0) = 1,

(19)

where

h(ν) ≡µ − G(ν) = µ −1

??m+1

Here β1, ···, βm+1, γ1, ···, γn+1 are actually β1,µ, ···, βm+1,µ, γ1,µ, ···, γn+1,µ, which are

(m + n + 2) real roots of the equation G(x) = µ satisfying the condition (15).

2σ2ν2− (r −1

i=1(βi− ν)?n+1

2σ2− λζ)ν − λ

m

?

i=1

piηi

ηi− ν+

n

?

j=1

qjθj

θj+ ν− 1

=

?σ2

2

j=1(−γj+ ν)

j=1(θj+ ν)

?m

i=1(ηi− ν)?n

.

(20)

Theorem 4.2. (A particular bounded solution to the OIDE (17)) If there is a nonneg-

ative random variable X such that H(ν) := E[Xν] satisfies the difference equation (19), then

the Laplace transform of X, i.e. E[e−sX], solves the nonhomogeneous OIDE (17).

Proof : See Appendix B. ?

Theorem 4.3. Under the HEM, we have

ATµ=d

2

σ2

Z(1,−γ1)?n

j=1Z(θj+ 1,−γj+1− θj)

Z(βm+1)?m

i=1Z(βi,ηi− βi)

,

(21)

where all the gamma and beta random variables on the RHS are independent, and therefore for

any ν ∈ (−1,β1),

E[Aν

?2

Tµ]

?νΓ(1 + ν)Γ(1 − γ1)

=

σ2

Γ(1 − γ1+ ν)

·

n

?

j=1

?Γ(θj+ 1 + ν)Γ(1 − γj+1)

Γ(1 − γj+1+ ν)Γ(θj+ 1)

?

·

m

?

i=1

?Γ(βi− ν)Γ(ηi)

Γ(ηi− ν)Γ(βi)

?

·Γ(βm+1− ν)

Γ(βm+1)

(22)

.

13

Page 14

Proof: Consider a random variable χ that is defined by the right side of (21). Then simple

algebra shows that E[χν] is given by the right side of (22). Moreover, we can verify that E[χν]

solves the difference equation (19). By Theorem 4.2, a∗(s) := Ee−sχ, for any s ≥ 0, is a

particular bounded solution to the OIDE (17), and the proof is terminated via Theorem 4.1. ?

4.2 Pricing Formulae for Asian Options and Hedging Parameters under the

HEM

Theorem 4.4. Under the HEM, for every µ and ν such that µ > 0 and ν ∈ (0,β1− 1), the

double-Laplace transform of XE(S0

?2S0

n

?

Therefore, the Asian option price is equal to

XAt− e−k)+w.r.t. t and k is given by:

?ν+1Γ(2 + ν)Γ(1 − γ1)

?Γ(θj+ 2 + ν)Γ(1 − γj+1)

L(µ,ν) =

X

µν(ν + 1)

Xσ2

Γ(2 − γ1+ ν)

?

·

j=1

Γ(2 − γj+1+ ν)Γ(θj+ 1)

·

m

?

i=1

?Γ(βi− ν − 1)Γ(ηi)

Γ(ηi− ν − 1)Γ(βi)

?

·Γ(βm+1− ν − 1)

Γ(βm+1)

. (23)

P(t,k) =e−rt

t

L−1(L(µ,ν))|k=ln(X/Kt).

And two common greeks delta ∆(P(t,k)) and gamma Γ(P(t,k)) can be calculated as follows

L−1?XSν

·

j=1

∆(P(t,k)) =

∂

∂S0P(t,k) =e−rt

n

?

t

0

µν

(

2

Xσ2)ν+1·Γ(2 + ν)Γ(1 − γ1)

Γ(2 − γ1+ ν)

?

i=1

?Γ(θj+ 2 + ν)Γ(1 − γj+1)

Γ(2 − γj+1+ ν)Γ(θj+ 1)

·

m

?

?Γ(βi− ν − 1)Γ(ηi)

Γ(ηi− ν − 1)Γ(βi)

?

·Γ(βm+1− ν − 1)

Γ(βm+1)

????k=ln(X/Kt)

Γ(P(t,k)) =∂2

∂S2

?

0

n

P(t,k) =e−rt

t

L−1?XSν−1

0

µ

(

2

Xσ2)ν+1·Γ(2 + ν)Γ(1 − γ1)

Γ(2 − γ1+ ν)

?

i=1

·

j=1

?Γ(θj+ 2 + ν)Γ(1 − γj+1)

Γ(2 − γj+1+ ν)Γ(θj+ 1)

·

m

?

?Γ(βi− ν − 1)Γ(ηi)

Γ(ηi− ν − 1)Γ(βi)

?

·Γ(βm+1− ν − 1)

Γ(βm+1)

????k=ln(X/Kt)

Proof. The proof is similar to that of Theorem 3.4. ?

5 Pricing Asian Options via a Two-Sided Euler Inversion Al-

gorithm with a Scaling Factor

In this section, we intend to price Asian options under both the BSM and the HEM by inverting

L(µ,ν) in (12) and (23) numerically. The algorithm used here is proposed in Petrella [37],

14

Page 15

which, as a generalization of the one-sided Euler inversion algorithm ([1] and [14]), introduces

a two-sided Euler inversion with a scaling factor.

The inversion formula in Petrella [37] to get f(t,k) from its Laplace transform L(µ,ν) is

exp(A1/2 + A2/2)

4tk

?

+2

(−1)sRe

?∞

s=0

?

−

f(t,k)=

×L

?A1

∞

?

∞

?

∞

2t,A2

2k

?

+ 2

∞

?

?A1

s=0

(−1)sRe

?

−L

?A1

,A2

2k

2t,A2

??

−ijπ

t

2k−iπ

k−isπ

k

??

s=0

?

−L

2t−iπ

t

−isπ

t

+2

j=0

(−1)jRe

?

?∞

s=0

(−1)sL

?A1

?A1

2t−iπ

t

,A2

2k−iπ

k−isπ

k

??

???

+2

j=0

d− e−

(−1)jRe

?

(−1)sL

2t−iπ

t

−ijπ

t

,A2

2k+iπ

k+isπ

k

e+

d,

(24)

where the two errors are given by

e+

d

=

∞

?

?

j2=1

∞

?

j1=0

e−(j1A1+j2A2)f((2j1+ 1)t,(2j2+ 1)k) +

∞

?

j1=1

e−j1A1f((2j1+ 1)t,k), (25)

e−

d

=

−1

j2=−∞

∞

?

j1=0

e−(j1A1+j2A2)f((2j1+ 1)t,(2j2+ 1)k),

(26)

and A1and A2(to be specified later) are some inversion parameters used to control the errors.

The inversion appears to be accurate, stable, and easy to implement. For example, under

the BSM, the prices produced via our algorithm highly agree with benchmarks generated by the

other three important methods by Linetsky, Geman-Yor-Shaw, and Vecer, and it is stable even

for low volatilities (e.g. σ = 0.05). The algorithm is easy to implement primarily because the

closed-form Laplace transform involves only gamma functions. We also study the algorithm’s

stability, derive a discretization error bound, and conduct a comparison with the Fourier and

Laplace inversion algorithm by Fusai [22]; see the online supplement.

In the inversion, we have to calculate alternating series of the form?∞

expression (24). To accelerate the convergence rate, we adopt the idea of Euler transformation

i=0(−1)iai in the

15

Page 16

(see [1] and [14]), and approximate?∞

i=0(−1)iaiby E(m,n) :=?m

k=0

m!

k!(m−k)!2−mSn+k, where

Sj:=?j

We use (n1,m1) and (n2,m2) to express parameters involved in Euler transformations for Euler

i=0(−1)iai. Since there are two transforms, Euler transformation will be used twice.

inversion w.r.t. t and k, respectively. More precisely, for the first infinite sum and the inner

series of both the third and fourth double sum on the RHS of (24), we use Euler transformation

with parameters (n2,m2); while for the second infinite sum and the outer series of both the

third and fourth double sum on the RHS of (24), we use Euler transformation with parameters

(n1,m1). As suggested in Abate and Whitt [1], we shall set m1= n1+ 15 and m2= n2+ 15.

In the inversion algorithm there are several parameters to be chosen: (1) (n1, n2). The larger

n1and n2will lead to better accuracy at the cost of computation time. In our experiments,

n1 = n2 = 35 seems to achieve excellent accuracy even for low volatilities (e.g. σ = 0.05);

usually even n1 = 15 and n2 = 35 can yield good accuracy if the volatility is not low. (2)

(A1, A2). (3) The scaling factor X. One appealing feature of the inversion algorithm is its

insensitivity to the choices of A1, A2, and X even for low volatilities. In other words, for

wide ranges of parameters A1, A2, and X, the numerical results are almost identical. This

is illustrated in Figure 1 in the online supplement. For convenience, in most cases we simply

select A1= 28, A2= 40, and, as suggested in Petrella [37],

X = Kt · ek= Kt · exp

?

min

?

A2

2/(σ√t),A2

4

??

.

(27)

for practical implementation, although one can freely choose other values.

5.1 Pricing Asian Options under the BSM

5.1.1 Comparison of Accuracy with Other Methods

To check the accuracy of our double-Laplace (DL) inversion algorithm, we consider seven test

cases in Table 1, which are frequently used in the literature ([21], [32], [43], [17], [48]). A

detailed comparison of accuracy between our double-Laplace inversion algorithm with those

obtained from six other existing methods is given in Table 2. From Table 2, we can see that our

DL prices highly agree with benchmarks of existing methods. Specifically, our DL prices agree

16

Page 17

Case No.

1

2

3

4

5

6

7

S0

2.0

2.0

2.0

1.9

2.0

2.1

2.0

K

2

2

2

2

2

2

2

r

0.02

0.18

0.0125

0.05

0.05

0.05

0.05

σ

0.10

0.30

0.25

0.50

0.50

0.50

0.50

t

1

1

2

1

1

1

2

Table 1: Seven test cases in the literature.

with both Geman-Yor-Shaw’s and Linetsky’s results to ten decimal points and with Vecer’s

results to six decimal points. Moreover, our DL prices also agree with Zhang’s PDE results to

six decimal points. These consistence indicates that our double-Laplace inversion algorithm is

very accurate and can also be used as benchmarks for pricing Asian options.

Case

1

2

3

4

5

6

7

DL Prices

0.0559860415

0.2183875466

0.1722687410

0.1931737903

0.2464156905

0.3062203648

0.3500952190

Linetsky

0.0559860415

0.2183875466

0.1722687410

0.1931737903

0.2464156905

0.3062203648

0.3500952190

GY-Shaw

0.0559860415

0.2183875466

0.1722687410

0.1931737903

0.2464156905

0.3062203648

0.3500952190

Vecer

0.055986

0.218388

0.172269

0.193174

0.246416

0.306220

0.350095

Zhang

0.055986

0.218388

0.172269

0.193174

0.246416

0.306220

0.350095

GYS-Mellin

0.0559856

0.218359

0.172263

0.193060

0.246522

0.306501

0.348924

MAE3

0.055986

0.218663

0.172263

0.193188

0.246382

0.306139

0.349909

Table 2: Comparison of accuracy with other existing methods. The parameters associated with our double-

Laplace (DL) inversion method are n1 = 35, n2 = 55, A1 = 28, A2 = 40, and X given by (27). Results of

Linetsky’s eigenfunction expansion method are taken from Table 3 in [32]. Vecer’s PDE results are from Table

A in of [48]. The “ GYS-Mellin” numbers are taken from a Mellin transformed-based approximation in [43]. The

other three columns, including “GY-Shaw,” “Zhang,” and “MAE3”, are all taken from the table on p. 383 in

[17], and correspond to the methods in [41], [51], and [17], respectively. Our numerical DL prices are calculated

using Matlab 7.1 on a desktop with Quad CPU 2.66 GHz.

Although Shaw’s GYS-Mellin results and Dewynne and Shaw’s MAE3 results seem less

accurate than other methods, they are still sufficiently accurate in practice and moreover, these

two methods have their own advantages. First, Shaw’s GYS-Mellin method turns out to be very

fast. For example, in Case 1 in Table 2 and on a desktop with Quad CPU 2.66 GHz, it takes

only 0.047 seconds to produce one result; whereas our pricing method requires 0.563 seconds

to match the GYS-Mellin result to five decimal points. Second, these two methods work better

than most other methods when the volatility is extremely low; for details, see Section 5.1.2.

Note that although in our theorems the dividend δ = 0, they can be easily extended to the

case of nonzero dividends. Indeed for the two families of the extended seven cases in Section 6.2

17

Page 18

(δ > r) and Section 6.3 (δ = r) in [17], our DL prices, GY-Shaw prices (or its variant CIBess

prices) and Zhang’s results agree with one another to six or seven decimal points, hence being

more accurate than MAE3; see Table 3.

Extension of Seven Cases in Table 1 as δ > r (see Section 6.2 in [17]).

Krδσt

DL Prices

2 0.020.040.11 0.0357853

2 0.18 0.360.310.0522607

2 0.01250.025 0.252

2 0.05 0.1 0.51

2 0.05 0.1 0.51

2 0.05 0.1 0.51

2 0.05 0.10.52

Extension of Seven Cases in Table 1 as δ = r (see Section 6.3 in [17]).

Krδσt

DL Prices

2 0.02 0.020.1 1 0.0451431

2 0.18 0.18 0.31

2 0.0125 0.01250.252

20.05 0.05 0.51

2 0.050.050.51

2 0.05 0.05 0.51

2 0.05 0.050.52

Case

1∗

2∗

3∗

4∗

5∗

6∗

7∗

S0

2.0

2.0

2.0

1.9

2.0

2.1

2.0

GY-Shaw

0.0357853

0.0522607

0.145308

0.147562

0.191747

0.242316

0.240495

MAE3

0.0357854

0.0522755

0.145310

0.147618

0.191760

0.242283

0.240564

Zhang

0.0357853

0.0522607

0.145308

0.147562

0.191747

0.242316

0.240495

0.145308

0.147562

0.191747

0.242316

0.240495

Case

1∗∗

2∗∗

3∗∗

4∗∗

5∗∗

6∗∗

7∗∗

S0

2.0

2.0

2.0

1.9

2.0

2.1

2.0

CIBess

0.0451431

0.115188

0.158380

0.169202

0.217815

0.272924

0.291315

MAE3

0.0451431

0.115188

0.158378

0.169238

0.217805

0.272869

0.291264

Zhang

0.0451431

0.115188

0.158380

0.169202

0.217815

0.272924

0.291315

0.115188

0.158380

0.169202

0.217815

0.272924

0.291315

Table 3: Comparison of accuracy in extended cases when δ > r or δ = r

5.1.2Comparison of Behaviors for Low Volatilities

It is well known that many numerical methods for Asian option pricing do not perform well for

low volatilities (see [15, 21, 42]). Here we would like to conduct cross-comparisons of behaviors

for reasonably low (e.g., σ = 0.05) and extremely low volatilities (e.g., σ ≤ 0.01) between our

double-Laplace inversion method and the three methods discussed in [17], GY-Shaw (or its

variants GYS-Full and CIBess), MAE3, and Zhang’s method, for three cases δ < r, δ > r and

δ = r, where δ denotes the dividend (in [17] the dividend is denoted by q).

To investigate the behaviors of these algorithms in the case of low volatilities, we modify

the test case 1 in Table 1 and the test cases 1∗and 1∗∗in Table 3 by letting the volatility σ

be small; see Table 4. Note that the parameter settings are the same as in Section 6.1–6.3 in

[17]. Table 4 demonstrates that when the volatility is reasonably small (= 0.05), the numerical

results obtained by the four methods coincide with one another to six or seven decimal points.

18

Page 19

Nonetheless, when the volatility is extremely small (≤ 0.01), MAE3 and Zhang’s results are

still available and highly agree with each other, but our double-Laplace inversion method and

GY-Shaw methods no longer work. Consequently, we conclude that our methods and GY-Shaw

methods are reliable in the case of normal or reasonably low volatilities (σ ≥ 0.05). However, we

may use Dewynne and Shaw’s asymptotic method or Zhang’s PDE method for the extremely

low volatilities (σ < 0.05).

Extension of Case 1 in Table 1 when δ < r and σ is extremely small (Section 6.1 in [17]).

Case

S0

Krδσt

DL Prices

1 2.02 0.020 0.11 0.0559860

1A 2.02 0.020 0.051 0.0339412

1B 2.02 0.020 0.011

1C 2.02 0.020 0.0051

1D2.02 0.020 0.0011

Extension of Case 1∗in Table 3 when δ > r and σ is extremely small (see Section 6.2 in [17]).

Case

S0

Krδσt

DL Prices

1∗

2.02 0.02 0.040.11 0.0357853

1A∗

2.02 0.02 0.040.051 0.0140247

1B∗

2.0 2 0.020.04 0.011 NA

1C∗

2.02 0.02 0.040.0051 NA

1D∗

2.02 0.020.04 0.0011 NA

Extension of Case 1∗∗in Table 3 when δ = r and σ is extremely small (see Section 6.3 in [17]).

Case

S0

Krδσt

DL Prices

1∗∗

2.02 0.020.02 0.110.0451431

1A∗∗

2.02 0.02 0.020.051 0.0225755

1B∗∗

2.02 0.020.02 0.011 NA

1C∗∗

2.0 2 0.020.02 0.0051 NA

1D∗∗

2.02 0.02 0.020.0011 NA

GY-Shaw

0.0559860

0.0339412

NA

NA

NA

MAE3

0.0559860

0.0339412

0.0199278

0.0197357

0.0197353

Zhang

0.0559860

0.0339412

0.0199278

0.0197357

0.0197353

NA

NA

NA

GYS-full

0.0357853

0.0140247

NA

NA

NA

MAE3

0.0357854

0.0140248

0.000190254

3.7991×10−7

o(10−70)

Zhang

0.0357853

0.0140247

0.000190254

3.7993×10−7

o(10−72)

CIBess

0.0451431

0.0225755

NA

NA

NA

MAE3

0.0451431

0.0225755

0.00451536

0.00225768

0.000451537

Zhang

0.0451431

0.0225755

0.00451536

0.00225768

0.000451537

Table 4: Comparison of accuracy for extremely low volatilities when δ < r, δ > r or δ = r

5.2Pricing Asian Options under the HEM

In this section, we price Asian options numerically under the HEM by inverting the double-

Laplace transform (23) via the two-sided Euler inversion algorithm. Without loss of generality,

we concentrate on Kou’s model, which along with the BSM are the most important special cases

of the HEM. Table 5 and Table 6 provide the numerical results of Asian option prices using

double-Laplace transform in the cases of λ = 3 and λ = 5, respectively. The double-Laplace

inversion method seems to be still accurate under Kou’s model. Furthermore, our algorithm is

19

Page 20

efficient because it takes only around 6 seconds to produce one numerical result on a desktop

with Quad CPU 2.66 GHz.

σ

0.05

0.05

0.05

0.05

0.05

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.5

K

90

95

100

105

110

90

95

100

105

110

90

95

100

105

110

90

95

100

105

110

90

95

100

105

110

90

95

100

105

110

DL Prices

13.41924

8.98812

4.95673

2.13611

0.83091

13.48451

9.20478

5.53662

2.88896

1.33809

14.03280

10.32293

7.21244

4.78516

3.02270

15.19639

11.92926

9.14769

6.86049

5.04029

16.68984

13.73384

11.17115

8.99114

7.16816

18.35379

15.62810

13.22860

11.13944

9.33799

MC Prices

13.42054

8.98730

4.95681

2.13453

0.82995

13.47574

9.20559

5.53619

2.88890

1.33781

14.03489

10.32461

7.21556

4.78822

3.02558

15.19689

11.93168

9.15063

6.86412

5.04400

16.69294

13.73747

11.17579

8.99645

7.17317

18.35851

15.63389

13.23485

11.14602

9.34396

Std Err

0.00048

0.00095

0.00162

0.00208

0.00193

0.00071

0.00135

0.00207

0.00249

0.00238

0.00193

0.00276

0.00343

0.00380

0.00380

0.00350

0.00431

0.00495

0.00533

0.00545

0.00506

0.00586

0.00649

0.00692

0.00716

0.00658

0.00738

0.00804

0.00853

0.00887

Abs Err

-0.00130

0.00082

-0.00008

0.00158

0.00096

0.00877

-0.00081

0.00043

0.00006

0.00028

-0.00289

-0.00168

-0.00312

-0.00306

-0.00288

-0.00050

-0.00242

-0.00294

-0.00363

-0.00331

-0.00310

-0.00363

-0.00464

-0.00531

-0.00501

-0.00472

-0.00579

-0.00625

-0.00658

-0.00597

Rel Err

0.0097%

0.0091%

0.0016%

0.0740%

0.1200%

0.0651%

0.0088%

0.0078%

0.0021%

0.0210%

0.0206%

0.0163%

0.0432%

0.0638%

0.0952%

0.0033%

0.0203%

0.0321%

0.0529%

0.0656%

0.0186%

0.0264%

0.0415%

0.0590%

0.0698%

0.0257%

0.0370%

0.0472%

0.0590%

0.0639%

Table 5: Numerical results of Asian option prices under Kou’s model with λ = 3. Other parameters of the

model are set as: S0 = 100, r = 0.09, t = 1.0, p1 = 0.6, q1 = 0.4, and η1 = θ1 = 25. Parameters of the

algorithm are set as: n1 = 35, n2 = 55, A1 = 38.9, A2 = 40, and X = Ktexp{min(A2/θ,A2/10)}, where

θ = 2/p(σ2+ 2p1λ/η2

0.0001; Std Err is the standard error of the MC price; Abs Err and Rel Err are absolute and relative errors,

respectively.

1+ 2q1λ/θ2

1)t; DL prices are obtained by double-Laplace inversion; MC price are Monte

Carlo simulation estimates obtained by simulating 1 million paths and setting the discretization step size to be

Appendix A. Proof of Theorem 4.1

Proof: For the HEM, the argument is similar to that for the BSM in Theorem 3.1 except that we

need to show the process {M(t)} is still a local martingale in the jump diffusion case. Indeed,

20

Page 21

σK

90

95

LL Price

13.47952

9.16588

5.38761

2.72681

1.28264

13.55964

9.41962

5.91537

3.35071

1.74896

14.17380

10.53795

7.48805

5.09001

3.32061

15.33688

12.10723

9.35336

7.08059

5.26109

16.81490

13.87995

11.33257

9.16131

7.34063

18.46259

15.75006

13.36027

11.27716

9.47826

MC Price

13.47729

9.16468

5.38519

2.72275

1.28034

13.56384

9.42350

5.91707

3.35124

1.74934

14.17586

10.53973

7.48864

5.09000

3.31967

15.33728

12.10732

9.35297

7.07908

5.25875

16.81475

13.87950

11.33142

9.15913

7.33721

18.46136

15.74865

13.35842

11.27410

9.47384

Std Err

0.00075

0.00135

0.00208

0.00252

0.00242

0.00102

0.00173

0.00246

0.00287

0.00281

0.00217

0.00300

0.00367

0.00405

0.00409

0.00367

0.00448

0.00511

0.00551

0.00565

0.00518

0.00598

0.00662

0.00706

0.00730

0.00667

0.00748

0.00814

0.00864

0.00898

Error

0.00223

0.00120

0.00242

0.00406

0.00230

-0.00420

-0.00388

-0.00170

-0.00053

-0.00038

-0.00206

-0.00178

-0.00059

0.00001

0.00096

-0.00040

-0.00009

0.00039

0.00151

0.00234

0.00015

0.00045

0.00115

0.00218

0.00342

0.00123

0.00141

0.00185

0.00306

0.00443

0.05

0.05

0.05

0.05

0.05

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.5

100

105

110

90

95

100

105

110

90

95

100

105

110

90

95

100

105

110

90

95

100

105

110

90

95

100

105

110

Table 6:

the model are set as: S0 = 100, r = 0.09, t = 1.0, p1 = 0.6, q1 = 0.4, and η1 = θ1 = 25. Parameters

of the algorithms are n1 = 35, n2 = 55, A1 = 35.9, A2 = 40, and X = Ktexp{min(A2/θ,A2/10)}, where

θ = 2/p(σ2+ 2p1λ/η2

by Ito’s formula for jump diffusions, we have

Numerical results of Asian option prices under Kou’s model with λ = 5.Other parameters of

1+ 2q1λ/θ2

1)t. DL prices are obtained by the double Laplace inversion; MC prices are

Monte Carlo simulation estimates by using 1 million paths and setting the discretization step size to be 0.0001.

da(S(t))=

a?(S(t−))dSc(t) +1

?

+d

[a(S(u)) − a(S(u−))].

2a??(S(t−))d?Sc,Sc?(t) + d

?

?

0<u≤t

[a(S(u)) − a(S(u−))]

=(r − λζ)S(t−)a?(S(t−)) +1

?

2σ2S2(t−)a??(S(t−))

dt + σS(t−)a?(S(t−))dW(t)

0<u≤t

21

Page 22

Since a(s) solves the OIDE (17), we have

da(S(t))= [(S(t−) + µ)a(S(t−)) − µ]dt + σS(t−)a?(S(t−))dW(t)

+d

[a(S(u)) − a(S(u−))] − λ

?

0<u≤t

?+∞

−∞

[a(S(t−)ey) − a(S(t−))]fY(y)dydt.

Note that

dM(t)= exp

?

?

−

?t

0

{µ + S(u)}du

?

{µ + S(u)}du

· da(S(t)) + a(S(t−)) · dexp

??

?

?t

−

?t

0

{µ + S(u)}du

?

+da(S(t), exp

?

−

?t

0

+ µexp

?

−

0

{µ + S(u)}du

?

dt.

Plugging d(a(S(t)) into dM(t) yields

dM(t)= exp

?

−

?

?t

−

?

0

{µ + S(u)}du

?t

?t

?

a?(S(t−))σS(t−)dW(t)

?

0<u≤t

{µ + S(u)}du

+exp

0

{µ + S(u)}dud

?

[a(S(u)) − a(S(u−))]

??+∞

−λexp

−

0

−∞

[a(S(t−)ey) − a(S(t−))]fY(y)dydt,

which is a local martingale. Then the same proof as in Theorem 3.1 applies. ?

Appendix B. Proof of Theorem 4.2

Proof: Similar algebra as in Theorem 3.2 yields that

?∞

where β1is the smallest positive root of G(x) = µ and z(x) = y(s) − 1. Plugging h(a) in (20)

into (28), we have that for all a ∈ (0,min(β1,1)),

?∞

?∞

i=1j=1

Using the same technique as in the proof of Theorem 3.2, we can show that the first integral

0 =

−∞

eax?e−x(z(x) + 1) + h(a)z(x)?dx,

for any a ∈ (0,min(β1,1)),

(28)

−∞

eax?

eax?

e−x(z(x) + 1) +

?

−1

2σ2a2−

?

r −σ2

2

− λζ

?

a + µ

?

z(x)

?

dx

+

−∞

− λ

m

?

piηi

ηi− a+

n

?

qjθj

θj+ a− 1

z(x)

?

dx = 0.

on the left hand side of the above is equal to

?∞

−∞

eax

?

e−x(z(x) + 1) −1

2σ2z??(x) +

?

r −σ2

2

− λζ

?

z?(x) + µz(x)

?

dx.

22

Page 23

In addition, we claim that the second integral is equal to

?∞

Indeed, this is because

?∞

=

−λ

−∞

=

−λ

−∞

=

−∞

?∞

where the second equality is via change of variable x − u = π, and the last equality holds

because 0 < a < β1< η1.

−∞

eax

?

−λ

?∞

−∞

[z(x − u) − z(x)]fY(u)du

?

dx.

−∞

eax

?∞

?∞

eaπz(π)

?

fY(u)

−λ

?∞

??∞

??∞

?

i=1

−∞

[z(x − u) − z(x)]fY(u)du

?

dx

−∞

eaxz(x − u)dx

?

?

?

qjθj

θj+ a− 1

du +

?∞

?∞

?∞

−∞

eaxλz(x)dx

fY(u)

−∞

?∞

m

?

ea(u+π)z(π)dπ du +

−∞

eaxλz(x)dx

?∞

−λ

−∞

eaufY(u)dudπ +

−∞

eaxλz(x)dx

=

−∞

eax?

− λ

piηi

ηi− a+

n

?

j=1

z(x)

?

dx,

Thus we have

?∞

λ

−∞

?∞

eax?

[z(x − u) − z(x)]fY(u)du

e−x(z(x) + 1) −1

2σ2z??(x) +

?

r −σ2

2

− λζ

?

z?(x) + µz(x)

−

−∞

?

dx = 0,

for all a ∈ (0,min(β1,1)).

By the uniqueness of the moment generating function we have an OIDE as follows

?

Now transferring the OIDE back to y(s), with s = e−xwe have

e−x(z(x)+1)−1

2σ2z??(x)+

r −σ2

2

− λζ

?

z?(x)+µz(x)−λ

?∞

−∞

[z(x − u) − z(x)]fY(u)du = 0.

z(x) = y(s) − 1, z?(x) = −sy?(s), z??(x) = sy?(s) + s2y??(s),

z(x − u) = z(−logs − u) = z(−log(seu)) = y(seu) − 1

and the OIDE becomes

−σ2

2s2y??(s) − (r − λζ)sy?(s) + (s + µ)y(s) − λ

?∞

−∞

[y(seu) − y(s)]fY(u)du = µ,

i.e., Ly(s) = (s + µ)y(s) − µ, from which the proof is completed. ?

23

Page 24

Appendix C. On the Recursion (19) under the HEM

In this section, we shall give an alternative proof that under the HEM, E[Aν

Tµ] solves the

recursion (19) using the Feynman-Kac formula. More precisely, under the HEM, E[Aν

Tµ] satisfies

the recursion (19), i.e.,

h(ν)E[Aν

Tµ] = νE[Aν−1

Tµ] for any ν ∈ (0,β1), (29)

where h(ν) is given by (20).

Proof: Under the HEM, define Y (t) =?t

price at the time u. Consider a function f∗(t,x,y) defined as follows:

0S(u)du, where S(u) represents the underlying stock

f∗(t,x,y) := E (Yν

T| S(t) = x,Y (t) = y),t ∈ [0,T],x ∈ R+,y ∈ R,

where T is any fixed positive real number and ν ∈ (−1,β1) is a constant. Note that {(S(t),Y (t)) :

t ≥ 0} is a two-dimensional Markov process under the HEM so that f∗(t,S(t),Y (t)) =

E (Yν

T| Ft). Accordingly, the multivariate Feynman-Kac theorem implies that f∗(t,x,y) solves

the following PIDE:

f∗

t(t,x,y) + (r − λζ)xf∗

?+∞

On the other hand, by the Markovian property, we can rewrite f∗(t,x,y) as follows.

??

Introduce a new function

x(t,x,y) + xf∗

y(t,x,y) +σ2

2x2f∗

xx(t,x,y)

+ λ

−∞

[f∗(t,xez,y) − f∗(t,x,y)]f∗

Y(z)dz = 0,t ∈ [0,T],x ∈ R+,y ∈ R.

f∗(t,x,y) = Ey + x

?T−t

0

e(r−λζ−σ2

2)u+σW(u)+PN(u)

i=1Yidu

?ν?

= E [(y + xAT−t)ν].

ˆf∗(t,x,y) := f∗(T − t,x,y) = E [(y + xAt)ν],t ∈ [0,T],x ∈ R+,y ∈ R,

which then satisfies the following PIDE

−ˆf∗

t(t,x,y) + (r − λζ)xˆf∗

?+∞

x(t,x,y) + xˆf∗

y(t,x,y) +σ2

2x2ˆf∗

xx(t,x,y)

+λ

−∞

[ˆf∗(t,xez,y) −ˆf∗(t,x,y)]f∗

Y(z)dz = 0,t ∈ [0,T],x ∈ R+,y ∈ R, (30)

24

Page 25

with the initial value

ˆf∗(0,x,y) = yν.

It is worth noting that the “T”in the PIDE (30) can be any positive real number.

Next, for any ν ∈ (0,β1), define h∗(x,y) = E??y + xATµ

R+, y ∈ R. Taking the Laplace transform w.r.t. t on both sides of the PIDE (30) and then

multiplying both sides by µ, we can show that h∗(x,y) solves a PIDE as follows.

?ν?=?∞

0µe−µtˆf∗(t,x,y)dt, x ∈

µyν− µh∗(x,y)+(r − λζ)xh∗

x(x,y) + xh∗

y(x,y) +σ2

2x2h∗

xx(x,y)

+λ

?+∞

−∞

[h∗(xez,y) − h∗(x,y)]f∗

Y(z)dz = 0, x ∈ R+,y ∈ R.

(31)

Interchanging the derivatives and integrals by using Theorem A. 12 on the pp. 203-204 in Schiff

[39], we have

xh∗

x(x,y)=

νE??y + xATµ

νxE

ν(ν − 1)E??y + xATµ

−ν(ν − 1)2yE

?ν?− νyE

??y + xATµ

?ν−1?

,

xh∗

y(x,y)=

??y + xATµ

?ν−1?

,

?ν?+ ν(ν − 1)y2E

x2h∗

xx(x,y)=

??y + xATµ

?ν−2?

??y + xATµ

?ν−1?

.

Substituting above into (31) yields

?

+ ν?x − (r − λζ)y − σ2(ν − 1)y?E

+ λ

−∞

In the special case x = 1 and y = 0, since

?+∞

we obtain that for any ν ∈ (0,β1),

µyν+

−µ + (r − λζ)ν +σ2

2ν(ν − 1)

?

E??y + xATµ

??y + xATµ

?ν?

?ν−1?

?ν??fY(z)dz = 0.

i=1

+σ2

2ν(ν − 1)y2E

??y + xATµ

?ν−2?

?+∞

?E??y + xezATµ

?ν?− E??y + xATµ

−∞

?

E

?

ezνAν

Tµ

?

− E

?

Aν

Tµ

??

fY(z)dz = E

?

Aν

Tµ

?

m

?

piηi

ηi− ν+

n

?

j=1

qjθj

θj+ ν− 1

,

σ2

which is exactly (29) and (19). The proof is completed. ?

2ν2+ (r − λζ −σ2

2)ν + λ

m

?

i=1

piηi

ηi− ν+

n

?

j=1

qjθj

θj+ ν− 1

− µ

E

?

Aν

Tµ

?

= −νE

?

Aν−1

Tµ

?

,

25

Page 26

Acknowledgements

We would like to thank the Area Editor Mark Broadie, the Associate Editor, two anonymous

referees, Gianluca Fusai, Guillermo Gallego, Paul Glasserman, Vadim Linetsky, Giovanni Pe-

trella, Jan Vecer, Ward Whitt, and participants at INFORMS annual meeting 2009, Workshop

on Optimal Stopping and Singular Stochastic Control Problems in Finance 2009 in Singapore,

Young Researchers Workshop on Finance 2010 in Tokyo, International Symposium on Financial

Engineering and Risk Management 2010 in Taipei, and Columbia-Oxford Risk Summit 2010, for

their helpful comments. The fist author’s research is partially supported by GRF of Hong Kong

RGC (Project No. 610709) and DAG (Project No. DAG11EG07G and DAG08/09.EG07).

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Online Supplement

Pricing Asian Options under a Hyper-Exponential Jump Diffusion Model

Ning Cai and S. G. Kou

HKUST and Columbia University

More Discussion on the Numerical Algorithm

1 Stability of the Method

To study the stability of the method, we perform some numerical experiments under the BSM

to show the absolute and relative errors of our double-Laplace inversion method against various

choices of parameters A1, A2and X. The “true” prices are obtained by using Monte Carlo sim-

ulation with a control variate being?t

0eX(s)ds, because it is easy to compute E

??t

0eX(s)ds

?

=

(ert−1)/r, and?t

Richardson extrapolation is also employed to reduce the discretization bias generated when we

0eX(s)ds has a high degree of correlation with the payoff function. In addition,

discretize the sample path to approximate the integral. More precisely, let M(h) be the Monte

Carlo estimator without Richardson extrapolation when the discretization step size is set to

be h. Then we use (4M(h) − M(2h))/3 rather than M(h) as the final estimator to achieve

the discretization bias reduction. For more details about the technique of control variates and

Richardson extrapolation, see Glasserman [24].

Figure 1 shows how the absolute and relative errors change as A1, A2and X vary in the

case of low volatility σ = 0.05, illustrating that our algorithm is insensitive to the selection of

parameters A1, A2and X. For normal volatilities, our method becomes even more stable and

associated plots can be obtained on request.

2 Discretization Error Bounds of Euler Inversion Algorithm un-

der the BSM

The discretization error bound of the Euler inversion algorithm was first studied by Abate and

Whitt [1], and was extended to a two-sided Laplace inversion case by Petrella [37]. In this

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30 4050 6070

−2

−1

0

1

2

x 10

−3

σ=0.05

Absolute errors

Parameter A1

30 40 5060 70

−1

0

1

x 10

−3

σ=0.05

Relative errors

Parameter A1

303540 4550 55 60

−2

−1

0

1

2

x 10

−3

σ=0.05

Absolute errors

Parameter A2

303540 45 505560

−1

0

1

x 10

−3

σ=0.05

Relative errors

Parameter A2

250300 350400450500 550 600

−2

−1

0

1

2

x 10

−3

σ=0.05

Absolute errors

Parameter X

250300350400 450 500550600

−1

0

1

x 10

−3

σ=0.05

Relative errors

Parameter X

Upper Limit of the 95% confidence interval

Upper Limit of the 95% confidence interval

Upper Limit of the 95% confidence interval

Lower Limit of the 95% confidence interval

Lower Limit of the 95% confidence interval

Lower Limit of the 95% confidence interval

Figure 1: The stability and accuracy of the algorithm as A1, A2 and X vary in the case of low volatility

σ = 0.05. The default choices for unvarying algorithm parameters are A1 = 28, A2 = 40 and X given by (27).

The absolute errors and relative errors are reported on the left and right graphs, respectively. For broad regions

of A1, A2 and X our algorithm appears to be stable and accurate, all within the 95% confidence intervals. In

fact, the relative errors are all smaller than 0.02%.

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subsection, by extending the results in Petrella [37], we provide discretization error bounds

of the inversion algorithm for our specific case of Asian option pricing under the BSM. The

discretization error bounds decay exponentially, therefore leading to a fast convergence.

Recall that what we want to invert is L(µ,ν) =?∞

XE(S0

0

?∞

−∞e−µte−νkf(t,k)dkdt, where f(t,k) =

XAt− e−k)+. Then we can prove the following theorem for the error bounds.

Theorem 2.1. Suppose t ∈ (0,

1+˜ r+σ2/2 > 0 and ˜ r = max(r−σ2

A1

2(θ1+θ2)) and k >A2

θ2, for some constant θ2> 0, where θ1=

2,0). Then the discretization error bounds e+

?

1

1 − e−(A1−2(θ1+θ2)t)

2/2+[(1+˜ r+σ2)t−1]θ2+θ1t.

dand e−

dsatisfy

e+

d≤

C+(θ1)

1 − e−(A1−2θ1t)

e−A2

1 − e−A2+ e−(A1−2θ1t)

?

,

(32)

e−

d≤ C−(θ1,θ2)

e−(θ2k−A2)

1 − e−(θ2k−A2),

(33)

with C+(θ1) := 2S0eθ1tand C−(θ1,θ2) := 2S0etσ2θ2

Before proving this theorem, we give an example to illustrate how to apply it in real sit-

uations.Consider the case where r = 0.09, σ = 0.2 and t = 1, and we use A1 = 50,

A2 = 40 and θ2 = 20. Then ˜ r = 0.07, θ1 = 1.09, t = 1 ∈

4 ∈

Plugging them into (32) and (33), we can get discretization error bounds: e+

?

0,A1

2θ1

?

≡ (0,1.19), and k =

?

A2

θ2,+∞

?

≡ (2,+∞). Simple algebra yields that C+(θ1) ≈ e6.39and C−(θ1,θ2) ≈ e16.59.

d≤ 2.53 × 10−15

and e−

d≤ 6.80 × 10−11. Hence, the discretization error for Asian option price is theoretically

no more than 6.80 × 10−11× e−rt/t ≈ 6.22 × 10−11.

Proof of Theorem 2.1: First, since the scaling factor X > S0, we have that

?S0

where k = log(X

f(t,k) = XE

XAt− e−k

?+

≤ XE

?S0

XAt−S0

Xe−k

?+

= S0E(At− e−k)+,

Kt). On the other hand, we can bound Atas follows

?t

At=

?t

0

e(r−σ2

2)s+σW(s)ds ≤

0

exp

?

˜ rt + σ max

{0≤s≤t}W(s)

?

ds = texp

?

˜ rt + σ max

{0≤s≤t}W(s)

?

,

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Page 33

where ˜ r := max(r −σ2

2,0). Since max{0≤s≤t}W(s)=d|W(t)|, it follows that

?

S0E

??

S0E

??

2S0E

f(t,k)

≤

=

S0Etexp{˜ rt + σ|W(t)|} − e−k?+

??

+S0E

??

+S0E

??

texp{˜ rt + σW(t)} − e−k?

texp{˜ rt − σW(t)} − e−k?

texp{˜ rt + σW(t)} − e−k?

texp{˜ rt − σW(t)} − e−k?

texp{˜ rt + σW(t)} − e−k?

I{W(t)≥0, texp(˜ rt+σW(t))>e−k}

?

I{W(t)<0, texp(˜ rt−σW(t))>e−k}

?

≤

I{texp(˜ rt+σW(t))>e−k}

?

I{texp(˜ rt−σW(t))>e−k}

?

=

I{texp(˜ rt+σW(t))>e−k}

?

,

via the symmetric property of standard Brownian motion.

Next, introduce a new measure¯P such thatd¯P

dP= eYt−(˜ r+σ2/2)t, where Yt:= ˜ rt + σW(t).

Then the change of measure leads to

f(t,k)

≤

=

2S0¯E

?

texp{˜ rt + σWt}I{texp(˜ rt+σW(t))>e−k}× e−Yt+(˜ r+σ2/2)t?

2S0te(˜ r+σ2/2)t¯P{texp(Yt) > e−k}

2S0e(1+˜ r+σ2/2)t¯P{Yt> −k − log(t)}

2S0eθ1t¯P{Yt> −k − log(t)},

≤

=

where θ1:= 1 + ˜ r + σ2/2 > 0 and the last inequality holds because t < etfor any t > 0.

Therefore, when j1≥ 0 and j2≥ 0, we have

f((2j1+ 1)t,(2j2+ 1)k)

≤

2S0eθ1(2j1+1)t¯P{Yt> −(2j2+ 1)k − log((2j1+ 1)t)}

2S0eθ1te2θ1j1t= C+(θ1)e2θ1j1t,

≤

where C+(θ1) := 2S0eθ1t. On the other hand when j1≥ 0 and j2≤ −1, we have that for any

θ2> 0,

f((2j1+ 1)t,(2j2+ 1)k)

≤

2S0eθ1(2j1+1)t¯P{Yt> −(2j2+ 1)k − log((2j1+ 1)t)}

2S0eθ1(2j1+1)t¯P{Yt> −j2k − log((2j1+ 1)t)}

2S0eθ1(2j1+1)t¯E

eθ2j2k+θ2log((2j1+1)t),

≤

≤

?

eθ2Yt?

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Page 34

where the second inequality holds as j2≤ −1 and the third inequality comes from Markov’s

inequality. Since x + 1 ≤ exfor any x > −1, we obtain that eθ2log((2j1+1)t)≤ eθ2[(2j1+1)t−1]and

2S0eθ1(2j1+1)t¯E

f((2j1+ 1)t,(2j2+ 1)k)

≤

=

?

eθ2Yt?

?

eθ2j2k+θ2[(2j1+1)t−1]

eθ2Yt?

2S0e(θ1+θ2)t−θ2 ¯E

· e2(θ1+θ2)j1teθ2j2k

=

C−(θ1,θ2)eθ2j2k+2(θ1+θ2)j1t,

where

C−(θ1,θ2) := 2S0e(θ1+θ2)t−θ2·¯E

?

eθ2Yt?

= 2S0e(θ1+θ2)t−θ2· E

?

e(θ2+1)Yt−(˜ r+σ2/2)t?

.

Recall that Yt= ˜ rt + σW(t). Simple algebra yields

C−(θ1,θ2) = 2S0etσ2θ2

?

2/2+[(1+˜ r+σ2)t−1]θ2+θ1t.

If we have t ∈

1)t,(2j2+ 1)k) obtained above, we can get

?

0,A1

2θ1

, according to the definition of e+

dand the bound of function f((2j1+

e+

d

≤

∞

?

C+(θ1)

j2=1

∞

?

j1=0

e−(j1A1+j2A2)C+(θ1)e2θ1j1t+

∞

?

j1=1

e−j1A1C+(θ1)e2θ1j1t

=

∞

?

e−A2

1 − e−A2

C+(θ1)

1 − e−(A1−2θ1t)

j2=1

∞

?

j1=0

e−(A1−2θ1t)j1−j2A2+ C+(θ1)

∞

?

e−(A1−2θ1t)

1 − e−(A1−2θ1t)

?

j1=1

e−(A1−2θ1t)j1

=

C+(θ1)

1

1 − e−(A1−2θ1t)+ C+(θ1)

?

=

e−A2

1 − e−A2+ e−(A1−2θ1t)

,

which is exactly (32).

For e−

dwe have for any t ∈

?

0,

A1

2(θ1+θ2)

?

and k >A2

θ2,

e−

d

≤

−1

?

j2=−∞

∞

?

j1=0

e−(j1A1+j2A2)C−(θ1,θ2)eθ2j2k+2(θ1+θ2)j1t

=

C−(θ1,θ2)

∞

?

j1=0

e−(A1−2(θ1+θ2)t)j1

−1

?

e−(θ2k−A2)

1 − e−(θ2k−A2),

j2=−∞

ej2(θ2k−A2)

=

C−(θ1,θ2)

1

1 − e−(A1−2(θ1+θ2)t)

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Page 35

from which (33) is proved. ?

Double-Laplace Inversion Method

2 decimal3 decimal

(n1,n2)=(35,35)∗

(3.5 secs)∗∗

(n1,n2)=(15;15)

(1.2 secs)

Fourier-Laplace Inversion Method

2 decimal

(nl,nf)=(35,115)

(17.5 secs)

(nl,nf)=(15,55)

(4.6 secs)

4 decimal

(35,35)

(3.5 secs)

(15;35)

(2.0 secs)

5 decimal

(35,35)

(3.5 secs)

(15;35)

(2.0 secs)

σ=0.05

(CPU time)

σ=0.1

(CPU time)

(35,35)

(3.5 secs)

(15;35)

(2.0 secs)

3 decimal

(35,135)

(20.1 secs)

(15,75)

(6.4 secs)

4 decimal

(35,195)

(28.6 secs)

(15,95)

(7.7 secs)

5 decimal

(35,195)

(28.6 secs)

(15,95)

(7.7 secs)

σ=0.05

(CPU time)

σ=0.1

(CPU time)

Table 7: Comparison of the efficiency between our double-Laplace inversion and Fusai’s Fourier-Laplace in-

version method. In this table, (n1,n2)=(35,35)∗means that (n1,n2) should be set roughly at least (35,35) to

achieve 2-decimal accuracy. (3.5 secs)∗∗below (n1,n2)=(35,35)∗means that the corresponding CPU time is 3.5

seconds. The CPU times associated with Fusai’s method are obtained using the code implemented by Matlab

7.1. All computations in Table 7 are conducted on an IBM laptop with a Pentium M 1.86GHz processor. We

can see that to achieve the same accuracy, our method is more efficient than Fusai’s.

1520 25 3035

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Absolute errors of FL prices vs. parameter Al when σ=0.1

Absolute errors of FL prices

Parameter Al

1520 253035

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Absolute errors of LL prices vs. parameter A1 when σ=0.1

Parameter A1

Figure 2: Comparison of the stability and accuracy between the double-Laplace inversion and the Fourier-

Laplace inversion method in the case of low volatility σ = 0.1 under Kou’s model, where other parameters are

K = 100, S0 = 100, t = 1, λ = 3, r = 0.09, p1 = 0.6, q1 = 0.4, and η1 = θ1 = 25;. The absolute errors of

LL prices (obtained by the double-Laplace inversion method) and FL prices (obtained by the Fourier-Laplace

inversion method) are reported on the left and right graphs, respectively. Other parameters for the left graph

are n1 = 35, n2 = 55, A2 = 40 and X = 5460; while other parameters for the right graph are nl= 35, nf = 135

and Af = 40. We can see that LL prices are quite accurate and stable when A1 varies between [22.2,38], but FL

prices are so desultory that we cannot decide which Al we should choose.

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Page 36

3Comparison with the Fourier and Laplace Inversion Algo-

rithm

Under the BSM, Fusai [22] gave a closed-form for the Fourier-Laplace transform of Asian option

price w.r.t. k = ln(σ2Kt/(4S0)) and h = σ2t/4, respectively. Despite some similarities, there

are some key differences between our method and Fusai’s method. (1) Our method performs

better for low volatility, e.g., σ = 0.05 or 0.1. Specifically, for Fusai’s method, a large number

of terms are needed to do the Euler inversion to achieve a desired accuracy. In comparison,

our algorithm in general requires far fewer terms in computation, especially for low volatility

(See Table 7). This is mainly because we use the latest inversion method with a scaling factor

in Petrella [37]. (2) Our method performs better in jump diffusion models. Specifically, the

Fourier-Laplace inversion method seems unstable in the case of low volatilities under Kou’s

model because it is sensitive to parameters. To illustrate the sensitivity, we fix Af= 40 and let

Alchange from 15 to 38. The right panel of Figure 2 illustrates how the difference between the

numerical result and the true value changes as Alvaries in the case of σ = 0.1 and K = 100.

In comparison with the double-Laplace inversion on the left panel of Figure 2, FL prices seem

unstable. Figure 2 seems to indicate that, with jumps, our double-Laplace inversion method

works in a more stable manner than the Fourier-Laplace method, especially in the case of low

volatility. (3) Under the BSM, we can derive a theoretical discretization error bound for the

double-Laplace inversion. See Section 2 in the online supplement. (4) In terms of the main

theoretical difference, note that the recursion used in Fusai’s paper, namely (8), has no unique

but infinitely many solutions. We spend considerable efforts to overcome this difficulty; see

Section 3.

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