Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach

• Published date: 01 February 2023
• Author: Kailin Ding,Zhenyu Cui,Xiaoguang Yang
• Date: 01 February 2023
• DOI: http://doi.org/10.1002/fut.22387

Received: 12 October 2021

|

Accepted: 2 November 2022

DOI: 10.1002/fut.22387

RESEARCH ARTICLE

Pricing arithmetic Asian and Amerasian options:

A diffusion operator integral expansion approach

Kailin Ding

1

|Zhenyu Cui

2

|Xiaoguang Yang

1

1

Academy of Mathematics and Systems

Science, Chinese Academy of Sciences,

Beijing, China

2

School of Business, Stevens Institute of

Technology, Hoboken, New Jersey, USA

Correspondence

Zhenyu Cui, School of Business, Stevens

Institute of Technology, Babbio Center

545, 1 Castle Point on Hudson, Hoboken,

NJ 07030, USA.

Email: zcui6@stevens.edu

Abstract

In this paper, we propose a new explicit series expansion formula for the price

of an arithmetic Asian option under the Black–Scholes model and Merton's

jump‐diffusion model. The method is based on an equivalence in law relation

together with the diffusion operator integral method proposed by Heath and

Platen. The method yields explicit series expansion formula for the Asian

options' prices. The theoretical convergence of the expansion to the true value

is established. We also consider the American Asian option (i.e., Amerasian

option) and derive the corresponding expansion formula through the early

exercise premium representation. Numerical results illustrate the accuracy

and efficiency of the method as compared with benchmarks in the literature.

KEYWORDS

American Asian options, Asian option, diffusion operator integral, series expansion

JEL CLASSIFICATION

G12, G13

1|INTRODUCTION

Arithmetic Asian options are financial derivatives whose payoffs depend on the time‐average of the underlying asset prices.

Asian options are commonly seen in the foreign exchange market and the commodity market (Heenk et al., 1990). Due to its

dependence on the time integral of the underlying stochastic process, the valuation of arithmetic Asian options is very

challenging, and there exists no closed‐form formula for the price of the arithmetic Asian option even in the Black–Scholes

model. Hence, the literature has mainly focused on developing analytical approximations and various computational

techniques for its valuation. The literature is vast and can roughly be classified as follows: Laplace transform method

(Geman & Yor, 1993;Kirkby,2016), Monte Carlo (MC) method (Kemna & Vorst, 1990), partial differential equation (PDE)

method (Vecer, 2002), continuous‐time Markov chain approximation method (Cai et al., 2015; Chatterjee et al., 2018;Cui

et al., 2018;Kirkby&Nguyen,2020), moment matching methods (Fusai & Tagliani, 2002), tight lower and upper bounds

(Choe & Kim, 2021; Fusai & Kyriakou, 2016), and so forth. A survey and summary of different valuation methods for Asian

options can be found in Boyle and Potapchik (2008)andFuetal.(1999). On the other hand, the literature on the American‐

style Asian options, or Amerasian options, for which it is possible to early exercise the option, is relatively thin. Existing

methods for Amerasian options include: analytical approximation (Hansen & Jørgensen, 2000;L.Wuetal.,1999), binomial

tree method (Dai, 2003), and MC simulation method (R. Wu & Fu, 2003).

The above‐mentioned literature all focus on numerical method or analytical approximations, and most of the time the

method is not guaranteed to converge to the true solution and is computationally intensive. An exception to the above

literature on European‐style arithmetic Asian options is an elegant spectral expansion approach first proposed in Linetsky

J Futures Markets. 2023;43:217–241. wileyonlinelibrary.com/journal/fut © 2022 Wiley Periodicals LLC.

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(2004) for the case of the Black–Scholes model, which yields a convergent series expansion for the price of an arithmetic

Asian option. The motivation of the current paper is to develop an alternative convergent series expansion for the arithmetic

Asian options as compared with the spectral expansion approach. The spectral expansion method is based on solving the

eigenfunctions of the corresponding diffusions, and involves special functions, such as the Whittaker function, which are

time‐consuming to compute. In contrast, our proposed formula does not involve special functions. The proposed approach

in this paper is based on the intuitive idea of expanding the “geometric Brownian motion with affine drift”around a base

model, the geometric Brownian motion (GBM). Hence we can fully utilize the closed‐form Black–Scholes formula in

developing the coefficients of the explicit series expansion. This idea of expanding one stochastic process around a base

stochastic process dates back to the diffusion operator integral (DOI) idea of Heath and Platen (2002), and see also

Kristensen and Mele (2011) for a recent application to options pricing. To the best of authors' knowledge, this is the fi rst

time that the method is developed for the case of arithmetic Asian options. In addition, our proposed method can be

naturally extended to the case of American Asian options, which also contributes to the literature.

To summarize, the contributions of the paper are:

1. Under Merton's jump‐diffusion (MJD) model, we propose novel explicit series expansion formulas for the prices of

both fixed‐strike and floating‐strike arithmetic Asian options. The formulas are explicit and easy to implement.

2. Under the Black–Scholes model, we propose the series expansion formula for the price of American‐style arithmetic

Asian options, that is, the Amerasian option, which is new to the literature.

3. We establish the theoretical convergence of the expansion formula and extensive numerical experiments confirm

the accuracy and efficiency of the formula as compared with the literature.

The remainder of the paper is organized as follows: Section 2gives the main expansion results for arithmetic Asian

options. Section 3discusses the corresponding expansion results for the American Asian options. Section 4presents

numerical experiments to illustrate the performance of the expansion formulas and compares them with the existing

literature. Section 5concludes the paper with a discussion of future research directions. Technical proofs are collected

in the appendix.

2|SERIES EXPANSIONS FOR ARITHMETIC ASIAN OPTIONS

2.1 |The model setup

Denote ≥

S

S=( )

tt 0as the asset price and consider the following process

≥

XX=( )

tt 0

:

≔∈

¦XaSbSduab+,,

,

tt

t

u

0

(1)

which is the linear combination of

S

t

and its time integral. For a fixed maturity T, consider the payoff function

≔

φ

XXK() ( −)

TT

+

and note that it includes the following specific types of Asian options as special cases:

(i) If a=

0

,

b

=T

1,

K

>0

, then 

()

φ

XSduK()= −

TT

T

u

1

0

+

is the payoff of a fixed‐strike arithmetic Asian call option.

(ii) If a

k

=−(k>0

),

b

=T

1,

K

=0

, then 

()

φ

XSdukS()= −

TT

T

uT

1

0

+

is the payoff of a floating‐strike arithmetic

Asian put option.

The fixed‐strike Asian put option and floating‐strike Asian call option can be similarly considered, utilizing the

symmetry results which are in analogous to the put–call parity (Vecer, 2002) for floating and fixed‐strike Asian options.

Under the risk‐neutral measure, consider the jump‐diffusion model, which is an exponential Lévy model of

the form:

S

Se=

,

tL

0t(2)

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DING ET AL.