An efficient and stable method for short maturity Asian options

• Author: Jiacheng Fan,Mingzhe Liu,Rupak Chatterjee,Zhenyu Cui
• Date: 01 December 2018
• DOI: http://doi.org/10.1002/fut.21956
• Published date: 01 December 2018

Received: 9 June 2017

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Revised: 7 June 2018

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Accepted: 12 June 2018

DOI: 10.1002/fut.21956

RESEARCH ARTICLE

An efficient and stable method for short maturity

Asian options

Rupak Chatterjee

1

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Zhenyu Cui

2

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Jiacheng Fan

2

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Mingzhe Liu

2

1

Department of Physics and Hanlon

Financial Systems Center, Stevens

Institute of Technology, Hoboken,

New Jersey

2

Department of Financial Engineering,

School of Business, Stevens Institute of

Technology, Hoboken, New Jersey

Correspondence

Zhenyu Cui, School of Business, Stevens

Institute of Technology, 1 Castle Point on

Hudson, Hoboken, NJ 07030.

Email: zcui6@stevens.edu

In this paper, we develop a Markov chain‐based approximation method to price

arithmetic Asian options for short maturities under the case of geometric

Brownian motion. It has the advantage of being a closed‐form approximation

involving only matrices. It is an accurate, efficient, and stable method for the

pricing and hedging of short maturity arithmetic Asian options for which

previous methods in the literature have shown either slower convergence or

instabilities in hedging parameters. We demonstrate that this method is as good

as and sometimes better than existing approximation methods in the literature.

KEYWORDS

arithmetic Asian option, Markov chain, stable Greeks, volatility regime

JEL CLASSIFICATION

91G80, 93E11, 93E20

1

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INTRODUCTION

Asian (average price) options are financial derivatives whose payoffs are dependent on the geometric–arithmetic averages

of the underlyingasset returns, such as stocks, commodities, or financial indices. It was first introduced into the academic

literature by Boyle and Emanuel (1980) (see Boyle, 1993, for a historical account of the development of Asian options).

Since then, it has been popular in financial markets due to the smoothing effect of the averaging feature. It is also less

expensive thanstandard European options becausethe volatility of the average asset is oftenlower. Another reason for the

popularity of the Asian option lies in the fact that its price is harder to manipulate by large market participants compared

to path‐independent European call or put options. This is particularly important for thinly traded commodities (see

relevant discussions in Linetsky,2004). In the literature of executivecompensations, there is also similar risk of stock price

manipulations by the executives to bump up their compensation packages (Johnson & Tian, 2000a,b; Hall & Murphy,

2003). Thus, there are some literature proposing to utilize this averaging feature to design executive stock options (Tian,

2013; Bernard, Boyle, & Chen, 2016) to better align the interests of the executives to the company. Asian option payoffs

resemble those of the variable annuities (Bernard, Cui, & Vanduffel, 2017; Cui, Feng, & MacKay, 2017a), and efficient

methods for pricing and hedging Asian options also can be applied to managing the growing variable annuity industry.

Geometric Asian options have closed‐form solutions in a handful of model settings (Angus, 1999; Wong & Cheung, 2004).

However, in practice, most commonly traded Asian option contracts are based on the arithmetic average. Even under a

geometric Brownian motion setting, exact closed‐form formulas have not been obtained (Kao & Lyuu, 2003). There are

several theoretical and numerical challenges in pricing arithmetic Asian options. First, the determination of the distribution

of the sum of log normal random variables is a notoriously difficult problem (Asmussen, Jensen, & Rojas‐Nandayapa, 2016;

Dufresne, 2008). Second, existing exact pricing representations are computationally expensive. For example, the triple

integral formula of Yor (1992) is exact, but direct numerical integration is difficult (see also the recent discussion in Lyasoff,

2016, and an alternative integral representation in Cui & Nguyen, 2017). In a pioneering paper, Geman and Yor (1993) first

J Futures Markets. 2018;38:1470–1486.wileyonlinelibrary.com/journal/fut1470

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© 2018 Wiley Periodicals, Inc.

obtained the Laplace transform of the price of the arithmetic Asian option in terms of Kummer confluent hypergeometric

functions. The numerical inversion of this Laplace transform suffers from numerical instabilities and is also subject to

parameter restrictions (Carr & Schröder, 2001, for a detailed discussion). Along the Laplace transform–based approach, there

are also recent literature proposing to characterize the Laplace transforms of Asian option prices as solutions to a related

functional equation, see Cai, Song, and Kou (2015) and Cui, Lee, and Liu (2018). There are continued interests in utilizing

the Markov chain approximations in pricing and hedging path‐dependent options, see for example Cui, Kirkby, and Nguyen

(2017b,c), Kirkby, Nguyen, and Cui (2017), Li and Zhang (2016), (2017b), etc. In the case of stochastic volatility models, there

is also a recent method of using a regime‐switching geometric Brownian motion to approximate the stochastic volatility

process with jumps, and then valuate Asian option prices (Kirkby & Nguyen, 2016). Finally, there are vast amounts of

literature devoted to developing analytical approximations to arithmetic Asian option prices, but for most methods there is

no guarantee that the approximation will converge (in a weak sense) to the exact solution. Some representative literature in

this category are Turnbull and Wakeman (1991), Rogers and Shi (1995), Zhang (2001), Chang and Tsao (2011), Chung,

Shackleton, and Wojakowski (2003), Tsao, Chang, and Lin (2003), Lo, Palmer, and Yu (2014), Lee (2014), Kahalé (2017), etc.

Although the above approximation methods work fairly well for the price of Asian options, most of the methods cannot be

directly extended to the Greeks of the Asian options.

This motivates us to develop a general approximation method that can be applied to both the pricing and hedging of

Asian options. The proposed method in this paper applies to the full range of model parameters and is particularly well‐

suited for short maturity Asian options due to its accuracy and stability. However, most methods in the literature are

numerically less efficient in the presence of small maturities, high volatilities, and/or very low volatilities. This issue has

been elaborated in Linetsky (2004), p. 865, it is stated:

…

the dimensionless time to expiration =∕τσT()

4

2is the crucial parameter that controls the numerical convergence of

the series

…

The larger the value of

τ

, the faster the convergence. For smaller values of this parameter,

convergence significantly slows down

…

.

Note that this issue has also been noted and discussed in Geman and Yor (1993), where the numerical Laplace

transform inversion requires some special care for small maturities (Shaw, 2002).

Low‐volatility investing has also drawn attention from both practice (Baker, Bradley, & Wurgler, 2011; Hsu & Li,

2013) and academic literature (Dutt & Humphery‐Jenner, 2013; Engle & Rangel, 2008; Stambaugh, Yu, & Yuan, 2012);

thus, it is important to have an accurate assessment of the derivative value and cost of the associated hedging program

in low‐volatility environments.

Small maturities and high volatility regime are of significant interest to financial industry where it is particularly important

to have accurate hedging methods for derivatives close to maturity during a turbulent and highly volatile market period. The

partial differential equation (PDE) method of Vecer (2001) is currently the most commonly used method for pricing Asian

options for all ranges of maturities and volatilities. However, due to inherent instability of the finite difference method for small

time‐steps, the PDE method of Vecer (2001) is unstable in computing gammas of the Asian option for small maturities. This

paper proposes an efficient and stable method based on the continuous‐time Markov chain (CTMC) approximation. It will be

shown that this method can handle all parameter ranges including both small maturities, and high or low‐volatility regimes.

Furthermore, it is shown to yield stable option Greeks. This method is not based on asymptotic expansions (see Foschi,

Pagliarani, & Pascucci, 2013; Pirjol & Zhu, 2016; Pirjol & Zhu, 2017, for related discussions), which often perform poorly for

mediumdatedoptions.Ourproposedmethodisvalidforanyfixedmaturity.

The contributions of this paper are as follows:

1. We obtain an explicit closed‐form matrix approximation formula for the price of arithmetic Asian options in the

Black–Scholes model, and we demonstrate that our method yield accurate results for a full range of option

parameters.

2. The method is accurate, efficient, and stable for both option prices and option Greeks. It provides accurate values

compared to the benchmark, Vecer’s PDE approach, and is more stable in computing Greeks than Vecer’s PDE

method in the high volatility and short maturity regimes.

The paper is organized as follows: Section 2 presents the main results on the Markov chain approximation method;

Section 3 presents the numerical examples and detailed comparison to previous literature; Section 5 concludes the

paper with future research directions.

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