Analytically pricing exchange options with stochastic liquidity and regime switching

• Published date: 01 May 2023
• Author: Xin‐Jiang He,Sha Lin
• Date: 01 May 2023
• DOI: http://doi.org/10.1002/fut.22403

Received: 19 November 2022

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Accepted: 20 January 2023

DOI: 10.1002/fut.22403

RESEARCH ARTICLE

Analytically pricing exchange options with stochastic

liquidity and regime switching

Xin‐Jiang He

1

|Sha Lin

2

1

School of Economics, Zhejiang

University of Technology, Hangzhou,

China

2

School of Finance, Zhejiang Gongshang

University, Hangzhou, China

Correspondence

Sha Lin, School of Finance, Zhejiang

Gongshang University, Hangzhou, China.

Email: linsha@mail.zjgsu.edu.cn

Abstract

We investigate the valuation of exchange options when the market is affected by

changing economic conditions as well as liquidity risks. The volatility and

expected returns of both stocks are assumed to be controlled by a continuous‐time

Markov chain to reflect the effects of varying economic conditions, and a liquidity

discounting factor is employed to capture the impact of market liquidity on stock

prices. Once the model has been established, we construct a risk‐neutral measure

with the use of regime‐switching Esscher transform, and the characteristic

function is then derived in an analytical form, so that a closed‐form formula for

exchange options can be presented. We further analyze the effects of the two

considered factors on exchange option prices numerically.

KEYWORDS

analytical solution, continuous‐time Markov chain, exchange options, liquidity risks

1|INTRODUCTION

An exchange option allows for two parties to exchange two assets, respectively, owned by them at the end of the

contract; such an exchange would be determined by the contract holder based on his/her own interest. It was actually

established to manage financial risks resulting from the relative fluctuation of two assets, and is now one of the most

widely traded exotic options, as it provides vital implications for a lot of common contracts in finance practice.

The pricing of exchange options can be dated back to 1978, when a closed‐form pricing formula was derived under the

Black–Scholes model (Margrabe, 1978). Under the same framework, power exchange options, constructed by using unique

features of power and exchange options, were analytically priced by Blenman and Clark (2005). Different stochastic factors

were introduced to revise the unrealistic assumptions used by the Black–Scholes model for exchange option pricing, including

the incorporation of stochastic volatility (Antonelli et al., 2010; Pasricha & Goel, 2022) and the addition of stochastic interest

rate (Bernard & Cui, 2010). Jumps have also been considered in the pricing of exchange options, including the Hawkes jump‐

diffusion model (Pasricha & Goel, 2021)andtheL

é

vy model (Chen & Wan, 2010). Furthermore, Cheang and Garces (2020)

combined stochastic volatility with jump diffusions when pricing exchange options, while exchange options taking credit risks

into consideration were studied by Wang (2016)andXuetal.(2019). In addition, varying economic condition is empirically

shown to be a vital factor that affects stock prices (Hamilton, 1990;He&Chen,2022;He&Lin,2022;Siu&Elliott,2022),

which has also been taken into consideration when pricing exchange options (Yoon et al., 2011).

Although these results are very appealing, liquidity risks, as one type of the most critical financial risks requiring

efficient management (Liu, 2006), were all neglected in these works. In fact, there is even no consensus reached in

terms of the method for introducing liquidity risks into financial derivative valuation, though several approaches have

been developed for pricing derivatives in the presence of liquidity risks. One strand of authors believes that liquidity

risks are caused by market practitioners taking different actions, and a typical example in this category is Liu and Yong

J Futures Markets. 2023;43:662–676.wileyonlinelibrary.com/journal/fut662

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