Analytical valuation of Asian options with counterparty risk under stochastic volatility models

• Author: Xingchun Wang
• DOI: http://doi.org/10.1002/fut.22064
• Published date: 01 March 2020
• Date: 01 March 2020

J Futures Markets. 2020;40:410–429.wileyonlinelibrary.com/journal/fut410

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© 2019 Wiley Periodicals, Inc.

Received: 31 March 2019

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Accepted: 19 September 2019

DOI: 10.1002/fut.22064

RESEARCH ARTICLE

Analytical valuation of Asian options with counterparty

risk under stochastic volatility models

Xingchun Wang

School of International Trade and

Economics, University of International

Business and Economics, Beijing, China

Correspondence

Xingchun Wang, School of International

Trade and Economics, University of

International Business and Economics,

Office 416, Qiuzhen Building, 100029

Beijing, China.

Email: xchwangnk@aliyun.com and

wangx@uibe.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Numbers: 11671084,

11701084; University of International

Business and Economics, Grant/Award

Number: 17YQ01

Abstract

In this paper, we consider Asian options with counterparty risk under stochastic

volatility models. We propose a simple way to construct stochastic volatility

models through the market factor channel. In the proposed framework, we

obtain an explicit pricing formula of Asian options with counterparty risk and

illustrate the effects of systematic risk on Asian option prices. Specially, the

U‐shaped and inverted U‐shaped curves appear when we keep the total risk of

the underlying asset and the issuer’s assets unchanged, respectively.

KEYWORDS

Asian options, counterparty risk, stochastic correlation, stochastic volatility

JEL CLASSIFICATION

G13

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INTRODUCTION

In this paper, we focus on the pricing issue of Asian options with counterparty risk under stochastic volatility models.

The payoff of Asian options depends on some form of averaging prices of the underlying asset, and the average can be a

geometric or arithmetic one. Both geometric and arithmetic Asian options have been intensively investigated in the

literature. For instance, Angus (1999) works under the Black–Scholes model to investigate the prices of geometric Asian

options. Kim and Wee (2014) obtain explicit prices of geometric Asian options with fixed and floating strikes under

Heston’s stochastic volatility models. Bayraktar and Xing (2011) find a sequence of functions that uniformly converge to

the price of an arithmetic Asian option, when the dynamics of the underlying asset follows a jump‐diffusion process.

Cai and Kou (2012) obtain a closed form for the double‐Laplace transform of arithmetic Asian options under the

hyperexponential jump‐diffusion model. Cai, Song, and Kou (2015) propose a general framework for pricing both

continuously and discretely monitored Asian options under one‐dimensional Markov processes and derive the double

transform of the Asian option price in terms of the unique bounded solution to a related functional equation. In this

paper, we mainly focus on geometric Asian options with counterparty risk under stochastic volatility models, where

leverage effects and stochastic correlation between assets are considered.

Counterparty risk has been considered when valuing the over‐the‐counter (OTC) derivatives, since these derivatives

are privately written and are not guaranteed by a third party, making holders of OTC contracts vulnerable to

counterparty risk. Credit‐sensitive OTC contracts include credit default swaps (CDS), forwards and European options.

European options with counterparty risk are first studied by Johnson and Stulz (1987) and then extended by Klein

(1996) and many other studies. For instance, Klein and Inglis (1999) and Liao and Huang (2005) focus on the effects of

stochastic interest rate on European option prices. Other factors, such as rare shocks (see, e.g., Tian, Wang, Wang, &

Wang, 2014; W. Xu, Xu, Li, & Xiao, 2012), stochastic volatility (see, e.g., Lee, Yang, & Kim, 2016; G. Wang, Wang &

Zhou, 2017; Yang, Lee, & Kim, 2014), and stochastic default barriers (see, e.g., Cao & Wei, 2001; Hui, Lo, & Ku, 2007;

Klein & Inglis, 2001; X. Wang, 2016), are also investigated. In addition, power exchange options, a generalization of

European options, have also been studied by taking counterparty risk into consideration (see, e.g., X. Wang, Song &

Wang, 2017; G. Xu, Shao, & Wang, 2019). In CDS markets, CDS dealers sell credit protection on a number of underlying

firms. Arora, Gandhi, and Longstaff (2012) investigate how dealers’default risk affects CDS prices using a cross‐

sectional data set. Brigo, Capponi, and Pallavicini (2014) and Crépey (2015a, 2015b) consider the counterparty risk

valuation of CDS.

In this paper, we aim to consider Asian options with counterparty risk under stochastic volatility models, since

Asian options are traded popularly in the OTC market. A typical example of Asian options traded in the OTC

market is Asian options on interest rates (see, e.g., Almeida & Vicente, 2012). These popular instruments are less

susceptible to market manipulation and offering simpler hedging strategies than regular interest rate options (see,

e.g., Chacko & Das, 2002). Another example is commodity options, and a variety of commodity options with

average underlying prices are traded OTC (see, e.g., Alexander & Venkatramanan, 2008). Specially, Asian options

are popular and commonly used for the price risk management (see, e.g., Kyriakou, Pouliasis, & Papapostolou,

2016). With the significant trading volume increases of Asian options, and the harrowing experience of the

subprime mortgage crisis in 2007, the counterparty risk should by no means be ignored when pricing Asian options

traded in the OTC market.

Actually, Asian options with counterparty risk have been investigated in the literature. For instance, Tsao and Liu

(2012) obtain approximation formulae for arithmetic Asian options subject to credit risk and show how the issuers’

characteristics affect the credit discount of Asian options. Jeon, Yoon, and Kang (2016) derive the closed‐form pricing

formula of vulnerable geometric Asian options for fixed and floating strike options with two assets captured by two

correlated geometric Brownian motions. However, different from these studies, we work under a general framework,

which captures leverage effects as well as stochastic correlation between assets. The proposed framework essentially

generalizes the model in X. Wang (2017a), and the differences in option prices implied by the proposed model and

X. Wang’s (2017a) model show the effects of stochastic volatilities of the market index, the underlying asset, and the

issuer’s assets. More specifically, we propose a simple way to construct stochastic volatility models and obtain an

explicit pricing formula for Asian options with counterparty risk in the proposed framework. Additionally, we break

down the risk of risky assets into idiosyncratic and systematic risk as in the capital asset pricing model (CAPM). We

begin with the dynamics of the market index, which represents the uncertainty stemming from systematic risk. Then

we connect the dynamics of the underlying asset and the issuer’s assets through the market factor channel, by using the

quantity betas to represent the asset’s sensitivity to systematic risk. In essence, asset prices are driven by two‐factor

stochastic volatility processes displaying leverage effects as well as stochastic correlation between two assets. To be

specific, the correlation coefficient between the underlying asset and the issuer’s assets is time‐varying and depends on

current levels of the variances of both assets and the market index as well. In the proposed framework, we derive the

closed‐form pricing formulae of Asian options with counterparty risk using the derived explicit expression of the

generating function. Furthermore, it should be noted that the proposed framework encompasses many existing models

as special cases, including X. Wang (2017a). In the numerical section, the U‐shaped curve appears when we investigate

Asian call option prices when the underlying asset has different proportions of systematic risk by keeping total risk

unchanged. In addition, the inverted U‐shaped curve appears when the issuer’s asset has different proportions of

systematic risk but unchanged total risk.

The rest of this paper is organized as follows. In Section 2, the theoretical framework is described and explicit pricing

formulae are derived. Section 3 is devoted to numerical results. Finally, Section 4 concludes the paper.

2

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STOCHASTIC VOLATILITY MODELS

In this section, we propose a stochastic volatility model and derive the explicit pricing formulae of Asian options with

counterparty risk. Motivated by the fact that the risk of a risky asset consists of systematic and idiosyncratic risk, we start

by specifying thedynamics of the market portfolio, andthen describe the dynamics of the underlying asset and the issuer’s

assets, respectively, using the quantity betas to represent the asset’s sensitivity to systematic risk as in the CAPM.

For valuation purposes, all processes are assumed under the risk neutral measure

Q

on a filtered probability space

Q

(

Ω,,)d

, which describes the uncertainty of the economy. On the probability space

Q

(

Ω,,)d

, the values of the market

portfolio are assumed to be governed by the following process:

WANG

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