The Valuation of Power Exchange Options with Counterparty Risk and Jump Risk

• Published date: 01 May 2017
• DOI: http://doi.org/10.1002/fut.21803
• Author: Yongjin Wang,Xingchun Wang,Shiyu Song
• Date: 01 May 2017

The Valuation of Power Exchange Options

with Counterparty Risk and Jump Risk

Xingchun Wang, Shiyu Song,*and Yongjin Wang

This study presents a pricing model for power exchange options, in which the possibility of

default by the risky counterparty as well as the arrival of important business information are

taken into consideration. The idiosyncratic and common jump components induced by the

arrival of business information are subsumed into all asset price processes whose dynamics

are correlated with each other. Employing the measure-change technique, we obtain a pricing

formula for the values of power exchange options with counterparty risk. At last, based on the

derived formula, we numerically analyze the impacts of counterparty risk and jump risk on

option prices. ©2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:499–521, 2017

1. INTRODUCTION

In this work, we consider a valuation model for power exchange options with counterparty

risk and jump risk. Power exchange options are a generalization of power options (see,

e.g., Tompkins (2000)) and Fischer–Margrabe exchange options (see, e.g., Fischer (1978)

Margrabe (1978)), both of which are rather powerful financial tools in fields of hedging non-

linear risk or compensation designs. Blenman and Clark (2005) investigate power exchange

options and explicitly solve for the price of European power exchange options under the

assumption that the underlying assets are governed by geometric Brownian motions. Wang

(2016a) extends the framework of Blenman and Clark (2005) to value power exchange op-

tions by incorporating correlated jump risk. Here, we study power exchange options with

counterparty risk in a more flexible framework.

Because an exchange clearinghouse does not take the other side of every over-the-

counter (OTC) transaction, holders of OTC contracts are vulnerable to counterparty risk,

which refers to the risk in a financial contract that one counterparty fails to make the pay-

ments in accordance with agreed terms. With the immense development of the OTC deriva-

tives markets 1over the last two decades, the significant counterparty risk confronted by

parties of OTC derivatives should by no means be ignored after the harrowing experience

Xingchun Wang is at the School of International Tradeand Economics, University of International Business

and Economics, Beijing 100029, China. Shiyu Song is at the School of Science, Tianjin University, Tianjin

300072, China. Yongjin Wang is at the School of Business, Nankai University, Tianjin 300071, China. This

study was supported by the National Natural Science Foundation of China (Nos. 11271203, 71532001 and

71503044).

JEL Classification: G13

*Correspondence author,School of Science, Tianjin University, Tianjin 300072, China. Tel: +8613612034872,

Fax: +022-23506423, e-mail: songshiyunk@aliyun.com

Received May 2015; Accepted June 2016

1The statistics in the ISDAsurvey “OTC Derivatives Market Analysis: Interest Rate Derivatives” published in January

2015, show that the total OTC derivatives notional outstanding approximated US$691.5 trillion at the end of June

2014.

The Journal of Futures Markets, Vol. 37, No.5, 499–521 (2017)

©2016 Wiley Periodicals, Inc.

Published online 21 August 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21803

500 Wang, Song, and Wang

of the subprime mortgage crisis in 2007. As a consequence, the issue of credit exposure

has to be taken seriously when pricing credit-sensitive OTC contracts such as credit default

swaps (CDS), forwards and options. For instance, Brigo and Mercurio (2006) show how

to consider the event that the counterparty may default in the risk neutral valuation of the

financial payoff. Brigo, Capponi, and Pallavicini (2014) develop an arbitrage-free valuation

framework for bilateral counterparty risk. Cr´

epey (2015a, 2015b) investigate the valuation

and hedging of bilateral counterparty risk on OTC derivatives. Gregory (2012) explains the

emergence of counterparty risk and considers portfolio management and hedging of credit

value adjustment and debit value adjustment. Brigo, Morini, and Pallavicini (2013) focus

on quantitative methods for the pricing and hedging of counterparty and funding risk for

all asset classes including interest rates, foreign exchange, commodities and equity. Wang

(2016b) investigates the valuation of catastrophe equity put options with counterparty risk.

In a word, after the subprime mortgage crisis, the subject of counterparty risk has become

an unavoidable issue, and hence taking counterparty risk into account is necessary when

valuing OTC contracts.

European vanilla options with counterparty risk have also been studied in the literature.

In Johnson and Stulz (1987), the authors obtain a pricing formula under the assumption that

the option holder will receive the total assets of the option writer in default at the expiration

date. Later, a more realistic assumption is made in Klein (1996), where the final payout in

default depends on the terminal market value of assets as well as the amount of other equally

ranking liabilities of the writer. Additionally, European options with counterparty risk have

also been considered by Hui, Lo, and Ku (2007), Hull and White (1995), Hung and Liu

(2005), Jarrow and Turnbull (1995), and Klein and Inglis (1999). However, all these studies

rule out jump risk. To the best of our knowledge, there are few papers which have considered

European options with counterparty risk under jump-diffusion models (see, e.g., Xu, Xu, Li,

& Xiao 2012 and Tian, Wang, Wang, & Wang 2014). Xu et al. (2013) price vulnerable call

options under a jump-diffusion model, but the lack of consideration on jump risk common

to firm values and stock prices may sometimes run counter to the reality. In Tian et al.

(2014), the authors obtain a closed-form valuation formula for vulnerable European options

under the assumption that the dynamics of asset prices are governed by jump-diffusions

with two sorts of assets correlated with each other. However, Tian et al. (2014) only focus

on the common jump times of the common jump component. Here, we consider not only

the common jump times but also the jump sizes. In addition, the differences in the effects

of common jump components on asset prices are also incorporated. Actually, a more flexible

framework is considered in this paper, and our pricing formula encompasses many existing

formulae as special cases.

Basically, incorporating jump risk into the dynamics of asset prices has its own pro-

nounced economic implications. Merton (1976) pioneers a compound Poisson process to

characterize the outliers in stock returns due to the arrival of new business information. From

then on, various jump diffusion models have been advocated as a class of the most favorable

alternative models to interpret the heavy tails suggested empirically in asset return distribu-

tions. Take for instance the representative models of the double exponential jump-diffusion

model in Kou (2002), the Markov-modulated jump-diffusion model in Elliott, Siu, Chan,

and Lau (2007), the mixed-exponential jump-diffusion model in Cai and Kou (2011) and

the stochastic volatility model with jumps in Scott (1997). Additionally, the option pricing

issue (see, e.g., Carr & Wu, 2004) and hedging strategies (see, e.g., Hubalek, Kallsen, & Kar-

wczyk, 2006 and Wang & Wang, 2014) have also been studied under general L ´

evy dynamics.

Taking counterparty risk and jump risk into account, we provide a valuation model

for power exchange options. Following previous literature on vulnerable options, we adopt

the structural approach to deal with counterparty risk. That is to say, default occurs when