Differences in the Prices of Vulnerable Options with Different Counterparties

• Date: 01 February 2017
• Author: Xingchun Wang
• DOI: http://doi.org/10.1002/fut.21789
• Published date: 01 February 2017

Differences in the Prices of Vulnerable

Options with Different Counterparties

Xingchun Wang*

In this paper, a new pricing model is proposed to investigate the differences in the prices

of vulnerable options with different counterparties. I start by specifying the dynamics of the

market portfolio, and then break down the risk of the underlying asset and the assets of

the counterparties into systematic and idiosyncratic risk, which allows me to distinguish the

counterparties from these two kinds of risk. Finally, the derived pricing formulae are used to

illustrate the differences between vulnerable option prices. ©2016 Wiley Periodicals, Inc. Jrl

Fut Mark 37:148–163, 2017

1. INTRODUCTION

In over-the-counter (OTC) markets, options are privately written and are not guaranteed

by a third party. Because of the possibility of default by the issuer, options traded in OTC

markets have lower values than otherwise identical exchange-listed options. This difference

is quite important for all participants in OTC markets and has been studied in the literature.

Johnson and Stulz (1987) first incorporate default risk into the option pricing model and

study the prices of vulnerable options. Klein (1996) extends the results in Johnson and

Stulz (1987) by allowing the option writer to hold other liabilities which rank equally with

payments under the option. Based on the work of Klein (1996), many other factors such

as stochastic interest rate, rare shocks, stochastic volatility, and stochastic default barriers

are considered to illustrate the impacts of these factors on vulnerable option prices (see

the coming section for detail). Additionally, after the subprime mortgage crisis, the issue of

credit exposure has been taken seriously when pricing other credit-sensitive OTC contracts.

Brigo, Capponi, and Pallavicini (2014) develop an arbitrage-free valuation framework for

credit default swaps (CDS) with bilateral default risk. Wang (2015a) considers the value of

catastrophe equity put options with default risk. Cr´

epey (2015a,b) investigate the valuation

and hedging of bilateral default risk on OTC derivatives. Wang, Song, and Wang (2015)

focus on power exchange options with jump risk and default risk.

In this paper, I investigate the differences in the prices of vulnerable options with

different counterparties. Options on the same underlying asset are usually issued by a set

of counterparties in OTC markets. For instance, Gain Capital offers OTC options on major

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

and Economics, Beijing 100029, China. This study was supported by the National Natural Science Foun-

dation of China (Nos. 11271203 and 71503044) and by Academic Discipline Project in UIBE. The author

would like to thank the anonymous referee and the editor, Robert I. Webb, for their helpful comments and

valuable suggestions that led to several important improvements. All errors are my responsibility.

JEL Classification: G13

*Correspondence author,School of International Trade and Economics, University of International Business

and Economics, Beijing 100029, China. Tel: +86(10)64493670, Fax: +86(10)64493042, e-mail: xchwangnk@

aliyun.com; wangx@uibe.edu.cn

Received March 2015; Accepted March 2016

The Journal of Futures Markets, Vol. 37, No.2, 148–163 (2017)

©2016 Wiley Periodicals, Inc.

Published online 19 May 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21789

Vulnerable Options with Different Counterparties 149

foreign exchange pairs including the USD/JPY pair. Saxo Bank also provides the USD/JPY

currency options, and the pricing model is based on an implied volatility surface for the

Black–Scholes model based on the interbank markets.1In CDS markets, different CDS

dealers sell credit protection on the same underlying firm. Arora, Gandhi, and Longstaff

(2012) investigate how dealers’ default risk affects CDS prices using a cross-sectional data

set. Here, I develop a pricing model to investigate the effects of issuer’s default risk on option

prices. Vulnerable option prices depend on the probability of default of the issuer, however,

the probability that issuers will default will likely differ, which makes a difference in option

prices. This difference is the focus of this paper and has never been studied in the literature.

To capture the correlation between assets, I begin with the dynamics of the market

portfolio, and then connect the dynamics of the underlying asset and the assets of the issuers

through a common factor.Actually, the risk of an asset comprises systematic and idiosyncratic

risk, also known as diversifiable risk. Over the long run, a well-diversified portfolio provides

returns which correspond with its exposure to systematic risk. Here I use the market portfolio,

and break down the risk of the assets into systematic and idiosyncratic risk. In this way, I

can distinguish the issuers from these two kinds of risk and investigate the differences in

the prices of vulnerable options issued by them. In addition, the quantity betas are used to

represent the asset’s sensitivity to systematic risk. In contrast to earlier studies, this paper

has three main characteristics. First, I break down the risk into idiosyncratic risk for each

asset and systematic risk that affects the prices of all assets. This assumption allows me to

investigate the impacts of systematic risk on vulnerable option prices. Second, the correlation

between two assets is connected and affected by the market betas, not just a given constant

as in Klein (1996) and the extended versions of Klein (1996). Third, the proposed framework

even allows me to investigate the case where the issuers have the same total risk but different

relative proportions of systematic and idiosyncratic risk.

The rest of this paper is organized as follows. In the coming section, I review the

literature on vulnerable options. In Section 3, the theoretical framework is described and

explicit pricing formulae of vulnerable options are derived. Section 4 is devoted to numerical

results. In Section 5, an extended version of the theoretical framework is given. Finally,

Section 6 summarizes and concludes the paper.

2. THE LITERATURE ON VULNERABLE OPTIONS

The effects of default risk on vulnerable option prices have been investigated in the literature.

Motivated by the fact that a large number of options are privately written and there have been

defaults on some insurance contracts, Johnson and Stulz (1987) first consider the valuation

of European options with default risk. Klein (1996) relaxes the assumption in Johnson and

Stulz (1987) that default events occur when the option value is greater than the value of the

issuer’s assets by allowing the option issuer to hold other liabilities which rank equally with

payments under the option.

Interest rate risk has been taken into account to illustrate the impact of interest rate

on vulnerable option prices. Assuming interest rate varies stochastically, Klein and Inglis

(1999) derive a closed-form pricing formula of vulnerable options via the partial differential

equation approach and present a lot of numerical examples to illustrate vulnerable option

1For further information about a wide range of OTC options they offers, please visit the Gain Capital web-

site http://www.gaincapital.com/otc-trading-solutions.shtml and the Saxo Bank website http://www.saxobank.com/

prices/fx-options/#saxoModal.