Pricing Vulnerable Options with Jump Clustering

• Date: 01 December 2017
• Published date: 01 December 2017
• DOI: http://doi.org/10.1002/fut.21843
• Author: Keshab Shrestha,Weidong Xu,Yong Ma

Pricing Vulnerable Options with

Jump Clustering

Yong Ma, Keshab Shrestha, and Weidong Xu *

This paper presents a valuation of vulnerable European options using a model with self-exciting

Hawkes processes that allow for clustered jumps rather than independent jumps. Many ex-

isting valuation models can be regarded as special cases of the model proposed here. Using

numerical analyses, this study also performs sensitivity analyses and compares the results to

those of existing models for European call options. The results show that jump clustering

has a significant impact on the option value. ©2017 Wiley Periodicals, Inc. Jrl Fut Mark

37:1155–1178, 2017

1. INTRODUCTION

When an option buyer is exposed to the option writer’s default risk, the option is referred

as a vulnerable option. In this case, the option writer’s default risk is an example of a coun-

terparty risk. Most over-the-counter options and other derivative securities are associated

with counterparty risks. Traditional research has, to a large extent, neglected this type of

risk. However, the recent financial crisis has brought the issue of counterparty risk to the

forefront of the discussion among policymakers, practitioners, and academic researchers.

Counterparty risks are also an important risk factor recognized by Basel III from the Bank of

International Settlements (BIS) and the Dodd–Frank Act in the United States, which was

signed into federal law by President Obama on July 21, 2010.

In order to price vulnerable options, we need to model the underlying asset price dy-

namics and the counterparty risk. Based on the default model of corporate bonds proposed by

Merton (1974), Johnson and Stulz (1987) present several vulnerable option pricing models,

in which the option itself is assumed to be the only liability of the option writer. Under the

assumption of independence between the underlying asset and the default risk of the option

writer, Hull and White (1995) and Jarrow and Turnbull (1995) derive analytical solutions

for vulnerable option values. However, they assume that the payout ratio is exogenous and

Yong Ma is an Assistant Professor at the College of Finance and Statistics, Hunan University, Changsha,

Hunan, China. Keshab Shrestha is a Professor of Banking and Finance at the School of Business, Monash

University Malaysia, Bandar Sunway, Selangor, Malaysia. Weidong Xu is an Associated Professor at the

School of Management, Zhejiang University, Hangzhou, Zhejiang, China. We are very grateful to the Editor

Robert I. Webb and the anonymous referee for their valuable comments and suggestions. This research

was supported by National Natural Science Foundation of China (71601075), Humanity and Social Science

YouthFoundation of the Ministry of Education of China (15YJC790071), and Zhejiang and Hunan Provincial

Natural Science Foundation of China (LY15G010002,2016JJ3047). All errors are our own.

*Correspondence author,School of Management, Zhejiang University, Hangzhou 310085, China. Tel: +86-571-

88206867, Fax: +86-571-88206867, e-mail: xwd1981@163.com; weidxu@zju.edu.cn

Received October 2015; Accepted December 2016

The Journal of Futures Markets, Vol. 37, No.12, 1155–1178 (2017)

©2017 Wiley Periodicals, Inc.

Published online 23 February 2017 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21843

1156 Ma, Shrestha, and Xu

the default event is modeled using a reduced-form approach. 1Klein (1996) extends these

models by allowing the option writer to have other liabilities, where the values of the assets

underlying the option and the option writer’s assets are allowed to be correlated. He also

allows the payout ratio or recovery rate to be endogenous. Subsequently, many extensions

and variants of the models used by Johnson and Stulz (1987) and Klein (1996) have been

proposed. For instance, Klein and Inglis (1999) extend the model proposed by Klein (1996)

by incorporating interest rate risk. In another study,Klein and Inglis (2001) allow the default

boundary to depend on the value of the option itself. Hung and Liu (2005) present a model

to value vulnerable options in an incomplete market. Chang and Hung (2006) investigate

vulnerable American options based on the two-point Geske and Johnson method. Finally,

Klein and Yang (2010) extend the models of Johnson and Stulz (1987) and Klein (1996) to

price vulnerable American options.

All the works mentioned above use Brownian-based diffusion processes to model the

stock price and firm-value dynamics. Even though diffusion models are used extensively in

the valuation of derivative securities, owing to their tractability, these models have some

limitations because they do not allow jumps. First, theoretically speaking, in an efficient

market, asset prices fully reflect all available relevant information. In addition, news arrives

randomly and, when it does so, we expect the stock price to exhibit jumps. Second, these

models are considered to be inconsistent with empirical observations in financial markets,

where jumps are clearly visible (Bakshi, Cao, & Chen, 1997; Bates, 1996, 2000; Eraker,2004;

Eraker, Johannes, & Polson, 2003; Pan, 2002). Finally, in such models, a firm will not default

unexpectedly because of the impossibility of sudden big drops in the firm’s value (Jones,

Mason, & Rosenfeld, 1984). In order to overcome these limitations, Xu, Xu, Li, & Xiao (2012)

and Tian, Wang, Wang, & Wang (2014) propose improved models in which the underlying

stock price and the firm value follow Poisson jump-diffusion processes. Specifically, Tina

et al. divide the jumps into an idiosyncratic component, which affects only one particular

asset price, and a systematic component, which affects all the asset prices.

However, Poisson jump-diffusion models do not allow for clustered jumps, where a

single jump increases the probability of future jumps occurring. When news initially arrives,

we expect further news to arrive that would clarify the initial news, or that would reveal the

extent of its impact. In this case, security prices would respond not only to the initial news,

but also to the ways in which market participants and firms react to the news. Therefore,

theoretically, we expect to observe jump clustering. The clustering of jumps is especially

expected during a crisis.AsA

¨

ıt-Sahalia, Cacho-Diaz, & Laeven (2015) point out, “what

makes a crisis worthy of that name is typically not the initial jump, but the amplification

that takes place subsequently over hours or days, and the fact that other markets become

1In the existing literature, there are two main approaches to modeling credit risk: (1) the structural approach; and

(2) the reduced-form approach. In the structural approach, the default event is modeled based on the fundamental

characteristics of the obligor (e.g., the values of the obligor’s assets and debt). The default event is assumed to take

place once the value of the asset falls below a threshold value (e.g., Black & Cox, 1976; Longstaff & Schwartz,

1995; Merton, 1974). The structural approach is intuitive because it links the default risk to the firm’s economic

fundamentals. The main shortcoming of this approach lies in the assumption that the value the obligor’s assets

can be observed directly. In the reduced-form approach, instead of modeling the value of the obligor’s assets

and its capital structure, a default time is modeled as a stopping time by exogenously specifying a hazard rate or

intensity of default (Duffie & Singleton, 1999; Jarrow,Lando, & Turnbull, 1997; Madan & Unal, 1998. Owing to its

tractability, the reduced-form approach to modeling credit risk is popular. Though there exist differences between

the two approaches, G¨

und¨

uz and Uhrig-Homburg (2014) have recently shown that their predictive powers are quite

close, on average. For more details about the two approaches, please refer to Bielecki and Rutkowski (2013), and

the references therein.