The Pricing of Catastrophe Equity Put Options with Default Risk

• DOI: http://doi.org/10.1111/irfi.12075
• Published date: 01 June 2016
• Date: 01 June 2016

The Pricing of Catastrophe Equity

Put Options with Default Risk*

XINGCHUN WANG

School of International Trade and Economics, University of International Business

Economics, Beijing 100029, China

ABSTRACT

In this paper, we present a pricing model for catastrophe equity put options

with default risk by assuming that the default of the option issuer may occur

at any time prior to maturity of the option. Catastrophic events are assumed

to occur according to a doubly stochastic Poisson process, and stock price is

affected by the catastrophe losses, which follow the compound doubly

stochastic Poisson process. As for default risk, we adopt typical structural

approaches, and we also allow the correlation between the underlying stock

and the assets of the option issuer. Under this framework, we derive a pricing

formula for catastrophe equity put options with default risk. Finally, numerical

analysis is presentedto illustrate effects of defaultrisk on catastrophe equity put

option prices.

JEL classification: G13

I. INTRODUCTION

In this study, we consider a valuation model for catastrophe equity put options

with default risk

1

and continuous default monitoring.

Catastrophe equity put options are a form of options that provide its holder

with the right to sell a specified amount of its stocks to investors at a

predetermined price. This kind of options allows its holder to have the right to

exercise the option when the accumulated losses surpass a specified trigger. Thus,

insurance companies could raise additional equity capital by selling their stocks

after they pay for catastrophe losses. In this way, insurance companies could

cover large losses caused by catastrophic events. Catastrophe equity put options

have been investigated extensively in the literature. Among them, Jaimungal

and Wang 2006 obtain closed-form formulae for the price of the option under

* This study was supported by the National Natural Science Foundation of China (No. 11271203)

and the Fundamental Research Funds for the Central Universities in UIBE (14QD03). The authors

would like to thank the anonymous referees and associate editor for their helpful comments and valu-

able suggestions that led to several important improvements.

1 Default risk is the risk in a financial contract that one counterparty defaults prior to maturity

and fails to make the agreed payments. Default risk has been one of the risks participants in

the over-the-counter markets have to face, for instance, Arora et al. 2012 show that default risk

is priced in the credit default swaps markets.

© 2016 International Review of Finance Ltd. 2016

International Review of Finance, 16:2, 2016: pp. 181–201

DOI: 10.1111/irfi.12075

the assumptions that asset prices are modeled through a jump-diffusion process,

which is correlated to the loss process, and the losses are generated by a

compound Poisson process. Chang et al.2008 price a richer set of catastrophe

options in an incomplete-market discrete-time setting. In Chang and Hung

2009, the authors provide explicit analytical formulae for catastrophe put option

prices under deterministic and stochastic interest rates when the underlying asset

price is modeled through a Lévy process with finite activity. Chang et al.2010

adopt the no-arbitrage martingale pricing methodology to consider an Asian-

style catastrophe option with claim arrival and loss uncertainties in a doubly bi-

nomial framework. Lin and Wang 2009 obtain the analytical expression for the

price of perpetual American catastrophe equity put options using the expected

discounted penalty function proposed by Gerber and Shiu 1998. Jiang et al.

2013 present a catastrophe option pricing model that considers default risk,

and the option issuer can default only at the maturity. It is also worth mention-

ing that catastrophe bonds and catastrophe swaps are also introduced to manage

catastrophe risk and have been studied in the literature, for instance, Burnecki

and Kukla 2003 calculate non-arbitrage prices of a zero-coupon and coupon ca-

tastrophe bond in the compound doubly stochastic Poisso n model framework.

Lee and Yu 2007 consider a reinsurance contract using a contingent-claim

framework and examine how a reinsurance company can reduce its default risk

by issuing catastrophe bonds. Braun 2011 analyzes the catastrophe swaps,

when catastrophe occurrence is modeled as a doubly stochastic Poisson process

with a mean-reverting Ornstein–Uhlenbeck intensity.

Default risk refers to the risk in a financial contract that one counterparty

defaults prior to maturity and has received much attention because of the

2007–2008 financial crisis. The most cited example is the largest insurance

company in the world, American International Group (AIG). AIG sold insurance

against bond defaults through derivative contracts called as credit default swaps

(CDS), a swap contract in which the buyer of the contract pays a specified

amount to the seller and receives a payoff if the bond defaults happen. CDS is

frequently traded in the over-the-counter (OTC) markets and is priced by taking

into consideration the possible defaults of the seller of the contract. Brigo et al.

2014 develop an arbitrage-free valuation framework for bilateral default risk.

Crépey 2015a, 2015b investigates the valuation and hedging of bilateral default

risk on OTC derivatives. Liang et al.2014 derive a pricing formula of CDS with

bilateral default risk in an intensity-based model with Markov regime switching.

By examining an extensive data set of CDS transaction prices, Arora et al.2012

show that default risk is priced in the CDS markets. Hence, taking default risk

into account is necessary when holding and valuing OTC contracts.

Because there is no organized exchange, holders of OTC contracts are

vulnerable to default risk. Actually, European options with default risk have

been studied in the literature. Johnson and Stulz 1987 first incorporate default

risk into option pricing model by assuming that the option is the sole liability

of the option issuer and default will happen when the value of the option is

greater than the value of the assets of the option issuer. Later, a more realistic

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