The Pricing of Catastrophe Equity Put Options with Default Risk

Published date01 June 2016
AuthorXingchun Wang
Date01 June 2016
The Pricing of Catastrophe Equity
Put Options with Default Risk*
School of International Trade and Economics, University of International Business
Economics, Beijing 100029, China
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 classication: G13
In this study, we consider a valuation model for catastrophe equity put options
with default risk
and continuous default monitoring.
Catastrophe equity put options are a form of options that provide its holder
with the right to sell a specied 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 specied 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 nancial 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. 181201
DOI: 10.1111/ir.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 nite 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 OrnsteinUhlenbeck intensity.
Default risk refers to the risk in a nancial contract that one counterparty
defaults prior to maturity and has received much attention because of the
20072008 nancial 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 specied
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 rst 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
International Review of Finance
© 2016 International Review of Finance Ltd. 2016182

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