Abstracts

Recently, the trade of financial derivatives in the over-the-counter markets has increased rapidly. Since there is no organized exchange to guarantee the promised payment in over-the-counter markets, the option holder is vulnerable to default risk. These options subject to the credit default risk are called vulnerable options. The value of a vulnerable option is less than a non-vulnerable option because of the possibility of default. Many researches subject to credit default risk have been studied extensively after Black and Scholes and Merton. Johnson and Stulz first proposed the option pricing formula for vulnerable European options, assuming that the option is the only liability of the counterparty. They assumed that an option holder receives all the assets of the option writer when the value of the option writer's assets is less than the value of the option. In the research of Hull and White, the other liabilities of the option writer were considered and the payment was determined by a proportion of the nominal claim when default occurs. But they did not consider the dependence between the value of the assets of the option writer and the asset underlying the option. Jarrow and Turnbull presented a new approach for pricing and hedging derivative securities with credit risk. Klein extended the study of Johnson and Stulz by allowing the counterparty to have other liabilities in the capital structure. Also, he obtained the formula under the assumption of recognizing the correlation between the option writer's asset and asset underlying the option, implying a payout ratio endogenous to the model is specified. He assumed that the option holder receives the proportion of nominal claim by the option writer in the event of financial distress and default boundary is fixed. In contrast to above assumptions, Klein considered the dependence of the total liabilities of the option writer on the value of the claim of the option holder. But the above papers assumed the volatility of the underlying asset is constant over the life of the vulnerable option. This simplified assumption is inappropriate to explain the volatility smile or skew of the implied volatility of the underlying asset. Hence a model must be developed to adapt to the real financial situation. Empirical evidence presents that volatility is a random process rather than a deterministic process. In the past two decades, many researches have been devoted to formulate these features of stochastic volatility models. For example, Heston model, the SABR model, the GARCH model and the Chen model. In this paper we consider the vulnerable option pricing with stochastic volatility extended the study of Klein to the case where the volatility of the underlying asset follows the mean-reverting OU process. The purpose of this paper is to provide asymptotic solutions of the vulnerable option pricing by applying singular perturbation method.