Pricing Vulnerable Options Under Stochastic Assets and Liabilities

Abstract

This paper presents both closed-form formulas and binomial tree algorithms to evaluate vulnerable derivatives. The payoff function extends mainly from the Klein (1996) and the Ammann (2001) credit risk frameworks. Three stochastic processes, the underlying stock price, the assets value of the option writer, and the liabilities value of the option writer, are suitably modeled. Closed-form solutions are derived for vulnerable European options under the suggested payoff function. A conditional binomial tree algorithm for two correlated stochastic processes, the underlying stock price and the asset-to-debt ratio process, are properly established. Moreover, adapting Rubinstein (1994) approach, a general binomial pyramid algorithm is set up. It is numerically illustrated that the proposed conditional binomial tree model contains the closed-form formula as a limiting case, for vulnerable European options.