A closed-form pricing formula for vulnerable European options under stochastic yield spreads and interest rates

This paper develops a three-factor valuation model of vulnerable European options incorporating stochastic yield spreads and interest rates, which extends a constant yield spread and deterministic interest rate proposed in Hull and White [12]. The dynamics of the short-term interest rate are represented implicitly by a stochastic bond price process. Therefore, the stochastic factors in the model are the spot price of the underlying asset that follows a geometrical Brownian motion process, the yield spreads and the default-free unit discount bond that are modeled by a mean-reverting Ornstein-Uhlenbeck stochastic process. Furthermore, we exploit Mellin transform techniques to derive a closed-form solution for vulnerable European options under the Black-Scholes model, which is simply computed using the standard normal cumulative distribution function so that the pricing and hedging of vulnerable European options can be computed very accurately and rapidly. Numerical experiments demonstrate how credit risk and interest rate risk affect the prices of European options.