Pricing Vulnerable Options with Stochastic Volatility and Stochastic Interest Rate

This paper considers the pricing issue of vulnerable European options when the price process of the underlying asset follows the GARCH diffusion model with stochastic interest rate. Based on the proposed model, we obtain an approximate solution for the vulnerable European option price via means of Fourier transform. In addition, the Greeks of vulnerable option price are derived explicitly. Besides, the approximate solution of vulnerable option price can be quickly computed by using the fast Fourier transform (FFT) algorithm. The results of Monte Carlo simulations indicate that FFT is accurate, fast and easy to implement. More important, the pricing model also reveals that: (i) a negative correlation of volatility with the spot return creates a fat left tail and thin right tail in the distribution of continuously compounded spot returns. Thus, for in-the-money options, the vulnerable option prices of the proposed model are higher than those of Klein (J Bank Finance 20(7):1211-1229, 1996). While for deep-out-of-the-money options, the vulnerable option prices of the proposed model are smaller; (ii) the higher long-run mean of the underlying asset price's instantaneous variance, the higher vulnerable option price; (iii) the long-run mean of the stochastic interest rate exerts a positive effect on the value of vulnerable European option.