A moment matching method for option pricing under stochastic interest rates

In this paper, we present a new and straightforward approximation methodology for pricing a call option in a Black and Scholes market, characterized by stochastic interest rates. The method relies on a Gaussian moment matching technique applied to a conditional Black and Scholes formula, used to disentangle the distributional complexity of the underlying price process. The problem then reduces to exploiting the Gaussian density and the expression of the bond price induced by the interest rate. To check its accuracy and computational time, we implement it for a CIR interest rate model correlated with the underlying, using Monte Carlo simulations as a benchmark. The method performance turns out to be quite remarkable, even when compared with similar results obtained by the affine approximation technique presented in Grzelak and Oosterlee, and by the expansion formula introduced in Kim and Kunimoto. In the last section, we apply the method also to the pricing of Forward-Starting options, to the evaluation of the credit spreads in the Merton structural approach to credit risk, and we outline a possible application to a stochastic volatility model.