Expansion method for pricing foreign exchange options under stochastic volatility and interest rates

TL;DR

This paper extends a known expansion method for pricing options to models with both stochastic volatility and interest rates, providing a second-order approximation and validating it against Monte Carlo simulations.

Contribution

It applies the smart expansion method to a more complex model incorporating stochastic interest rates and derives a second-order pricing formula.

Findings

The second-order expansion formula performs comparably to characteristic function-based methods.

Numerical results validate the accuracy of the proposed approximation.

The method offers a practical alternative for pricing in complex stochastic models.

Abstract

Some expansion methods have been proposed for approximately pricing options which has no exact closed formula. Benhamou et al. (2010) presents the smart expansion method that directly expands the expectation value of payoff function with respect to the volatility of volatility, then uses it to price options in the stochastic volatility model. In this paper, we apply their method to the stochastic volatility model with stochastic interest rates, and present the expansion formula for pricing options up to the second order. Then the numerical studies are performed to compare our approximation formula with the Monte-Carlo simulation. It is found that our formula shows the numerically comparable results with the method proposed by Grzelak et al. (2012) which uses the approximation of characteristic function.