Currency Option Pricing: Mean Reversion and Multi-Scale Stochastic Volatility

Abstract

This paper investigates the valuation of currency options when the underlying currency follows a mean-reverting lognormal process with multi-scale stochastic volatility. A closed-form solution is derived for the characteristic function of the log-asset price. European options can then be valued by means of Fourier inversion formula. The proposed model enables us to simultaneously calibrate to observed currency futures and implied volatility surface of currency options within a unified framework. The fractional fast Fourier transform (FFT) is adopted to implement the Fourier inversion so that the grid spacing restriction of the standard FFT can be relaxed, and it results in a more efficient computation. Using Monte Carlo simulation as a benchmark, our numerical examples show that the derived option pricing formula is accurate and efficient for practical use.

Keywords: Currency Option, Mean Reversion, Multiscale Stochastic Volatility, Implied Volatility