Option Prices and Risk-Neutral Densities for Currency Cross-Rates

Abstract

The theoretical relationship between the risk-neutral density (RND) of the euro/pound cross-rate and the bivariate RND of the dollar/euro and the dollar/pound rates is derived; the required bivariate RND is defined by the dollar-rate marginal RNDs and a copula function. The cross-rate RND can be used by banks, international businesses and central bankers to assess market expectations, to measure risks and to value options, without relying on over-the-counter markets which may be either non-existent or illiquid.

Empirical comparisons are made between cross-rate RNDs estimated from several datasets. Five one-parameter copula functions are evaluated and it is found that the Gaussian copula is the only one-parameter copula function which is ranked highly in all of the comparisons we have made.

Keywords: Option pricing, density estimation, exchange rates, cross-rate, copulas

JEL Classification: F31, G13, G15