Using Copulas to Construct Bivariate Foreign Exchange Distributions with an Application to the Sterling Exchange Rate Index

We model the joint risk-neutral distribution of the euro-sterling and the dollar-sterling exchange rates using option-implied marginal distributions that are connected via a copula function that satisfies the triangular no-arbitrage condition. We then derive a univariate distribution for a simplified sterling effective exchange rate index. Our results indicate that standard parametric copula functions, such as the commonly used Normal and Frank copulas, fail to capture the degree of asymmetry observed in the data. We overcome this problem by using a non-parametric dependence function in the form of a Bernstein copula which is shown to produce a very close fit. We further give an example of how our approach can be used to price currency index options