Quanto Pricing with Copulas

Abstract

We study the practical problem of pricing a particular multi-asset option, a quanto FX option. The Black model, which corresponds to a jointly lognormal distribution of asset prices at expiry, is inconsistent with the implied volatility smile for each of the three relevant currency pairs. We demonstrate a practical methodology for constructing a model for the joint distribution that is calibrated to all relevant implied volatilities. The margins of this distribution are determined separately in an initial stage. To calibrate the joint distribution to the implied volatility smile on the remaining FX rate, we perturb the dependence structure associated with the Black model (the Normal copula) in order to influence the tail dependence characteristics of the resulting joint distribution.

We calibrate our model to a number of real-life scenarios corresponding to several maturities and currency set-ups. We find that a well-known ad-hoc adjustment to the Black pricing formula often gives lower quanto call prices than those calculated under our transformed copula model. The relative difference in quanto prices with strikes furthest away from at-the-money is occasionally large (10-15%).

JEL Classification: G130, C190