Repricing the cross smile: an analytic joint density

Derivatives contracts on multiple foreign exchange rates must be priced to avoid arbitrage by contracts on the cross-rates. Given the triangle of smiles for two underlyings and their cross, Peter Austing provides an analytic formula for a joint probability density

When valuing a derivatives contract whose payout depends on two assets, the correlation between the random processes followed by those two assets must be taken into account. In most asset classes, there is no liquid instrument to determine that correlation. This makes the exposure to correlation hard to hedge, but straightforward from a modelling point of view since a single number, perhaps calculated from a historic time series of spot returns, can be used.

Foreign exchange is different. Let us