Path-Dependent Multicurrency Interest Rate Derivatives

This paper derives the arbitrage-free pricing formulae and hedging ratios of path-dependent multicurrency interest rate derivatives in continuous time. The exchange rates and the zero-coupon bond prices are modeled by multivariate log-normal processes with an arbitrary number of random factors. The exact pricing formulae are obtained by solving the time-evolution equation of the contingent claims described by a multi-dimensional Kolmogorov field equation. Applications to the Asian options and Bermudan options are given as examples. Several European options on currency swaps, forwards and futures contracts are also illustrated as special cases. These exact solutions are factor-independent and explicitly depend on the instantaneous covariances of the return rates of the exchange rates and the zero-coupon bonds. By scaling the historical covariances obtained from the market data, one can easily calibrate the volatility surface to match the vanilla options