Equity and Foreign Exchange Hybrid Models for Pricing Long-Maturity Financial Derivatives

Abstract

Modelling derivative products in Finance usually starts with the specification of a system of Stochastic Differential Equations (SDEs), that corresponds to state variables like stock, interest rate, Foreign Exchange (FX) rate and volatility. By correlating the SDEs for the different asset classes one can define the hybrid models, and use them for pricing multi-asset derivatives. Even if each of the individual SDEs yields a closed-form solution, a non-zero correlation structure between the processes may cause difficulties for efficient product pricing. Typically, a closed-form solution of hybrid models is not known, and numerical approximation by means of Monte Carlo (MC) simulation or discretization of the corresponding Partial Differential Equations (PDEs) has to be employed for pricing. The speed of pricing European derivative products is crucial, especially for the calibration of the SDEs. Several theoretically attractive SDE models, that cannot fulfil the speed requirements, are not used in practice.