Variational Methods In Derivatives Pricing

When underlying financial variables follow a Markov jump-diffusion process, the value function of a derivative security satisfies a partial integro-differential equation (PIDE) for European-style exercise or a partial integro-differential variational inequality (PIDVI) for American-style exercise. Unless the Markov process has a special structure, analytical solutions are generally not available, and it is necessary to solve the PIDE or the PIDVI numerically. In this chapter we briefly survey a computational method for the valuation of options in jump-diffusion models based on: (1) converting the PIDE or PIDVI to a variational (weak) form; (2) discretizing the weak formulation spatially by the Galerkin finite element method to obtain a system of ODEs; and (3) integrating the resulting system of ODEs in time. To introduce the method, we start with the basic examples of European, barrier, and American options in the Black-Scholes-Merton model, then describe the method in the general setting of multi-dimensional jump-diffusion processes, and conclude with a range of examples, including Merton's and Kou's one-dimensional jump-diffusion models, Duffie-Pan-Singleton two-dimensional model with stochastic volatility and jumps in the asset price and its volatility, and multi-asset American options.