Variational Methods in Derivatives Pricing
نویسندگان
چکیده
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 This research was supported by the National Science Foundation under grants DMI-0422937 and DMI-0422985.
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