13 Numerical Methods for the Solution of PIDEs

Several models including jumps have been presented in Chapter 12. They provide extensions of flat volatility, local volatility and stochastic volatility models. In this chapter, we present an additional way for solving contingent claims under such jump models. Instead of using the characteristic function of the underlying probability density (mainly limited to the plain vanilla case) or the Monte Carlo method (see Chapter 9), the SDE can also be transformed into a partial integro-differential equation (PIDE). The discretization of such PIDEs requires special care: a naïve discretization of the integral term using finite differences or finite elements would lead to a dense coefficient matrix in the corresponding linear system of equations. We restrict the discussion to one-dimensional PIDEs; in our opinion, (Q)MC should be used for higher-dimensional problems.

Extensions to valuate American options and a survey of finite difference methods for option pricing under finite activity jump diffusion models are given in (Salmi and Toivanen, 2012). Recently, Kwon and Lee have presented a method based on finite differences with three time levels with a second-order convergence rate (Kwon and Lee, 2011). A finite element discretization of the PIDE resulting from a stochastic volatility jump diffusion model is discussed in (Zhang, Pang, Feng and Jin, 2012). Matache and Schwab used a wavelet Galerkin-based method for the pricing of American ...

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