An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed point iteration scheme are presented. For typical model parameters, it is shown that the fixed point iteration reduces the error by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform (FFT) method. Techniques are developed for avoiding wrap-around effects. Numerical tests of convergence for a variety of options are presented.

The fair price for an American option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem. We develop an implicit discretization method for pricing such American options. The jump diffusion correlation integral term is computed using an iterative method coupled with an FFT while the American constraint is imposed by using a penalty method. We derive sufficient conditions for global convergence of the discrete penalized equations at each timestep. Finally, we present numerical tests which illustrate such convergence. Subject Classification (1991): 65M12, 65M60, 91B28 Mathematics

A semi-Lagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semi-Lagrangian timestepping. Crank-Nicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. With low volatility values, it is observed that the non-smoothness at the strike in the payoff affects the convergence rate; sub-quadratic convergence rate is observed.

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Many debt issues contain an embedded call option that allows the issuer to redeem the bond at specified dates for a specified price. The issuer is typically required to provide advance notice of a decision to exercise this call option. The valuation of these contracts is an interesting numerical exercise because discontinuities may arise in the bond value or its derivative at call and/or notice dates. Recently, it has been suggested that finite difference methods cannot be used to price callable bonds requiring notice (Büttler, 1995;Büttler and Waldvogel, 1996). Poor accuracy was attributed to discontinuities and difficulties in handling boundary conditions. As an alternative, a semi-analytical method using Green's functions for valuing callable bonds with notice was proposed (Büttler and Waldvogel, 1996). Unfortunately, the Green's function method is limited to special cases. Consequently, it is desirable to develop a more general approach. We provide this by using more advanced techniques such as flux limiters to obtain an accurate numerical partial differential equation method. Finally, in a typical pricing model (Cox et al., 1985) an inappropriate financial condition is required in order to properly specify boundary conditions for the associated PDE. We show that a small perturbation of such a model is free from such artificial conditions.