A Numerical Approach to Pricing American Call Options under SVJD

In Chapter 3 we considered the simpler case of geometric Brownian motion plus jump-diffusion dynamics. In Chapter 4 we considered the case of stochastic volatility and jump-diffusion dynamics but in that case the volatility process was of the Heston type and the jumps were normally distributed. We showed, in particular how the transform method can be applied to this situation. In Chapter 5 we considered numerical approaches to the problem under Heston stochastic volatility dynamics. There we focused on the transform approach. In this chapter we consider the case, which combines stochastic volatility and jump-diffusion, as was postulated in Chapter 2 and more extensively in Chapter 4. This will allow a far more general structure to be investigated. In Sec. 7.1 we give some general considerations to the solution of the problem using various techniques and boundary solutions. In Sec. 7.2 we discuss the solution using the projected successive overrelaxation (PSOR) method to determine the “exact” solution. In Sec. 7.3 we discuss the componentwise splitting approach, which decomposes the discretised problem into three linear complementarity problems with tridiagonal matrices. These problems can be efficiently solved using the Brennan and Schwartz (1977, 1978) algorithm. The accuracy of the componentwise splitting (CS) method is increased by applying the Strang (1968) symmetrization. In Sec. 7.4, we give an outline of the method of lines solution to the problem. Finally, Sec. 7.5 discusses a numerical comparison between the different methods and shows that the method of lines (MOL) is indeed superior, probably because it calculates the price, the delta, the gamma and the free surface all at the same time.