High-order compact finite difference schemes for option pricing in stochastic volatility jump-diffusion models

This thesis focusses on the derivation and implementation of high-order compact finite difference schemes to price a variety of options under various stochastic volatility and jump models, with the inclusion of further studies relating to the derivatives of options and the practice of hedging. First, we derive a implicit-explicit high-order compact finite difference scheme for pricing European options under the Bates model. The resulting scheme is fourth order accurate in space and second order accurate in time. In the numerical study this scheme is compared to both a second order finite difference scheme and high-order finite element methods, where it outperforms both in terms of convergence, computational speed and required memory allocation. A numerical stability study is conducted which indicates unconditional stability of the scheme. Second, we introduce the practice of hedging and give examples of hedging strategies created from a combination of option payoffs, we show the important role the derivatives of the option price play in forming profitable strategies. We go on to complete a study of the convergence of derivatives of the option price, the so-called Greeks. We conduct studies into Delta, vega and gamma hedging, where the derived high-order compact scheme outperforms a second-order finite difference method. Examples are provided to display how this increase in computational efficiency may assist financial practictoners. Third, we extend the high-order compact scheme to price European options under the stochastic volatility with comtemporaneous jumps model. The derived scheme is fourth order accurate in space and second order accurate in time. We conduct numerical studies to test the new high-order compact schemes convergence, computational speed and required memory allocation against a second-order finite difference scheme, where the results show improvements in convergence at the expense of computational time. Further studies of numerical stability indicate unconditional stability of the high-order compact scheme.