Computing Journal : Stochastic-volatility Models with Optional Jumps

In this paper we compare different models describing the implied volatility surface. In particular we analyse in-depth the numerical properties of the Heston model and we extend it to include Jump processes. In our approach, we first identify a closed-form solution or a closest proxy of it, we then price contingent claims testing correctness and efficiency of the numerical solution against the closed-form solution, finally the numerical solution is adapted to American-option pricing.For each model, the closed-form solution is obtained from the stochastic differential equation and the numerical solution is worked out from the integropartial-differential equation. In all cases the scaling behavior and the effective parameter range in practical cases is addressed