Alternating Direction Implicit methods for solutions of the Heston stochastic volatility model

This is a numerical study of ADI schemes applied to problems in finance. The work carried out centers on the Heston stochastic volatility model, which gained its popularity from the existence of closed form solutions. A detailed outline of the analytical procedure for applying the Fourier transform method is presented. ADI schemes Douglas-Rachford, Craig-Sneyd (CS), modified CS and Hundsdorfer-Verwer are studied in detail and the numerics are implemented in Matlab. The option sensitivities (Greeks) are also introduced in theory and some are briefly studied via their closed form solution and finite difference method.