Stochastic models in finance

Stochastic models for pricing financial securities are developed. First, we consider the Black Scholes model, which is a classic example of a complete market model and finally focus on Lévy driven models. Jumps may render the market incomplete and are induced in a model by inclusion of a Poisson process. Lévy driven models are more realistic in modelling of asset price dynamics than the Black Scholes model. Martingales are central in pricing, especially of derivatives and we give them the desired attention in the context of pricing. There are an increasing number of important pricing models where analytical solutions are not available hence computational methods come in handy, see Broadie and Glasserman (1997). It is also important to note that computational methods are also applicable to models with analytical solutions. We computationally value selected stochastic financial models using C++. Computational methods are also used to value or price complex financial instruments such as path dependent derivatives. This pricing procedure is applied in the computational valuation of a stochastic (revenue based) loan contract. Derivatives with simple pay of functions and models with analytical solutions are considered for illustrative purposes. The Black-Scholes P.D.E is complex to solve analytically and finite difference methods are widely used. Explicit finite difference scheme is considered in this thesis for computational valuation of derivatives that are modelled by the Black-Scholes P.D.E. Stochastic modelling of asset prices is important for the valuation of derivatives: Gaussian, exponential and gamma variates are simulated for the valuation purposes