Convergence in incomplete market models
The problem of pricing and hedging of contingent claims in incomplete markets has lead to the development of various valuation methodologies. This thesis examines the mean-variance and variance-optimal approaches to risk-minimisation and shows that these are robust under the convergence from discrete- to continuous-time market models. This property yields new convergence results for option prices, trading strategies and value processes in incomplete market models.Techniques from nonstandard analysis are used to develop new results for the lifting property of the minimal martingale density and risk-minimising strategies. These are applied to a number of incomplete market models:The restriction of hedging dates in a general class of discrete- and continuous-time models is studied and it is shown that the convergence of the underlying models implies the convergence of strategies and value processes.Similar results are obtained for multinomial models and approximations of the Black-Scholes model by direct observation of the price process. The concept of D 2-convergence is extended to these classes of models, including the construction of discretisation schemes. This yields new convergence results for these models as well as for option prices in a jump-diffusion model.The computational aspects of these approximations are examined and numerical results are provided in the case of European and Asian options.For ease of reference a summary of the main results from nonstandard analysis in the context of mathematical finance is given as well as a brief introduction to mean-variance hedging and variance-optimal pricing.