Convergence in Incomplete Market Models

The problem of pricing and hedging of contingent claims inincomplete markets has led to the development of various valuation methodologies. Thispaper examines the mean-variance approach to risk-minimisation and shows that it is robustunder the convergence from discrete- to continuous-time market models. This propertyyields new convergence results for option prices, trading strategies and value processesin incomplete market models. Techniques from nonstandard analysis are used to develop new results for the liftingproperty of the minimal martingale density and risk-minimising strategies. These areapplied to a number of incomplete market models: It is shown that the convergence of the underlying models implies the convergence ofstrategies and value processes for multinomial models and approximations of theBlack-Scholes model by direct discretisation of the price process. The concept of D²-convergenceis extended to these classes of models, including the construction of discretisationschemes. This yields new standard convergence results for these models. For ease of reference a summary of the main results from nonstandard analysis in thecontext of stochastic analysis is given as well as a brief introduction to mean-variancehedging and pricing.