Convergence in Incomplete Market Models

Abstract

The problem of pricing and hedging of contingent claims in incomplete markets has led to the development of various valuation methodologies. This paper examines the mean-variance approach to risk-minimisation and shows that it is 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: It is shown that the convergence of the underlying models implies the convergence of strategies and value processes for multinomial models and approximations of the Black-Scholes model by direct discretisation 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 standard convergence results for these models. For ease of reference a summary of the main results from nonstandard analysis in the context of stochastic analysis is given as well as a brief introduction to mean-variance hedging and pricing.