### Convergence in Incomplete Market Models

**P. Ekkehard Kopp**

*(University of Hull)*

**Volker Wellmann**

*(BNP Paribas)*

#### Abstract

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.

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Pages: 1-26

Publication Date: May 26, 2000

DOI: 10.1214/EJP.v5-71

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