Convergence in Incomplete Market Models
P. Ekkehard Kopp, University of Hull
Volker Wellmann, BNP Paribas
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
D2-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.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.