Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 2, Pages 113-120

Robust option replication for a Black-Scholes model extended with nondeterministic trends

John G. M. Schoenmakers1 and Peter E. Kloeden2

1Delft University of Technology, Department of Applied Analysis, Mekelweg 4, Delft 2628 CD, The Netherlands
2Johan Wolfgang Goethe Universitit, Fachbereich Mathematik, Frankfurt D-60054, Germany

Received 1 December 1997; Revised 1 August 1998

Copyright © 1999 John G. M. Schoenmakers and Peter E. Kloeden. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Statistical analysis on various stocks reveals long range dependence behavior of the stock prices that is not consistent with the classical Black and Scholes model. This memory or nondeterministic trend behavior is often seen as a reflection of market sentiments and causes that the historical volatility estimator becomes unreliable in practice. We propose an extension of the Black and Scholes model by adding a term to the original Wiener term involving a smoother process which accounts for these effects. The problem of arbitrage will be discussed. Using a generalized stochastic integration theory [8], we show that it is possible to construct a self financing replicating portfolio for a European option without any further knowledge of the extension and that, as a consequence, the classical concept of volatility needs to be re-interpreted.