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

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, 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.