Tommi Sottinen, University of Helsinki,
tommi.sottinen@helsinki.fi
Esko Valkeila, University of Turku and University of Helsinki,
esko.valkeila@helsinki.fi
Abstract: In the classical Black & Scholes pricing model the randomness of the log-returns of financial indices is modeled by Brownian motion. However, in this model the log-returns are independent and several authors have proposed to replace the Brownian motion by a fractional Brownian motion, which allows to model the dependency with one additional parameter, so-called Hurst index H.
We review some recent work of the authors and others, where it is shown that this leads to arbitrage in the fractional pricing model, even in some natural binomial tree like approximation of this model. On the other hand, it is possible to find a pricing measure, which allows to compute the analog of the fair price for European options in the fractional model. It is possible to modify the stochastic integration theory for fractional Brownian motion in such a way that the above mentioned fair price is also a hedging price.
We give also some results on models, where the randomness comes from a mixture of ordinary Brownian motion and a fractional one.