Fractional Diffusions and portfolio optimisation in finance

The project analyses the deviations from the classical Black-Scholes theory when the differential operators involved in the Black-Scholes PDE are replaced by their fractional versions. From a practical point of view it is highly desirable to develop a good understanding of these effects as it is common knowledge that the Black- Scholes approach suffers from several severe over-simplifications, e.g. dramatic under-estimations of the probabilities of extreme events. There is a fruitful interplay between the PDE approach and the probabilistic approach. In the present context, the passage from the Black-Scholes PDE (which simply is the multiplicative version of the heat equation) to the fractional equations corresponds to the passage from (geometric) Brownian motion to fractional Brownian motion and to alpha-stable Lévy processes. These variations are not of a superficial nature, but involve a rather drastic change of the basic paradigms, as the passage to fractional Brownian motion makes it necessary to leave the well-established framework of semi- martingale theory. The technical difficulties are highly challenging. In order to eventually arrive at tractable and practically relevant results we propose an asymptotic analysis of the above deviations of the Black-Scholes model.

The classical framework of mathematical finance is given by frictionless financial markets. Under No-Arbitrage hypotheses the price process should be a semi-martingale. This rules out some important models, e.g., models based on fractional Brownian motion, which is proposed by Mandelbrot and strongly supported by empirical findings. However, the introduction of transaction costs makes the arbitrage possibilities of fractional models disappear. This allows using these models as price processes for portfolio optimisation under transaction costs. We show that, for a market model based on the fractional Brownian motion, there exists a so-called shadow price, i.e., a least favorable frictionless market extension which lies within the bid-ask spread of the original market with transaction costs, such that trading in this fictitious market leads to the same maximal expected utility and optimal strategy. The behavior of an economic agent in the market with transactions costs can be explained by passing to a suitable frictionless shadow market. This result establishes a bridge between the world of frictionless semi-martingale and the world of fractional models under transaction costs.