Mathematisches Kolloquium: Christoph Czichowsky
Speaker: Christoph Czichowsky, London School of Economics
Title: Portfolio Optimisation, Transaction Costs, Shadow Prices and Fractional Brownian Motion
Abstract: In financial mathematics, one classically works with so-called frictionless markets, where arbitrary amounts of stocks can be bought and sold at each time for the same price. Under this assumption, the absence of arbitrage opportunities (riskless profits) implies that price processes have to be semimartingales, that is, stochastic processes which are "good" integrators. While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingale models based on fractional Brownian motion can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. Such models have been proposed by Benoit Mandelbrot more than fifty years ago because of their natural fractional scaling and related statistical properties. In this talk, I will present an overview over several results that provide a way how to use non-semimartingale price processes such as the fractional Black-Scholes model in portfolio optimisation under proportional transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs.