Portfolio Optimisation, Transaction Costs, Shadow Prices and Fractional Brownian Motion

Résumé

While absence of arbitrage in frictionless financial markets (i.e. without transaction costs) requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. 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. The talk is based on joint work with Walter Schachermayer.