Schachermayer Walter. Asymptotic Theory of Transaction Costs

European Mathematical Society, 2017. — 162 p. — (Zurich Lectures in Advanced Mathematics 23). — ISBN 978-3-03719-173-6.A classical topic in Mathematical Finance is the theory of portfolio optimization. Robert Merton's work from the early seventies had enormous impact on academic research as well as on the paradigms guiding practitioners.
One of the ramifications of this topic is the analysis of (small) proportional transaction costs, such as a Tobin tax. The lecture notes present some striking recent results of the asymptotic dependence of the relevant quantities when transaction costs tend to zero.
An appealing feature of the consideration of transaction costs is that it allows for the first time to reconcile the no arbitrage paradigm with the use of non-semimartingale models, such as fractional Brownian motion. This leads to the culminating theorem of the present lectures which roughly reads as follows: for a fractional Brownian motion stock price model we always find a shadow price process for given transaction costs. This process is a semimartingale and can therefore be dealt with using the usual machinery of mathematical finance.Models on finite probability spaces
Utility maximization under transaction costs: The case of finite
Growth-optimal portfolio in the Black–Scholes model
General duality theory
Local duality theory
Portfolio optimization under transaction costs
Shadow price process
Case study: Fractional Brownian motion
Appendix
Polar sets
Polyhedral sets
Legendre transformation