Market models with transaction costs

Recent research has shown that admitting transaction costs in mathematical models of financial markets not only gives a better approximation of reality but also widens the class of processes which can be used for modelling purposes. It turns out that in the framework of models with transaction costs the by now classical semimartingale theory is not the natural degree of generality any more (as it is in the frictionless theory). For instance, fractional Brownian motion can be taken as the stochastic driving factor as it does not create arbitrage opportunities if transaction costs have to be considered (although it is not a semimartingale and, without friction, allows for arbitrage possibilities). The main objectives of the research outlined in this proposal are the investigation of the following topics in the framwork of models with proportional transaction costs: To build up an appropriate duality theory for markets in continuous time. To prove an equivalence between an appropriate concept of absence of arbitrage and the existence of dual variables ("consistent price systems") in continuous time. To solve the problem of maximising the expected terminal utility of an agent whose preferences are described by a concave increasing utility function. To characterize the minimizer in the dual problem and use it for pricing of contingent claims "by marginal utility" in the present context. To find a description of all these quantities in concrete models (e.g. driven by fractional Brownian motion but also for models driven by classical Brownian motion or Levy processes) and to study its performance as a function of the level of transaction costs. The emphasis will be on the asymptotic behaviour as transaction costs tend to zero.

Transaction costs, e.g. a Tobin Tax, have important implications on financial markets. In the present project we investigate how the behavior of rational (i.e. utility maximizing) investors change when transaction costs occur or are increased/decreased. The focus lies on an asymptotic theory which describes these changes in the spirit of differential calculus when the level of transaction costs is small. Applying convex optimization and the corresponding duality theory, we are able to give very precise expressions of the resulting effects, e.g. on the size of the no trade region or the loss of expected utility.