# Asymptotic Theory of Transaction Costs

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*Walter Schachermayer*

A publication of the European Mathematical Society

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.

These 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 us 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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and researchers interested in portfolio optimization.