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Hedging, arbitrage and optimality with superlinear frictions. (English) Zbl 1403.91311

The authors study the phenomenon of price impact, or, that is the same, market depth. This phenomena means that trading moves prices against the trader: buying faster increases execution prices, and selling faster decreases them. In the models with price impact and with continuous time, the following questions arise: what is the analogue of a martingale measure, what about optimality conditions for utility maximization, which contingent claims are hedgeable and at what price? The paper is devoted to the answers for these questions. Models with multiple assets are considered and superhedging prices, absence of arbitrage and utility maximizing strategies are characterized under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. The notion of superlinear friction is involved, and superlinear frictions struggle with buying or selling too fast. Such frictions induce a duality between feasible trading strategies and shadow execution prices with a martingale measure. In such framework it is established that utility maximization strategies exist even if arbitrage is present.

MSC:

91G10 Portfolio theory
60G44 Martingales with continuous parameter
91B16 Utility theory
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