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Arbitrage with fractional Brownian motion? (English) Zbl 1152.91028

Absence of arbitrage, i.e. the impossibility of receiving a riskless gain by trading into a market, is a basic equilibrium condition at the financial mathematics. The problem discussed in this paper can be posed as: Is it possible to construct a class of economically meaningful strategies, that does not contain arbitrage and is at the same time sufficiently rich to be interesting from the perspective of pricing?
The authors discuss several results on existence or absence of arbitrage in models driven by a fractional Brownian motion for different classes of strategies. It is shown that in the purely fractional Black-Scholes models there is no subclass of self-financing strategies known that is arbitrage-free and sufficiently large to hedge relevant options. The purely fractional Black-Scholes models become arbitrage-free with Wick-self-financing strategies. But the notion of a Wick self-financing portfolios seems to be void of a sound economic interpretation, if it is to be interpreted in a real world sense. The authors show, that if one adds a Brownian component and considers mixed models, the class of regular portfolios is arbitrage free and includes hedges for many practically relevant options.

MSC:

91B28 Finance etc. (MSC2000)
60G15 Gaussian processes
60G18 Self-similar stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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