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Pricing by hedging and no-arbitrage beyond semimartingales. (English) Zbl 1199.91170
The authors construct a class of strategies for possibly non-semimartingale models that have the same quadratic variation as the Black-Scholes model or, more generally, as Brownian models with local volatility structures. It is shown that the aforementioned class of allowed strategies is free of arbitrage for a large class of non-semimartingale models, in particular, for the mixed fractional Brownian models. The non-arbitrage result demonstrates that some non-smooth functional behavior is required to construct arbitrage via distributional properties, which is not inherent in hedges of many interesting options. It is also shown that the no-arbitrage result still holds if a portfolio is changed abruptly at stopping times from a reasonably large class. It is emphasized that the option prices essentially depend only on the quadratic variation which can be viewed as a path property. Therefore option prices are robust with respect to probabilistic properties.

MSC:
91G10 Portfolio theory
91G20 Derivative securities (option pricing, hedging, etc.)
60G15 Gaussian processes
60G48 Generalizations of martingales
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