Exact Barrier Option Valuation with Arbitrary Functions for the Volatility

Estevão Rosalino Jr., Allan Jonathan Silva, Jack Baczynski, Dorival Leão


Focus, in the past four decades, has been obtaining closed-form expressions for the no-arbitrage prices and hedges of modified versions of the Europeanoptions, allowing the dynamic of the underlying assets to have non-constant pa-rameters.In this paper, we obtain a closed-form expression for the price and hedge of an up-and-out European barrier option, assuming that the volatility in the dynamicof the risky asset is an arbitrary deterministic function of time. Setting a con-stant volatility, the formulas recover the Black and Scholes results, which suggestsminimum computational effort.We introduce a novel concept of relative standard deviation for measuring the ex-posure of the practitioner to risk (enforced by a strategy). The notion that is found in the literature is different and looses the correct physical interpreta-tion. The measure serves aiding the practitioner to adjust the number of rebalancesduring the option’s lifetime.

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DOI: https://doi.org/10.5540/tema.2015.016.01.0061

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