Exact Barrier Option Valuation with Arbitrary Functions for the Volatility

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

Abstract


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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References


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

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Trends in Computational and Applied Mathematics

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