Exact Barrier Option Valuation with Deterministic Volatility^†

Focus, in the past four decades, has been obtaining closed-form expressions for the noarbitrage prices and hedges of modified versions of the European options, allowing the dynamic of the underlying assets to have non-constant parameters. 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 dynamic of the risky asset is an arbitrary deterministic function of time. Setting a constant 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 exposure of the practitioner to risk (enforced by a strategy). The notion that is found in the literature is different and looses the correct physical interpretation. The measure serves aiding the practitioner to adjust the number of rebalances during the option's lifetime.

barrier option; no-arbitrage pricing; hedging; Martingale measure; time-change for Martingales