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.

近四十年来，学界的研究焦点集中于求解欧式期权修正版本的无套利定价与对冲策略的闭式解，且允许标的资产的动态演化过程具备非恒定参数。本文针对向上敲出欧式障碍期权（up-and-out European barrier option），在风险资产动态波动率为任意确定性时间函数的假设下，推导得到其定价与对冲策略的闭式解。当波动率设定为恒定值时，本研究得到的公式可退化为布莱克-斯科尔斯（Black and Scholes）模型的经典结果，这表明该方法的计算开销极低。本文提出一种新颖的相对标准差概念，用于衡量实务从业者因执行某一策略所面临的风险敞口。现有文献中已提出的同类概念则与之不同，且无法保留准确的经济内涵。该测度可帮助实务从业者在期权存续期内调整对冲再平衡的次数。