Volatility risk structure for options depending on extrema

Tomonori Nakatsu

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We obtain a formula to compute the vega index (sensitivity with respect to changes in volatility) for options depending on extrema.
A one-dimensional perturbed diffusion process is considered in this article.
As a numerical application, we compute the vega index for options under the Black-Scholes model perturbed with a CEV (constant elasticity of variance model) type perturbation.
The values of the vega index obtained with the perturbed diffusion process and with non-perturbed Black-Scholes model significantly differ.

Abstract

In this paper, we give a decomposition formula to calculate the vega index (sensitivity with respect to changes in volatility) for options with prices that depend on the extrema (maximum or minimum) and terminal value of the underlying stock price; this is assumed to follow a one-dimensional perturbed diffusion process. As a numerical application, we compute the vega index for lookback, European and up-in call options under the Black–Scholes model perturbed with a constant elasticity of variance model-type perturbation. We compare these values with the standard nonperturbed Black–Scholes model, which, interestingly, turn out to be very different.

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