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From arbitrage to arbitrage-free implied volatilities

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  • Cornelis W. Oosterlee
  • Lech A. Grzelak

Abstract

ABSTRACT We propose a method for determining an arbitrage-free density implied by the Hagan;formula. (We use the wording "Hagan formula" as an abbreviation of the Hagan-Kumar-Le´sniewski-Woodward model.) Our method is based on the stochastic collocation;method. The principle is to determine a few collocation points on the implied;survival distribution function and project them onto the polynomial of an arbitrage free;variable for which we choose the Gaussian variable. In this way, we have equality;in probability at the collocation points while the generated density is arbitrage free.;Analytic European option prices are available, and the implied volatilities stay very;close to those initially obtained by the Hagan formula. The proposed method is very;fast and straightforward to implement, as it only involves one-dimensional Lagrange;interpolation and the inversion of a linear system of equations. The method is generic;and may be applied to other variants or other models that generate arbitrage.

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Handle: RePEc:rsk:journ0:2463775
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