Journal of Computational Finance

No-arbitrage SABR

Paul Doust


The Hagan et al SABR implied volatility approximation formula can result in a negative probability density function when applied to long-dated options. This paper presents a methodology for avoiding that problem by writing the density function as the sum of two components that are always positive. One component corresponds to the absorbing boundary at F = 0, which can be a significant part of the density function for long-dated options. An upper bound on the time to expiry for the Hagan et al formula is also derived, which shows that their formula cannot be expected to work as this upper bound is approached.

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