Arbitrage‐Free SABR. Issue 69 (January 2014) - Hagan, Patrick S.; Kumar, Deep; Lesniewski, Andrew; Woodward, Diana
Abstract Smile risk is often managed using the explicit implied volatility formulas developed for the SABR model [1]. These asymptotic formulas are not exact, and this can lead to arbitrage for low strike options. Here we provide an alternate method for pricing options under the SABR model: We use asymptotic techniques to reduce the SABR model from two dimensions to one dimension. This leads to an effective one‐dimensional forward equation for the probability density which has the same asymptotic order of accuracy as the explicit implied volatility formulas. We obtain arbitrage‐free option prices by numerically solving this PDE. The implied volatilities obtained from the numerical solutions closely match the explicit implied volatility curves, apart from a boundary layer at very low rates. For very low‐rate environments, or for very low strikes, the implied absolute (normal) volatility dips downward, closely matching market observations. We also show how negative rates can be accommodated by replacing the Fβ factor with (F + a)β.