Implied Volatilities for Mean Reverting SABR Models. Issue 108 (30th July 2020) - Hagan, Patrick S.; Lesniewski, Andrew S.; Woodeard, Diana E.
Abstract : Here we analyze the mean‐reverting SABR model: We use asympotic methods to derive an effective forward equation for the marginal density This effective forward equation is not exact, but is accurate through O(ϵ 2 ), the same accuracy as the SABR implied volatility formulas. Since the effective forward equation has one spatial dimension (F) instead of two (F, A), it can be solved efficiently numerically to obtain the density Q(T ex, F).European optionpricescan then be obtained by integrating to find the expected value of the option's payoff. We then use an effective media analysis to show that for any expiry T ex, we can obtain an effective forward equation with time‐independent coefficients that yields the same density Q(T ex, F) to within O(ϵ 2 ). This yields effective SABR parameters α eff, β eff, ν eff, and shows that to within O(ϵ 2 ), the implied volatilities of European options under the λ‐SABR model are given by the implied volatility formulas for the standard SABR model using these effective SABR parameters.