A volatility-of-volatility expansion of the option prices in the SABR stochastic volatility model

Abstract

We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for $\beta = 1$, namely $\partial_{\tau}u = \sigma^2 \big [ \frac{1}{2} (\partial^2_xu - \partial_xu) + \nu \rho \partial_x\partial_\sigma u + \frac{1}{2} \nu^2 \partial^2_\sigma u \, \big ] + \kappa (\theta - \sigma) \partial_\sigma$, by choosing $\nu$ as small parameter. This yields $u = u_0 + \nu u_1 + \nu^2 u_2 + \ldots$, with $u_j$ independent of $\nu$. The terms $u_j$ are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of $u$ that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" $\tau$ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of $\nu$, similar to Hagan's formula, but including also the {\em mean reverting term.} We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility.