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 β = 1, namely ∂τ u = σ 2 1 2 (∂ 2The 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" τ 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 ν, similar to Hagan's formula, but including also the 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. Contents32 5.2. The SABR model (with general β) 34 References 35

Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility Model

“…We close by observing that similar commutator estimates were obtained in [9,10,12,22]. The main difficulty addressed in this work is that L 0 is not an elliptic operator.…”

Approximate Solutions to Second-Order Parabolic Equations: evolution systems and discretization

“…The results of this paper are based in great part and extend some results in [9] and an unpublished 2011 IMA preprint [11]. See [20,33,45] for some recent, related results to that preprint. However, Section 3 is essentially new.…”

“…Here the list of papers that one could quote is truly Gargantuan, but let nevertheless mention the papers [6,7,13,14,17,22,25,26] among earlier papers most closely related to our work [12,10,9,11] (in chronological order), in which we have developed the Dyson-Taylor commutator method used in this paper. Let us mention also the more recent papers [16,18,20,21,23,24,33,45], where the reader will be able to find further references. Some general related monographs include [15,27,30].…”

Asymptotics of the time-discretized log-normal SABR model: The implied volatility surface

“…For a survey of existing results see [1]. A mean-reverting version of the model called λ-SABR was introduced in [24]; the asymptotics of options prices was recently studied in [17]. The simulation and pricing under the SABR model have been studied extensively.…”