A VOLATILITY-OF-VOLATILITY EXPANSION OF THE OPTION PRICES IN THE SABR STOCHASTIC VOLATILITY MODEL

Olesya Grishchenko (), Xiao Han () and Victor Nistor
Additional contact information
Olesya Grishchenko: Division of Monetary Affairs, Federal Reserve Board, Washington, DC 20551, USA
Xiao Han: Barclays Capital, Quantitative Analytics, New York City, USA
Victor Nistor: Université de Lorraine, UFR, MIM, 57000 Metz, France

International Journal of Theoretical and Applied Finance (IJTAF), 2020, vol. 23, issue 03, 1-49

Abstract: We propose a new type of asymptotic expansion for the transition probability density function (or heat kernel) of certain parabolic partial differential equations (PDEs) that appear in option pricing. As other, related methods developed by Costanzino, Hagan, Gatheral, Lesniewski, Pascucci, and their collaborators, among others, our method is based on the computation of the truncated asymptotic expansion of the heat kernel with respect to a “small” parameter. What sets our method apart is that our small parameter is possibly different from the time to expiry and that the resulting commutator calculations go beyond the nilpotent Lie algebra case. In favorable situations, the terms of this asymptotic expansion can quickly be computed explicitly leading to a “closed-form” approximation of the solution, and hence of the option price. Our approximations tend to have much fewer terms than the ones obtained from short time asymptotics, and are thus easier to generalize. Another advantage is that the first term of our expansion corresponds to the classical Black-Scholes model. Our method also provides equally fast approximations of the derivatives of the solution, which is usually a challenge. A full theoretical justification of our method seems very difficult at this time, but we do provide some justification based on the results of (Siyan, Mazzucato, and Nistor, NWEJ 2018). We therefore mostly content ourselves to demonstrate numerically the efficiency of our method by applying it to the solution of the mean-reverting SABR stochastic volatility model PDE, commonly referred to as the λSABR PDE, by taking the volatility of the volatility parameter ν (vol-of-vol) as a small parameter. For this PDE, we provide extensive numerical tests to gauge the performance of our method. In particular, we compare our approximation to the one obtained using Hagan’s formula and to the one obtained using a new, adaptive finite difference method. We provide an explicit asymptotic expansion for the implied volatility (generalizing Hagan’s formula), which is what is typically needed in concrete applications. We also calibrate our model to observed market option price data. The resulting values for the parameters σ, ρ, and ν are realistic, which provides more evidence for the conjecture that the volatility is mean-reverting.

Keywords: SABR PDE; implied volatility; asymptotic expansion (search for similar items in EconPapers)
Date: 2020
References: Add references at CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.worldscientific.com/doi/abs/10.1142/S0219024920500181
Access to full text is restricted to subscribers

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:ijtafx:v:23:y:2020:i:03:n:s0219024920500181

Ordering information: This journal article can be ordered from

DOI: 10.1142/S0219024920500181

Access Statistics for this article

International Journal of Theoretical and Applied Finance (IJTAF) is currently edited by L P Hughston

More articles in International Journal of Theoretical and Applied Finance (IJTAF) from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().