A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

13 Pages Posted: 3 Jan 2014 Last revised: 6 Jan 2014

See all articles by Olesya V. Grishchenko

Olesya V. Grishchenko

Board of Governors of the Federal Reserve System

Xiao Han

University of Texas at Austin - Red McCombs School of Business

Victor Nistor

Department of Mathematics; Pennsylvania State University

Date Written: January 2, 2014

Abstract

We study the SABR stochastic volatility model with the volatility-of-volatility parameter ν. We provide a method to expand the price CSABR(S, K, ν, σ, τ) of a European call in this model as a Taylor series in ν, CSABR(S, K, ν, σ, τ) = CBS(S,K, σ, τ) νC1 2C2 . . . νkCk O(νk 1), where CBS is the usual Black-Scholes pricing formula. As an illustration of the method, we compute the terms C1 and C2. The formulas for the correction terms C1 and C2 are fully explicit and do not require any integration or any special functions (unlike CBS). We also provide approximate formulas for the implied volatility and for the hedging parameter Δ. Formulas for other "greeks" can be obtained similarly. We provide numerical tests to show the accuracy of our formula for small ν and τ being not too large.

Keywords: SABR model, stochastic volatility, Duhamel-Dyson series, commutator method

JEL Classification: C60, G13

Suggested Citation

Grishchenko, Olesya V. and Han, Xiao and Nistor, Victor, A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model (January 2, 2014). Available at SSRN: https://ssrn.com/abstract=2374004 or http://dx.doi.org/10.2139/ssrn.2374004

Olesya V. Grishchenko

Board of Governors of the Federal Reserve System ( email )

20th Street and Constitution Avenue NW
Washington, DC 20551
United States
202-452-2981 (Phone)

Xiao Han

University of Texas at Austin - Red McCombs School of Business ( email )

Austin, TX 78712
United States

Victor Nistor (Contact Author)

Department of Mathematics ( email )

Ile du Saulcy
57045 Metz, cedex 1
France

HOME PAGE: http://iecl.univ-lorraine.fr/~Victor.Nistor/index.htm

Pennsylvania State University ( email )

University Park
State College, PA 16802
United States

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