Stochastic Volatility Models and Kelvin Waves

Journal of Physics A: Mathematical and Theoretical, Vol. 41, 2008

27 Pages Posted: 22 Sep 2012

See all articles by Alex Lipton

Alex Lipton

Hebrew University of Jerusalem; Massachusetts Institute of Technology (MIT)

Artur Sepp

Sygnum Bank's Asset Management

Date Written: January 1, 2008

Abstract

We use stochastic volatility models to describe the evolution of the asset price, its instantaneous volatility, and its realized volatility. In particular, we concentrate on the Stein-Stein model (SSM) (1991) for the stochastic asset volatility and the Heston model (HM) (1993) for the stochastic asset variance. By construction, the volatility is not sign-definite in SSM and is non-negative in HM. It is well-known that both models produce closed-form expressions for the prices of vanilla options via the Lewis-Lipton formula. However, the numerical pricing of exotic options by means of the Finite Difference and Monte Carlo methods is much more complex for HM than for SSM. Until now, this complexity was considered to be an acceptable price to pay for ensuring that the asset volatility is non-negative. We argue that having negative stochastic volatility is a psychological rather than financial or mathematical problem, and advocate using SSM rather than HM in most applications. We extend SSM by adding volatility jumps and obtain a closed-form expression for the density of the asset price and its realized volatility. We also show that the current method of choice for solving pricing problems with stochastic volatility (via the affine ansatz for the Fourier-transformed density function) can be traced back to the Kelvin method designed in the nineteenth century for studying wave motion problems arising in fluid dynamics.

Keywords: Stochastic volatility, Stein-Stein model, Heston model

JEL Classification: C00

Suggested Citation

Lipton, Alex and Sepp, Artur, Stochastic Volatility Models and Kelvin Waves (January 1, 2008). Journal of Physics A: Mathematical and Theoretical, Vol. 41, 2008, Available at SSRN: https://ssrn.com/abstract=2150644

Alex Lipton

Hebrew University of Jerusalem ( email )

Mount Scopus
Jerusalem, Jerusalem 91905
Israel

Massachusetts Institute of Technology (MIT) ( email )

77 Massachusetts Avenue
50 Memorial Drive
Cambridge, MA 02139-4307
United States

Artur Sepp (Contact Author)

Sygnum Bank's Asset Management ( email )

Uetlibergstrasse 134a
Zurich, 8045
Switzerland

HOME PAGE: http://artursepp.com

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