A Lognormal Type Stochastic Volatility Model With Quadratic Drift

26 Pages Posted: 18 Jul 2019

See all articles by Peter Carr

Peter Carr

New York University (NYU) - Finance and Risk Engineering Department

Sander Willems

Ecole Polytechnique Fédérale de Lausanne; Swiss Finance Institute

Date Written: July 17, 2019

Abstract

This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the instantaneous volatility level. The steady-state distribution of the instantaneous volatility belongs to the class of Generalized Inverse Gaussian distributions. We show that the quadratic term in the drift is crucial to avoid moment explosions and to preserve the martingale property of the stock price process. Using a conveniently chosen change of measure, we relate the model to the class of polynomial diffusions. This remarkable relation allows us to develop a highly accurate option price approximation technique based on orthogonal polynomial expansions.

Suggested Citation

Carr, Peter and Willems, Sander, A Lognormal Type Stochastic Volatility Model With Quadratic Drift (July 17, 2019). Available at SSRN: https://ssrn.com/abstract=3421304 or http://dx.doi.org/10.2139/ssrn.3421304

Peter Carr

New York University (NYU) - Finance and Risk Engineering Department ( email )

6 Metrotech Center
New York, NY 11201
United States

Sander Willems (Contact Author)

Ecole Polytechnique Fédérale de Lausanne ( email )

Station 5
Odyssea 1.04
1015 Lausanne, CH-1015
Switzerland

Swiss Finance Institute ( email )

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

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