Quadratic-Drift Stochastic Volatility for Stable Option Pricing

Summary

This paper introduces a one-factor stochastic volatility model in which instantaneous volatility follows a diffusion with quadratic drift and linear dispersion. Volatility mean-reverts toward a constant level, while the speed of mean reversion varies affinely with the volatility level. The model’s steady-state volatility distribution is in the Generalized Inverse Gaussian family.

The authors argue that the quadratic drift helps prevent moment explosions and preserve the stock price’s martingale property. They also use a change of measure to connect the model to polynomial diffusions. That connection supports an option-pricing approximation based on orthogonal polynomial expansions, which the paper describes as highly accurate. The provided description offers theoretical properties and a pricing technique, but gives no numerical benchmarks, calibration results, or comparison with other pricing methods, so practical performance cannot be assessed from this text alone.

Key ideas

• Instantaneous volatility is modeled as a diffusion with quadratic drift and linear dispersion.
• The mean-reversion speed changes affinely with the volatility level.
• The stationary volatility distribution belongs to the Generalized Inverse Gaussian family.
• Quadratic drift is presented as a way to avoid moment explosions and maintain the stock price martingale property.
• A change of measure links the model to polynomial diffusions and enables orthogonal-polynomial option-price approximation.

Tags

# A lognormal type stochastic volatility model with quadratic drift


# A lognormal type stochastic volatility model with quadratic drift









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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