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A lognormal type stochastic volatility model with quadratic drift

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  • Peter Carr
  • Sander Willems

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.

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  • Peter Carr & Sander Willems, 2019. "A lognormal type stochastic volatility model with quadratic drift," Papers 1908.07417, arXiv.org.
    • Handle: RePEc:arx:papers:1908.07417
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    References listed on IDEAS

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    Cited by:

    1. Kaustav Das & Nicolas Langren'e, 2020. "Explicit approximations of option prices via Malliavin calculus in a general stochastic volatility framework," Papers 2006.01542, arXiv.org, revised Jan 2024.
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    3. Vimal Raval & Antoine Jacquier, 2021. "The Log Moment formula for implied volatility," Papers 2101.08145, arXiv.org.

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