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Exact Solutions for a GBM-type Stochastic Volatility Model having a Stationary Distribution

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  • Alan L. Lewis

Abstract

We find various exact solutions for a new stochastic volatility (SV) model: the transition probability density, European-style option values, and (when it exists) the martingale defect. This may represent the first example of an SV model combining exact solutions, GBM-type volatility noise, and a stationary volatility density.

Suggested Citation

  • Alan L. Lewis, 2018. "Exact Solutions for a GBM-type Stochastic Volatility Model having a Stationary Distribution," Papers 1809.08635, arXiv.org, revised May 2019.
    • Handle: RePEc:arx:papers:1809.08635
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    File URL: http://arxiv.org/pdf/1809.08635
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    References listed on IDEAS

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    1. Alan L. Lewis, 1998. "Applications of Eigenfunction Expansions in Continuous‐Time Finance," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 349-383, October.
    2. Peter Christoffersen & Kris Jacobs & Karim Mimouni, 2010. "Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns, and Option Prices," The Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 3141-3189, August.
    3. Yiannis A. Papadopoulos & Alan L. Lewis, 2018. "A First Option Calibration of the GARCH Diffusion Model by a PDE Method," Papers 1801.06141, arXiv.org.
    4. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    5. Martin Tegnér & Rolf Poulsen, 2018. "Volatility Is Log-Normal—But Not for the Reason You Think," Risks, MDPI, vol. 6(2), pages 1-16, April.
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    Cited by:

    1. Peter Carr & Sander Willems, 2019. "A lognormal type stochastic volatility model with quadratic drift," Papers 1908.07417, arXiv.org.
    2. 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.
    3. Sigurd Emil Rømer & Rolf Poulsen, 2020. "How Does the Volatility of Volatility Depend on Volatility?," Risks, MDPI, vol. 8(2), pages 1-18, June.
    4. Kaustav Das & Nicolas Langren'e, 2018. "Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility," Papers 1812.07803, arXiv.org, revised Oct 2021.

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