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Switching to non-affine stochastic volatility: A closed-form expansion for the Inverse Gamma model

Author

Listed:
• Nicolas Langren'e
• Geoffrey Lee
• Zili Zhu

Abstract

This paper introduces the Inverse Gamma (IGa) stochastic volatility model with time-dependent parameters, defined by the volatility dynamics $dV_{t}=\kappa_{t}\left(\theta_{t}-V_{t}\right)dt+\lambda_{t}V_{t}dB_{t}$. This non-affine model is much more realistic than classical affine models like the Heston stochastic volatility model, even though both are as parsimonious (only four stochastic parameters). Indeed, it provides more realistic volatility distribution and volatility paths, which translate in practice into more robust calibration and better hedging accuracy, explaining its popularity among practitioners. In order to price vanilla options with IGa volatility, we propose a closed-form volatility-of-volatility expansion. Specifically, the price of a European put option with IGa volatility is approximated by a Black-Scholes price plus a weighted combination of Black-Scholes greeks, where the weights depend only on the four time-dependent parameters of the model. This closed-form pricing method allows for very fast pricing and calibration to market data. The overall quality of the approximation is very good, as shown by several calibration tests on real-world market data where expansion prices are compared favorably with Monte Carlo simulation results. This paper shows that the IGa model is as simple, more realistic, easier to implement and faster to calibrate than classical transform-based affine models. We therefore hope that the present work will foster further research on non-affine models like the Inverse Gamma stochastic volatility model, all the more so as this robust model is of great interest to the industry.

Suggested Citation

• Nicolas Langren'e & Geoffrey Lee & Zili Zhu, 2015. "Switching to non-affine stochastic volatility: A closed-form expansion for the Inverse Gamma model," Papers 1507.02847, arXiv.org, revised Mar 2016.
• Handle: RePEc:arx:papers:1507.02847
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File URL: http://arxiv.org/pdf/1507.02847

References listed on IDEAS

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1. Kenichiro Shiraya & Akihiko Takahashi, 2014. "Pricing Multiasset Cross‐Currency Options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 34(1), pages 1-19, January.
2. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
3. Fornari, Fabio & Mele, Antonio, 2006. "Approximating volatility diffusions with CEV-ARCH models," Journal of Economic Dynamics and Control, Elsevier, vol. 30(6), pages 931-966, June.
4. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
5. Fornari, Fabio & Mele, Antonio, 2001. "Recovering the probability density function of asset prices using garch as diffusion approximations," Journal of Empirical Finance, Elsevier, vol. 8(1), pages 83-110, March.
6. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
7. Andrey Itkin, 2013. "New solvable stochastic volatility models for pricing volatility derivatives," Review of Derivatives Research, Springer, vol. 16(2), pages 111-134, July.
8. Peter Christoffersen & Kris Jacobs & Karim Mimouni, 2010. "Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns, and Option Prices," Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 3141-3189, August.
9. Kenichiro Shiraya & Akihiko Takahashi, 2011. "Pricing average options on commodities," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 31(5), pages 407-439, May.
10. Barone-Adesi, Giovanni & Rasmussen, Henrik & Ravanelli, Claudia, 2005. "An option pricing formula for the GARCH diffusion model," Computational Statistics & Data Analysis, Elsevier, vol. 49(2), pages 287-310, April.
11. Kaeck, Andreas & Alexander, Carol, 2012. "Volatility dynamics for the S&P 500: Further evidence from non-affine, multi-factor jump diffusions," Journal of Banking & Finance, Elsevier, vol. 36(11), pages 3110-3121.
12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
13. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
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Citations

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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. 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.
3. Yuri F. Saporito & Xu Yang & Jorge P. Zubelli, 2017. "The Calibration of Stochastic-Local Volatility Models - An Inverse Problem Perspective," Papers 1711.03023, arXiv.org.
4. Kaustav Das & Nicolas Langren'e, 2018. "Closed-form expansions with respect to the mixing solution for option pricing under stochastic volatility," Papers 1812.07803, arXiv.org, revised Aug 2019.

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