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Relative Pricing of Options with Stochastic Volatility

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  • Ledoit, Olivier
  • Santa-Clara, Pedro
  • Yan, Shu

Abstract

This paper offers a new approach for pricing options on assets with stochastic volatility. We start by taking as given the prices of a few simple, liquid European options. More specifically, we take as given the “surface” of Black-Scholes implied volatilities for European options with varying strike prices and maturities. We show that the Black-Scholes implied volatilities of at-the-money options converge to the underlying asset’s instantaneous (stochastic) volatility as the time to maturity goes to zero. We model the stochastic processes followed by the implied volatilities of options of all maturities and strike prices as a joint diffusion with the stock price. In order for no arbitrage opportunities to exist in trading the stock and these options, the drift of the processes followed by the implied volatilities is constrained in such a way that it is fully characterized by the volatilities of the implied volatilities. Finally, we suggest how to use the arbitrage-free joint process for the stock price and its volatility to price other derivatives, such as standard but illiquid options as well as exotic options, using numerical methods. Our approach simply requires as inputs the stock price and the implied volatilities at the time the exotic option is to be priced, as well as estimates of the volatilities of the implied volatilities.

Suggested Citation

  • Ledoit, Olivier & Santa-Clara, Pedro & Yan, Shu, 2002. "Relative Pricing of Options with Stochastic Volatility," University of California at Los Angeles, Anderson Graduate School of Management qt7jp8f42t, Anderson Graduate School of Management, UCLA.
    • Handle: RePEc:cdl:anderf:qt7jp8f42t
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    Cited by:

    1. Aït-Sahalia, Yacine & Li, Chenxu & Li, Chen Xu, 2021. "Closed-form implied volatility surfaces for stochastic volatility models with jumps," Journal of Econometrics, Elsevier, vol. 222(1), pages 364-392.
    2. Bas Peeters, 2012. "Risk premiums in a simple market model for implied volatility," Quantitative Finance, Taylor & Francis Journals, vol. 13(5), pages 739-748, January.
    3. Itkin, Andrey, 2015. "To sigmoid-based functional description of the volatility smile," The North American Journal of Economics and Finance, Elsevier, vol. 31(C), pages 264-291.
    4. Ang, Andrew & Hodrick, Robert J. & Xing, Yuhang & Zhang, Xiaoyan, 2009. "High idiosyncratic volatility and low returns: International and further U.S. evidence," Journal of Financial Economics, Elsevier, vol. 91(1), pages 1-23, January.
    5. Sebastian Herrmann & Johannes Muhle-Karbe, 2017. "Model Uncertainty, Recalibration, and the Emergence of Delta-Vega Hedging," Papers 1704.04524, arXiv.org.
    6. Bak, Yuhyeon & Park, Cheolbeom, 2022. "Exchange rate predictability, risk premiums, and predictive system," Economic Modelling, Elsevier, vol. 116(C).
    7. Tim Bollerslev & Hao Zhou, 2003. "Volatility puzzles: a unified framework for gauging return-volatility regressions," Finance and Economics Discussion Series 2003-40, Board of Governors of the Federal Reserve System (U.S.).
    8. Carol Alexander & Leonardo M. Nogueira, 2004. "Hedging with Stochastic and Local Volatility," ICMA Centre Discussion Papers in Finance icma-dp2004-10, Henley Business School, University of Reading, revised Dec 2004.
    9. Aït-Sahalia, Yacine & Amengual, Dante & Manresa, Elena, 2015. "Market-based estimation of stochastic volatility models," Journal of Econometrics, Elsevier, vol. 187(2), pages 418-435.
    10. Yu, Jialin, 2007. "Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the Chinese Yuan," Journal of Econometrics, Elsevier, vol. 141(2), pages 1245-1280, December.
    11. Ai[diaeresis]t-Sahalia, Yacine & Kimmel, Robert, 2007. "Maximum likelihood estimation of stochastic volatility models," Journal of Financial Economics, Elsevier, vol. 83(2), pages 413-452, February.
    12. Alexey MEDVEDEV & Olivier SCAILLET, 2004. "A Simple Calibration Procedure of Stochastic Volatility Models with Jumps by Short Term Asymptotics," FAME Research Paper Series rp93, International Center for Financial Asset Management and Engineering.
    13. Moriggia, V. & Muzzioli, S. & Torricelli, C., 2009. "On the no-arbitrage condition in option implied trees," European Journal of Operational Research, Elsevier, vol. 193(1), pages 212-221, February.
    14. Carol Alexander & Leonardo Nogueira, 2004. "Stochastic Local Volatility," ICMA Centre Discussion Papers in Finance icma-dp2008-02, Henley Business School, University of Reading, revised Mar 2008.
    15. Carey, Alexander, 2008. "Natural volatility and option pricing," MPRA Paper 6709, University Library of Munich, Germany.
    16. Yacine Ait-Sahalia & Robert Kimmel, 2004. "Maximum Likelihood Estimation of Stochastic Volatility Models," NBER Working Papers 10579, National Bureau of Economic Research, Inc.
    17. Sebastian Herrmann & Johannes Muhle-Karbe, 2017. "Model uncertainty, recalibration, and the emergence of delta–vega hedging," Finance and Stochastics, Springer, vol. 21(4), pages 873-930, October.
    18. Martin Schweizer & Johannes Wissel, 2008. "Term Structures Of Implied Volatilities: Absence Of Arbitrage And Existence Results," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 77-114, January.
    19. Martin Schweizer & Johannes Wissel, 2008. "Arbitrage-free market models for option prices: the multi-strike case," Finance and Stochastics, Springer, vol. 12(4), pages 469-505, October.

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