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Implied And Local Volatilities Under Stochastic Volatility

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  • ROGER W. LEE

    (Department of Mathematics, Stanford University, Stanford, CA 94305, USA)

Abstract

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some naïve pricing and hedging schemes.We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a symmetric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou's asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slow-variation asymptotics against what we callsmall-variation asymptotics, and against Fouque, Papanicolaou, and Sircar'srapid-variation asymptotics.We apply the slow-variation asymptotics to approximate the biases of two naïve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied-volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies forhedgingunder stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk.

Suggested Citation

  • Roger W. Lee, 2001. "Implied And Local Volatilities Under Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 45-89.
    • Handle: RePEc:wsi:ijtafx:v:04:y:2001:i:01:n:s0219024901000870
    DOI: 10.1142/S0219024901000870
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    Citations

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    Cited by:

    1. Recchioni, Maria Cristina & Iori, Giulia & Tedeschi, Gabriele & Ouellette, Michelle S., 2021. "The complete Gaussian kernel in the multi-factor Heston model: Option pricing and implied volatility applications," European Journal of Operational Research, Elsevier, vol. 293(1), pages 336-360.
    2. Park, Sang-Hyeon & Kim, Jeong-Hoon, 2013. "A semi-analytic pricing formula for lookback options under a general stochastic volatility model," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2537-2543.
    3. Michele Azzone & Roberto Baviera, 2021. "A fast Monte Carlo scheme for additive processes and option pricing," Papers 2112.08291, arXiv.org, revised Jul 2023.
    4. Lan Zhang, 2012. "Implied and realized volatility: empirical model selection," Annals of Finance, Springer, vol. 8(2), pages 259-275, May.
    5. Elisa Al`os & David Garc'ia-Lorite & Makar Pravosud, 2022. "On the skew and curvature of implied and local volatilities," Papers 2205.11185, arXiv.org, revised Sep 2023.
    6. 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.
    7. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    8. Alexander, Carol & Nogueira, Leonardo M., 2007. "Model-free hedge ratios and scale-invariant models," Journal of Banking & Finance, Elsevier, vol. 31(6), pages 1839-1861, June.
    9. Carol Alexander & Alexander Rubinov & Markus Kalepky & Stamatis Leontsinis, 2012. "Regime‐dependent smile‐adjusted delta hedging," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 32(3), pages 203-229, March.
    10. Archil Gulisashvili & Elias M. Stein, 2009. "Implied Volatility In The Hull–White Model," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 303-327, April.
    11. Stephane Crepey, 2004. "Delta-hedging vega risk?," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 559-579.
    12. Vinicius Albani & Adriano De Cezaro & Jorge P. Zubelli, 2017. "Convex Regularization Of Local Volatility Estimation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-37, February.
    13. Dan Pirjol & Lingjiong Zhu, 2020. "Asymptotics of the time-discretized log-normal SABR model: The implied volatility surface," Papers 2001.09850, arXiv.org, revised Mar 2020.

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