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Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern

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  • Li, Minqiang

Abstract

It is known that actual option prices deviate from the Black-Scholes formula using the same volatility for different strikes. For the S&P 500 index options, we find that these deviations follow a stable pattern and are described by a simple function of at-the-money-forward total volatility. This im plies that the term structure of at-the-money-forward volatilities is su±cient to determine the entire volatility surface. We also find that the implied risk-neutral density is bimodal. The patterns we find are useful in predicting future implied volatilities.

Suggested Citation

  • Li, Minqiang, 2008. "Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern," MPRA Paper 11530, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:11530
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    References listed on IDEAS

    as
    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components1," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 39-55, October.
    3. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    4. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    5. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
    6. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    7. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    8. Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. "Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-1632, December.
    9. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    10. Joshua Rosenberg, 1999. "Implied Volatility Functions: A Reprise," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-027, New York University, Leonard N. Stern School of Business-.
    11. repec:bla:jfinan:v:59:y:2004:i:2:p:711-753 is not listed on IDEAS
    12. Bondarenko, Oleg, 2003. "Estimation of risk-neutral densities using positive convolution approximation," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 85-112.
    13. repec:bla:jfinan:v:59:y:2004:i:3:p:1405-1440 is not listed on IDEAS
    14. Peter Carr & Liuren Wu, 2003. "What Type of Process Underlies Options? A Simple Robust Test," Journal of Finance, American Finance Association, vol. 58(6), pages 2581-2610, December.
    15. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    16. Hentschel, Ludger, 2003. "Errors in Implied Volatility Estimation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 38(4), pages 779-810, December.
    17. Nicolae Garleanu & Lasse Heje Pedersen & Allen M. Poteshman, 2009. "Demand-Based Option Pricing," The Review of Financial Studies, Society for Financial Studies, vol. 22(10), pages 4259-4299, October.
    18. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    19. Gurdip Bakshi & Nikunj Kapadia & Dilip Madan, 2003. "Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options," The Review of Financial Studies, Society for Financial Studies, vol. 16(1), pages 101-143.
    20. Newey, Whitney & West, Kenneth, 2014. "A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 33(1), pages 125-132.
    21. Melick, William R. & Thomas, Charles P., 1997. "Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 32(1), pages 91-115, March.
    22. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    23. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    24. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    25. S. R. Hurst & Eckhard Platen & S. T. Rachev, 1999. "Option pricing for a logstable asset price model," Published Paper Series 1999-2, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    26. Charles J. Corrado & Tie Su, 1996. "Skewness And Kurtosis In S&P 500 Index Returns Implied By Option Prices," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 19(2), pages 175-192, June.
    27. Dennis, Patrick & Mayhew, Stewart, 2002. "Risk-Neutral Skewness: Evidence from Stock Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 37(3), pages 471-493, September.
    28. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    29. Melino, Angelo & Turnbull, Stuart M., 1995. "Misspecification and the pricing and hedging of long-term foreign currency options," Journal of International Money and Finance, Elsevier, vol. 14(3), pages 373-393, June.
    30. Jackwerth, Jens Carsten, 2000. "Recovering Risk Aversion from Option Prices and Realized Returns," The Review of Financial Studies, Society for Financial Studies, vol. 13(2), pages 433-451.
    31. David Backus & Silverio Foresi & Liuren Wu, 2002. "Accouting for Biases in Black-Scholes," Finance 0207008, University Library of Munich, Germany.
    32. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    33. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    34. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    35. Whaley, Robert E, 1986. "Valuation of American Futures Options: Theory and Empirical Tests," Journal of Finance, American Finance Association, vol. 41(1), pages 127-150, March.
    36. Brandt, Michael W. & Wu, Tao, 2002. "Cross-sectional tests of deterministic volatility functions," Journal of Empirical Finance, Elsevier, vol. 9(5), pages 525-550, December.
    37. 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.
    38. Buraschi, Andrea & Jackwerth, Jens, 2001. "The Price of a Smile: Hedging and Spanning in Option Markets," The Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 495-527.
    39. Campa, Jose M. & Chang, P. H. Kevin & Reider, Robert L., 1998. "Implied exchange rate distributions: evidence from OTC option markets1," Journal of International Money and Finance, Elsevier, vol. 17(1), pages 117-160, February.
    40. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    41. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    42. Charles J. Corrado & Tie Su, 1996. "Skewness And Kurtosis In S&P 500 Index Returns Implied By Option Prices," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 19(2), pages 175-192, June.
    43. Jose M. Campa & P.H. Kevin Chang & Robert L. Reider, 1997. "Implied Exchange Rate Distributions: Evidence from OTC Option Markets," NBER Working Papers 6179, National Bureau of Economic Research, Inc.
    44. Heston, Steven L, 1993. "Invisible Parameters in Option Prices," Journal of Finance, American Finance Association, vol. 48(3), pages 933-947, July.
    45. Sheikh, Aamir M., 1991. "Transaction Data Tests of S&P 100 Call Option Pricing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(4), pages 459-475, December.
    46. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    47. Jingzhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time- Changed Levy Processes," Finance 0401002, University Library of Munich, Germany.
    48. repec:bla:jfinan:v:53:y:1998:i:2:p:499-547 is not listed on IDEAS
    49. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    50. Bates, David S., 2000. "Post-'87 crash fears in the S&P 500 futures option market," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 181-238.
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    More about this item

    Keywords

    Black Scholes formula; Implied volatility skew; Stable pattern; Risk-neutral density;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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