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Option pricing with maximum entropy densities: The inclusion of higher‐order moments

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  • Omid M. Ardakani

Abstract

Entropy pricing applies notions of information theory to derive the theoretical value of options. This paper employs the maximum entropy (ME) formulation of option pricing, given risk‐neutral moment constraints computed directly from the observed prices. First, higher‐order moments are used to generate option prices. Then a generalization of Shannon entropy, known as Renyi entropy, is studied to account for extreme events. This ME problem provides a class of heavy‐tailed distributions. Examples and Monte Carlo simulations are provided to examine the effects of moment constraints on option prices. The call option values are then constructed using daily Standard and Poor's 500 index options. The findings suggest that entropy pricing with higher‐order moment constraints provides higher forecasting accuracy.

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  • Omid M. Ardakani, 2022. "Option pricing with maximum entropy densities: The inclusion of higher‐order moments," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(10), pages 1821-1836, October.
    • Handle: RePEc:wly:jfutmk:v:42:y:2022:i:10:p:1821-1836
    DOI: 10.1002/fut.22361
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    1. Hutchinson, James M & Lo, Andrew W & Poggio, Tomaso, 1994. "A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks," Journal of Finance, American Finance Association, vol. 49(3), pages 851-889, July.
    2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    3. M. Ryan Haley & Todd B. Walker, 2010. "Alternative tilts for nonparametric option pricing," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 30(10), pages 983-1006, October.
    4. Stutzer, Michael, 1996. "A Simple Nonparametric Approach to Derivative Security Valuation," Journal of Finance, American Finance Association, vol. 51(5), pages 1633-1652, December.
    5. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    6. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    7. Cassio Neri & Lorenz Schneider, 2012. "Maximum entropy distributions inferred from option portfolios on an asset," Finance and Stochastics, Springer, vol. 16(2), pages 293-318, April.
    8. Rompolis, Leonidas S. & Tzavalis, Elias, 2008. "Recovering Risk Neutral Densities from Option Prices: A New Approach," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 43(4), pages 1037-1053, December.
    9. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    10. Gurdip Bakshi & Nikunj Kapadia & Dilip Madan, 2003. "Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options," Review of Financial Studies, Society for Financial Studies, vol. 16(1), pages 101-143.
    11. Brennan, M J, 1979. "The Pricing of Contingent Claims in Discrete Time Models," Journal of Finance, American Finance Association, vol. 34(1), pages 53-68, March.
    12. John M. Cozzolino & Michael J. Zahner, 1973. "The Maximum-Entropy Distribution of the Future Market Price of a Stock," Operations Research, INFORMS, vol. 21(6), pages 1200-1211, December.
    13. Omid M. Ardakani & Nader Ebrahimi & Ehsan S. Soofi, 2018. "Ranking Forecasts by Stochastic Error Distance, Information and Reliability Measures," International Statistical Review, International Statistical Institute, vol. 86(3), pages 442-468, December.
    14. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    15. Eugene F. Fama, 1963. "Mandelbrot and the Stable Paretian Hypothesis," The Journal of Business, University of Chicago Press, vol. 36, pages 420-420.
    16. P. Carr & D. Madan, 2001. "Optimal positioning in derivative securities," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 19-37.
    17. Yatchew, Adonis & Hardle, Wolfgang, 2006. "Nonparametric state price density estimation using constrained least squares and the bootstrap," Journal of Econometrics, Elsevier, vol. 133(2), pages 579-599, August.
    18. Buchen, Peter W. & Kelly, Michael, 1996. "The Maximum Entropy Distribution of an Asset Inferred from Option Prices," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 31(1), pages 143-159, March.
    19. Amin, Kaushik I & Ng, Victor K, 1993. "Option Valuation with Systematic Stochastic Volatility," Journal of Finance, American Finance Association, vol. 48(3), pages 881-910, July.
    20. Philip Gray & Scott Newman, 2005. "Canonical valuation of options in the presence of stochastic volatility," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 25(1), pages 1-19, January.
    21. Rockinger, Michael & Jondeau, Eric, 2002. "Entropy densities with an application to autoregressive conditional skewness and kurtosis," Journal of Econometrics, Elsevier, vol. 106(1), pages 119-142, January.
    22. Philip Gray & Shane Edwards & Egon Kalotay, 2007. "Canonical valuation and hedging of index options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 27(8), pages 771-790, August.
    23. Les Gulko, 1999. "The Entropy Theory Of Stock Option Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 2(03), pages 331-355.
    24. 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.
    25. 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.
    26. Maasoumi, Esfandiar & Racine, Jeff, 2002. "Entropy and predictability of stock market returns," Journal of Econometrics, Elsevier, vol. 107(1-2), pages 291-312, March.
    27. Cox, John C & Ross, Stephen A, 1976. "A Survey of Some New Results in Financial Option Pricing Theory," Journal of Finance, American Finance Association, vol. 31(2), pages 383-402, May.
    28. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    29. Goldenberg, David H., 1991. "A unified method for pricing options on diffusion processes," Journal of Financial Economics, Elsevier, vol. 29(1), pages 3-34, March.
    30. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    31. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    32. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    33. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    34. Melanie Birke & Kay F. Pilz, 2009. "Nonparametric Option Pricing with No-Arbitrage Constraints," Journal of Financial Econometrics, Oxford University Press, vol. 7(2), pages 53-76, Spring.
    35. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    36. Rompolis, Leonidas S., 2010. "Retrieving risk neutral densities from European option prices based on the principle of maximum entropy," Journal of Empirical Finance, Elsevier, vol. 17(5), pages 918-937, December.
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    Cited by:

    1. Thomas P. Davis, 2023. "Information Theory and the Pricing of Contingent Claims: An Alternative Derivation of the Black–Scholes–Merton Formula," JRFM, MDPI, vol. 16(12), pages 1-7, December.
    2. Ardakani, Omid M., 2023. "Capturing information in extreme events," Economics Letters, Elsevier, vol. 231(C).
    3. Ardakani, Omid M., 2023. "Coherent measure of portfolio risk," Finance Research Letters, Elsevier, vol. 57(C).

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