Journal of Applied Mathematics

The Implied Risk Neutral Density Dynamics: Evidence from the S&P TSX 60 Index

Nessim Souissi

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The risk neutral density is an important tool for analyzing the dynamics of financial markets and traders’ attitudes and reactions to already experienced shocks by financial markets as well as the potential ones. In this paper, we present a new method for the extraction information content from option prices. By eliminating bias caused by daily variation of contract maturity through a completely nonparametric technique based on kernel regression, we allow comparing evolution of risk neutral density and extracting from time continuous indicators that detect evolution of traders’ attitudes, risk perception, and belief homogeneity. This method is useful to develop trading strategies and monetary policies.

Article information

J. Appl. Math., Volume 2017 (2017), Article ID 3156250, 10 pages.

Received: 14 March 2017
Accepted: 30 April 2017
First available in Project Euclid: 19 July 2017

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Souissi, Nessim. The Implied Risk Neutral Density Dynamics: Evidence from the S&P TSX 60 Index. J. Appl. Math. 2017 (2017), Article ID 3156250, 10 pages. doi:10.1155/2017/3156250.

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  • J. M. Harrison, “Martingales and arbitrage in multiperiod securities markets,” Journal of Economic Theory, vol. 20, no. 3, pp. 381–408, 1979.
  • G. Gemmill and A. Saflekos, “How useful are implied distributions? Evidence from stock-index options,” Journal of Derivatives, vol. 7, no. 3, pp. 83–98, 2000.
  • N. Souissi and A. Aloulou, “A Long memory and volatility behaviour in Paris option market,” International Journal of Bonds and Derivatives, vol. 1, no. 4, 2015.
  • D. P. Lynch and N. Panigirtzoglou, “Summary Statistics of Option-Implied Probability Density Functions and Their Properties,” SSRN Electronic Journal, vol. 345, Bank of England, 2008.
  • S. Taylor, P. Yadav, and Y. Zhang, “Cross-sectional analysis of risk-neutral skewness,” Journal of Derivatives, vol. 16, no. 4, pp. 38–52, 2009.
  • J. Birru and S. Figlewski, “Anatomy of a meltdown: The risk neutral density for the S&P 500 in the fall of 2008,” Journal of Financial Markets, vol. 15, no. 2, pp. 151–180, 2012.
  • D. T. Breeden and R. H. Litzenberger, “Prices of state-contingent claims implicit in option prices,” Journal of Business, vol. 51, no. 4, pp. 621–651, 1978.
  • J. C. Jackwerth, Option-implied risk-neutral distributions and risk aversion, Research Foundation of AIMR, Charlottesville, Va, USA, 2004.
  • S. Figlewski, Estimating the implied risk neutral density for the U.S. market portfolio, Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle, Oxford University Press, Oxford, UK, 2009.
  • R. Jarrow and A. Rudd, “Approximate valuation for arbitrary stochastic processes,” Journal of Financial Economics, vol. 10, pp. 349–369, 1982.
  • C. Corrado and T. Su, “Implied volatility skews and stock return skewness and kurtosis implied by stock option prices,” European Journal of Finance, vol. 3, no. 1, pp. 73–85, 1997.
  • S. Heston, “A closed-from solution for options with stochastic volatility with applications to bond and currency options,” The Review of Financial Studies, vol. 6, no. 2, pp. 327–343, 1993.
  • M. Rubinstein, “Implied binomial trees,” Journal of Finance, vol. 49, no. 3, pp. 771–818, 1994.
  • J. C. Jackwerth and M. Rubinstein, “Recovering probability distributions from option prices,” Journal of Finance, vol. 51, no. 5, pp. 1611–1631, 1996.
  • D. B. Madan and F. Milne, “Contingent claims valued and hedged by pricing and investing in a basis,” Mathematical Finance, vol. 4, no. 3, pp. 223–245, 1994.
  • P. Abken, D. Madan, and S. Ramamurtie, Estimation of Risk-Neutral and Statistical Densities by Hermite Polynomial Approximation : With an Application to Eurodollar Futures Options, 1996.
  • C. Chiarell, “Evaluation of american option prices in a path integral framework using fourier-hermite series expansions,” Journal of Economic Dynamics and Control, vol. 23, no. 10, pp. 1387–1424, 1998.
  • B. Bahra, “Probability distributions of future asset prices implied by option prices,” Bank of England Quarterly Bulletein, pp. 299–311, 1996.
  • B. J. Sherrick, P. Garcia, and V. Tirupattur, “Recovering probabilistic information from option markets: tests of distributional assumptions,” Journal of Futures Markets, vol. 16, no. 5, pp. 545–560, 1996.
  • W. R. Melik and C. Thomas, “Precovering an asset's implied pdf from options prices: applications to crude oil during the gulf crisis,” Journal of Financial and Quantitative Analysis, vol. 32, no. 1, pp. 91–116, 1997.
  • K. Adir and M. Rockinger, “Density-embedding functions,” HEC, 1997.
  • D. K. Backus, Accounting for Biases in Black-Scholes, CRIF, working paper series, 1997.
  • Y. Ait-Sahalia, “Nonparametric pricing of interest rate derivative securities,” Econometrica, Econometric Society, vol. 64, no. 3, pp. 527–560, 1996.
  • Y. Ait-Sahalia and A. Lo, “Non-parametric estimation state-price densities implied in financial asset prices,” Journal of Finance, vol. 53, no. 2, pp. 499–517, 1998.
  • Y. Ait-Sahalia and A. Lo, “Non-parametric risk management and implied risk aversion,” The Journal of Econometrics, vol. 94, pp. 5–9, 2000.
  • O. Bondarenko, “Estimation of risk-neutral densities using positive convolution approximation,” Journal of Econometrics, vol. 116, no. 1-2, pp. 85–112, 2003.
  • L. S. Rompolis and E. Tzavalis, “Retrieving risk neutral densities based on risk neutral moments through a gram–charlier series expansion,” Mathematical and Computer Modelling, vol. 46, no. 2, pp. 225–234, 2006.
  • A. M. Monteiro and L. s. Vicente, “Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring non negativity,” European Journal of Operational Research, vol. 187, no. 2, pp. 525–542, 2007.
  • L. S. Rompolis, “Retrieving risk neutral densities from european option prices based on the principle of maximum entropy,” Journal of Empirical Finance, vol. 17, no. 5, pp. 918–937, 2010.
  • F. J. Fabozzi, A. Leccadito, and R. S. Tunaru, “Extracting market information from equity options with exponential Lévy processes,” Journal of Economic Dynamics and Control, vol. 38, pp. 125–141, 2013.
  • E. Jondeau, “Reading the smile: the message conveyed by methods which infer risk neutral densities,” Journal of Internationnal Money and Finance, vol. 19, no. 6, pp. 885–915, 2000.
  • S. Coutant, “Reading PIBOR futures options smile: the 1997 snap election,” Journal of Banking and Finance, vol. 25, no. 11, pp. 1957–1987, 2001.
  • R. Bliss and N. Panigirtzoglou, “Testing the stability of implied probability density functions,” Journal of Banking and Finance, vol. 26, pp. 381–422, 2002.
  • G. Humphreys, “A quantitative mirror on the euribor market using implied probability density functions,” European Central Bank, vol. 1281, 2010.
  • C. Butler and H. S. Davies, “Assessing market views on monetary policy: the use of implied risk-neutral probability distributions,” in Proceedings of the BIS/CEPR conference on Asset Prices and Monetary Policy, 1998.
  • N. Panigirtzoglou and J. Proudman, “Recent developments in extracting information from options markets,” Bank of England Quarterly Bulletin, 2000.
  • R. Bliss and N. Panigirtzoglou, “Option-implied risk aversion estimates,” The Journal of Finance, vol. 59, no. 1, pp. 407–446, 2004.
  • N. Panigirtzoglou and G. Skiadopoulos, “A new approach to modeling the dynamics of implied distributions: theory and evidence from the s&p 500 options,” Journal of Banking & Finance, vol. 28, no. 7, pp. 1499–1520, 2004.
  • L. Kermiche, Dynamics of Implied Distributions: Evidence from the CAC 40 Options Market finance, 2009.
  • E. Nadaraya, “On estimating regression,” Theory of Probability and Its Applications, vol. 9, no. 1, pp. 141-142, 1964. \endinput