1932

Abstract

Trading in options with a wide range of exercise prices and a single maturity allows a researcher to extract the market's risk-neutral density (RND) over the underlying price at expiration. The RND contains investors’ beliefs about the true probabilities blended with their risk preferences, both of which are of great interest to academics and practitioners alike. With a particular focus on US equity options, I review the historical development of this powerful concept, practical details of fitting an RND to options market prices, and the many ways in which investigators have tried to distill true expectations and risk premia from observed RNDs. I briefly discuss areas of active current research including the pricing kernel puzzle and the volatility surface, and offer thoughts on what has been learned about RNDs so far and fruitful directions for future research.

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2018-11-01
2024-12-09
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Literature Cited

  1. Abken PA, Madan DB, Ramamurtie S 1996. Estimation of risk-neutral and statistical densities by Hermite polynomial approximation: with an application to eurodollar futures options Work. Pap. 96-5, Fed. Res. Bank Atlanta
    [Google Scholar]
  2. Aït-Sahalia Y, Lo AW 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. J. Finance 53:499–547
    [Google Scholar]
  3. Aït-Sahalia Y, Lo AW 2000. Nonparametric risk management and implied risk aversion. J. Econ. 94:9–51
    [Google Scholar]
  4. Andersen TG, Fusari N, Todorov V 2015. The risk premia embedded in index options. J. Financ. Econ. 117:558–84
    [Google Scholar]
  5. Aparicio SD, Hodges S 1998. Implied risk-neutral distribution: a comparison of estimation methods FORC Preprint PP 88–95 Univ. Warwick Coventry, UK:
    [Google Scholar]
  6. Bahra B 1997. Implied risk-neutral probability density functions from options prices: theory and application Work. Pap. 66 Bank Engl. London:
    [Google Scholar]
  7. Bakshi G, Cao C, Chen Z 1997. Empirical performance of alternative option pricing models. J. Finance 52:2003–49
    [Google Scholar]
  8. Bakshi G, Kapadia N 2003.a Delta-hedged gains and the negative market volatility risk premium. Rev. Financ. Stud. 16:527–66
    [Google Scholar]
  9. Bakshi G, Kapadia N 2003.b Volatility risk premiums embedded in individual equity options: some new insights. J. Deriv. 11:45–54
    [Google Scholar]
  10. Bakshi G, Kapadia N, Madan D 2003. Stock return characteristics, skew laws, and the differential pricing of individual equity options. Rev. Financ. Stud. 16:101–43
    [Google Scholar]
  11. Bakshi G, Madan D 2000. Spanning and derivative-security valuation. J. Financ. Econ. 55:205–38
    [Google Scholar]
  12. Bakshi G, Madan D, Panayotov G 2010. Returns of claims on the upside and the viability of U-shaped pricing kernels. J. Financ. Econ. 97:130–54
    [Google Scholar]
  13. Banz RW, Miller MH 1978. Prices for state-contingent claims: some estimates and applications. J. Bus. 51:653–72
    [Google Scholar]
  14. Barone-Adesi G, Mancini L, Shefrin H 2013. Behavioral finance and the pricing kernel puzzle: estimating sentiment, risk aversion and time preference Work. Pap. Swiss Finance Inst.
    [Google Scholar]
  15. Bates DS 1991. The crash of ’87: Was it expected? The evidence from options markets. J. Finance 46:1009–44
    [Google Scholar]
  16. Bates DS 1996. Jumps and stochastic volatility: exchange rate process implicit in Deutsche mark options. Rev. Financ. Stud. 9:69–107
    [Google Scholar]
  17. Bates DS 2000. Post-’87 crash fears in the S&P 500 futures option market. J. Econ. 94:181–238
    [Google Scholar]
  18. Bates DS 2003. Empirical option pricing: a retrospection. J. Econ. 116:387–404
    [Google Scholar]
  19. Bates DS 2008. The market for crash risk. J. Econ. Dyn. Control 32:2291–321
    [Google Scholar]
  20. Bekaert G, Hoerova M 2014. The VIX, the variance premium and stock market volatility. J. Econ. 183:181–92
    [Google Scholar]
  21. Berkowitz J 2001. Testing density forecasts, with applications to risk management. J. Bus. Econ. Stat. 19:465–74
    [Google Scholar]
  22. Bhattacharya M 1983. Transactions data tests of efficiency of the Chicago Board Options Exchange. J. Financ. Econ. 12:161–85
    [Google Scholar]
  23. Birru J, Figlewski S 2012. Anatomy of a meltdown: the risk neutral density for the S&P 500 in the fall of 2008. J. Financ. Mark. 15:151–80
    [Google Scholar]
  24. Black F 1976. Studies of stock price volatility changes. Proceedings of the 1976 Meeting of the Business and Economic Statistics Section, American Statistical Association, Washington, DC177–81 Washington, DC: Am. Stat. Assoc.
    [Google Scholar]
  25. Black F, Scholes M 1972. The valuation of option contracts and a test of market efficiency. J. Finance 27:399–418
    [Google Scholar]
  26. Black F, Scholes M 1973. The pricing of options and corporate liabilities. J. Political Econ. 81:637–59
    [Google Scholar]
  27. Bliss RR, Panigirtzoglou N 2002. Testing the stability of implied probability density functions. J. Bank. Finance 26:381–422
    [Google Scholar]
  28. Bliss RR, Panigirtzoglou N 2004. Option-implied risk aversion estimates. J. Finance 59:407–46
    [Google Scholar]
  29. Bollen NPB, Whaley RE 2004. Does net buying pressure affect the shape of implied volatility functions. J. Finance 49:711–53
    [Google Scholar]
  30. Bollerslev T 1987. A conditional heteroskedastic time series model for speculative prices and rates of return. Rev. Econ. Stat. 69:542–47
    [Google Scholar]
  31. Bollerslev T, Tauchen G, Zhou H 2009. Expected stock returns and variance risk premia. Rev. Financ. Stud. 22:4463–92
    [Google Scholar]
  32. Bollerslev T, Todorov V 2011. Tails, fears, and risk premia. J. Finance 66:2165–211
    [Google Scholar]
  33. Bollerslev T, Todorov V, Xu L 2015. Tail risk premia and return predictability. J. Financ. Econ. 118:113–34
    [Google Scholar]
  34. Borovička J, Hansen L, Scheinkman J 2016. Misspecified recovery. J. Finance 71:2493–544
    [Google Scholar]
  35. Breeden D, Litzenberger R 1978. Prices of state-contingent claims implicit in option prices. J. Bus. 51:621–52
    [Google Scholar]
  36. Brennan M 1979. The pricing of contingent claims in discrete time models. J. Finance 24:53–68
    [Google Scholar]
  37. Buchen PW, Kelly M 1996. The maximum entropy distribution of an asset inferred from option prices. J. Financ. Quant. Anal. 31:143–59
    [Google Scholar]
  38. Canina L, Figlewski S 1993. The informational content of implied volatility. Rev. Financ. Stud. 6:659–81
    [Google Scholar]
  39. Cao J, Han B 2013. Cross section of option returns and idiosyncratic volatility. J. Financ. Econ. 108:231–49
    [Google Scholar]
  40. Carr P, Geman H, Madan D, Yor M 2002. The fine structure of asset returns: an empirical investigation. J. Bus. 75:305–32
    [Google Scholar]
  41. Carr P, Wu L 2009. Variance risk premia. Rev. Financ. Stud. 22:1311–41
    [Google Scholar]
  42. Carr P, Wu L 2016. Analyzing volatility risk and risk premium in option contracts: a new theory. J. Financ. Econ. 120:1–20
    [Google Scholar]
  43. Chabi-Yo F 2012. Pricing kernels with stochastic skewness and volatility risk. Manag. Sci. 58:624–40
    [Google Scholar]
  44. Chicago Board Options Exchange. 2003. VIX CBOE Volatility Index Chicago: CBOE http://www.cboe.com/micro/vix/vixwhite.pdf
    [Google Scholar]
  45. Chicago Board Options Exchange. 2009. CBOE S&P 500® Implied Correlation Index Chicago: CBOE https://www.cboe.com/micro/impliedcorrelation/impliedcorrelationindicator.pdf
    [Google Scholar]
  46. Chicago Board Options Exchange. 2011. The CBOE skew index—SKEW Tech. Rep. CBOE Chicago: https://www.cboe.com/micro/skew/documents/skewwhitepaperjan2011.pdf
    [Google Scholar]
  47. Christensen BJ, Prabhala N 1998. The relation between implied and realized volatility. J. Financ. Econ. 50:125–50
    [Google Scholar]
  48. Christoffersen P, Heston S, Jacobs K 2009. The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manag. Sci. 55:1914–32
    [Google Scholar]
  49. Christoffersen P, Heston S, Jacobs K 2013. Capturing option anomalies with a variance-dependent pricing kernel. Rev. Financ. Stud. 26:1962–2006
    [Google Scholar]
  50. Christoffersen P, Jacobs K, Chang BY 2013. Forecasting with option-implied information. Handb. Econ. Forecast. 2:581–656
    [Google Scholar]
  51. Clews R, Panigirtzoglou N, Proudman J 2000. Recent developments in extracting information from options markets. Bank Engl. Q. Bull. 40:5060
    [Google Scholar]
  52. Cont R, Da Fonseca J 2002. Dynamics of implied volatility surfaces. Quant. Finance 2:45–60
    [Google Scholar]
  53. Cont R, Da Fonseca J, Durrleman V 2002. Stochastic models of implied volatility surfaces. Econ. Notes 31:361–77
    [Google Scholar]
  54. Corrado CJ, Su T 1996. Skewness and kurtosis in S&P 500 index returns implied by option prices. J. Financ. Res. 19:175–92
    [Google Scholar]
  55. Corrado CJ, Su T 1997. Implied volatility skews and stock return skewness and kurtosis implied by stock option prices. Eur. J. Finance 3:73–85
    [Google Scholar]
  56. Cox JC, Ross S 1976. The valuation of options for alternative stochastic processes. J. Financ. Econ. 3:145–66
    [Google Scholar]
  57. Cox JC, Ross SA, Rubinstein M 1979. Option pricing: a simplified approach. J. Financ. Econ. 7:229–73
    [Google Scholar]
  58. Cuesdeanu H, Jackwerth JC 2017.a The pricing kernel puzzle in forward looking data. Rev. Deriv. Res. In press. https://link.springer.com/article/10.1007/s11147-017-9140-8
    [Google Scholar]
  59. Cuesdeanu H, Jackwerth JC 2017.b The pricing kernel puzzle: survey and outlook. Ann. Finance 14:289–329
    [Google Scholar]
  60. Das SR, Sundaram RK 1999. Of smiles and smirks: a term structure perspective. J. Financ. Quant. Anal. 34:211–39
    [Google Scholar]
  61. Dennis P, Mayhew S 2002. Risk-neutral skewness: evidence from stock options. J. Financ. Quant. Anal. 37:471–93
    [Google Scholar]
  62. Dennis P, Mayhew S, Stivers C 2006. Stock returns, implied volatility innovations, and the asymmetric volatility phenomenon. J. Financ. Quant. Anal. 41:381–406
    [Google Scholar]
  63. Derman E, Kani I 1994. Riding on a smile. RISK 7:32–39
    [Google Scholar]
  64. Duffie D, Pan J, Singleton K 2000. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68:1343–76
    [Google Scholar]
  65. Dumas B, Fleming J, Whaley RE 1998. Implied volatility functions: empirical tests. J. Finance 53:2059–106
    [Google Scholar]
  66. Dupire B 1994. Pricing with a smile. RISK 7:18–20
    [Google Scholar]
  67. Egloff D, Leippold M, Wu L 2010. The term structure of variance swap rates and optimal variance swap investments. J. Financ. Quant. Anal. 45:1279–310
    [Google Scholar]
  68. Emanuel DC, MacBeth JD 1982. Further results on the constant elasticity of variance call option pricing model. J. Financ. Quant. Anal. 17:533–54
    [Google Scholar]
  69. Engle R 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987–1008
    [Google Scholar]
  70. Eriksson A, Ghysels E, Wang F 2009. The normal inverse Gaussian distribution and the pricing of derivatives. J. Deriv. 16:23–37
    [Google Scholar]
  71. Fed. Reserve Bank Minneap. 2014. Estimates of the future behavior of asset prices. https://www.minneapolisfed.org/banking/mpd/
  72. Feldman R, Heinecke K, Kocherlakota N, Schulhofer-Wohl S, Tallarini T 2015. Market-based probabilities: a tool for policymakers Work. Pap., Fed. Res. Bank Minneap.
    [Google Scholar]
  73. Figlewski S 2009. Estimating the implied risk neutral density. Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle T Bollerslev, JR Russell, M Watson 323–53 Oxford, UK: Oxford Univ. Press
    [Google Scholar]
  74. Figlewski S, Freund S 1994. The pricing of convexity risk and time decay in options markets. J. Bank. Finance 18:73–91
    [Google Scholar]
  75. Figlewski S, Malik MF 2014. Options on leveraged ETFs: a window on investor heterogeneity Work. Pap. New York Univ http://people.stern.nyu.edu/sfiglews/documents/Figlewski-Malik%202015FEB15.pdf
    [Google Scholar]
  76. Figlewski S, Webb G 1993. Options, short sales, and market completeness. J. Finance 48:761–77
    [Google Scholar]
  77. Fleming J 1998. The quality of market volatility forecasts implied by S&P 100 index option prices. J. Empir. Finance 5:317–45
    [Google Scholar]
  78. Gârleanu N, Pedersen LH, Poteshman AM 2009. Demand-based option pricing. Rev. Financ. Stud. 22:4259–99
    [Google Scholar]
  79. Gatheral J 2006. The Volatility Surface: A Practitioner's Guide Hoboken, NJ: John Wiley & Sons
    [Google Scholar]
  80. Gemmill G 2017. Behavioural biases and the pricing-kernel puzzle. https://ssrn.com/abstract=3019108
  81. Gemmill G, Saflekos A 2000. How useful are implied distributions? Evidence from stock index options. J. Deriv. 7:83–98
    [Google Scholar]
  82. Harrison JM, Kreps DM 1979. Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20:381–408
    [Google Scholar]
  83. Hens T, Reichlin C 2013. Three solutions to the pricing kernel puzzle. Rev. Finance 17:1029–64
    [Google Scholar]
  84. Heston S 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ Stud. 6:327–43
    [Google Scholar]
  85. Hull J, White A 1987. The pricing of options on assets with stochastic volatility. J. Finance 42:281–300
    [Google Scholar]
  86. Hull J, White A 2017. Optimal delta hedging for options. J. Bank. Finance 82:180–90
    [Google Scholar]
  87. Israelov R, Kelly B 2017. Forecasting the distribution of option returns Work. Pap Univ. Chicago
    [Google Scholar]
  88. Jackwerth JC 1999. Implied binomial trees: a literature review. J. Deriv. 7:66–82
    [Google Scholar]
  89. Jackwerth JC 2000. Recovering risk aversion from option prices and realized returns. Rev. Financ. Stud. 13:433–51
    [Google Scholar]
  90. Jackwerth JC 2004. Option-Implied Risk-Neutral Distributions and Risk Aversion Charlottesville, VA: Res. Foundation AIMR
    [Google Scholar]
  91. Jackwerth JC, Menner M 2017. Does the Ross recovery theorem work empirically Work. Pap. Univ. Konstanz Inst. Finance Ger.:
    [Google Scholar]
  92. Jackwerth JC, Rubinstein M 1996. Recovering probability distributions from option prices. J. Finance 51:1611–31
    [Google Scholar]
  93. Jackwerth JC, Vilkov G 2018. Asymmetric volatility risk: evidence from option markets. Work. Pap. Univ. Konstanz Inst. Finance Ger.:
    [Google Scholar]
  94. Jarrow R, Rudd A 1982. Approximate option valuation for arbitrary stochastic processes. J. Financ. Econ. 10:347–69
    [Google Scholar]
  95. Jensen CS, Lando D, Pedersen LH 2018. Generalized recovery. CEPR Discuss. Pap. DP12665
  96. Konstantinidi E, Skiadopoulos G 2016. How does the market variance risk premium vary over time? Evidence from S&P 500 variance swap investment returns. J. Bank. Finance 62:62–75
    [Google Scholar]
  97. Konstantinidi E, Skiadopoulos G, Tzagkaraki E 2008. Can the evolution of implied volatility be forecasted? Evidence from European and US implied volatility indices. J. Bank. Finance 32:2401–11
    [Google Scholar]
  98. Linn M, Shive S, Shumway T 2018. Pricing kernel monotonicity and conditional information. Rev. Financ. Stud. 31:493–531
    [Google Scholar]
  99. MacBeth J, Merville L 1980. Tests of the Black-Scholes and Cox call option valuation models. J. Finance 35:285–301
    [Google Scholar]
  100. Madan D, Milne F 1994. Contingent claims valued and hedged by pricing and investing in a basis. Math. Finance 4:223–45
    [Google Scholar]
  101. Markose S, Alentorn A 2011. The generalized extreme value distribution, implied tail index, and option pricing. J. Deriv. 18:35–60
    [Google Scholar]
  102. Melick WR, Thomas CP 1997. Recovering an asset's implied PDF from option prices: an application to crude oil during the Gulf Crisis. J. Financ. Quant. Anal. 32:91–115
    [Google Scholar]
  103. Merton RC 1976. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3:125–44
    [Google Scholar]
  104. OptionMetrics. 2008. Ivy DB File and Data Reference Manual Version 2.5.10 New York: OptionMetrics LLC
    [Google Scholar]
  105. Orosi G 2015. Estimating option-implied risk-neutral densities: a novel parametric approach. J. Deriv. 23:41–61
    [Google Scholar]
  106. Pan J 2002. The jump-risk premia implicit in options: evidence from an integrated time-series study. J. Financ. Econ. 63:3–50
    [Google Scholar]
  107. Poon SH, Granger C 2003. Forecasting volatility in financial markets: a review. J. Econ. Lit. 41:478–539
    [Google Scholar]
  108. Rompolis L, Tzavalis E 2007. Retrieving risk neutral densities based on risk neutral moments through a Gram–Charlier series expansion. Math. Comput. Model. 46:225–34
    [Google Scholar]
  109. Rosenberg J, Engle R 2002. Empirical pricing kernels. J. Financ. Econ. 64:341–72
    [Google Scholar]
  110. Ross S 1976. Options and efficiency. Q. J. Econ. 90:75–89
    [Google Scholar]
  111. Ross S 2015. The recovery theorem. J. Finance 70:615–48
    [Google Scholar]
  112. Rubinstein M 1976. The valuation of uncertain income streams and the pricing of options. Bell J. Econ. 7:407–25
    [Google Scholar]
  113. Rubinstein M 1985. Nonparametric tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE option classes from August 23, 1976 through August 31, 1978. J. Finance 40:455–80
    [Google Scholar]
  114. Rubinstein M 1994. Implied binomial trees. J. Finance 49:771–818
    [Google Scholar]
  115. Rubinstein M 1998. Edgeworth binomial trees. J. Deriv. 5:20–27
    [Google Scholar]
  116. Scott LO 1987. Option pricing when the variance changes randomly: theory, estimation, and an application. J. Financ. Quant. Anal. 22:419–38
    [Google Scholar]
  117. Shefrin H 2008. A Behavioral Approach to Asset Pricing Boston: Elsevier Academic. , 2nd ed..
    [Google Scholar]
  118. Shimko D 1993. The bounds of probability. RISK 6:33–37
    [Google Scholar]
  119. Söderlind P, Svensson L 1997. New techniques to extract market expectations from financial instruments. J. Monetary Econ. 40:383–429
    [Google Scholar]
  120. Stapleton RC, Subrahmanyam MG 1984. The valuation of multivariate contingent claims in discrete time models. J. Finance 39:207–28
    [Google Scholar]
  121. Stein EM, Stein JC 1991. Stock price distributions with stochastic volatility: an analytic approach. Rev. Financ. Stud. 4:727–52
    [Google Scholar]
  122. Stutzer M 1996. A simple nonparametric approach to derivative security valuation. J. Finance 51:1633–52
    [Google Scholar]
  123. Toft K, Prucyk B 1997. Options on leveraged equity: theory and empirical tests. J. Finance 52:1151–80
    [Google Scholar]
  124. Wiggins JB 1987. Option values under stochastic volatility: theory and empirical estimates. J. Financ. Econ. 19:351–77
    [Google Scholar]
  125. Wu D, Liu T 2018. New approach to estimating VIX truncation error using corridor variance swaps. J. Deriv. 25:54–70
    [Google Scholar]
  126. Xiu D 2014. Hermite polynomial based expansion of European option prices. J. Econ. 179:158–77
    [Google Scholar]
  127. Zhou H 2010. Variance risk premia, asset predictability puzzles, and macroeconomic uncertainty Work. Pap. Fed. Reserve Board Governors Finance Econ. Discuss. Ser. Washington, DC:
    [Google Scholar]
  128. Ziegler A 2007. Why does implied risk aversion smile. Rev. Financ. Stud. 20:859–904
    [Google Scholar]
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