Direct Semi-Parametric Estimation of the State Price Density Implied in Option Prices

Arbitrage-free estimates; Inverse problem; Penalized composite link model; State price density; Statistics and Probability; Social Sciences (miscellaneous); Economics and Econometrics; Statistics, Probability and Uncertainty

Abstract :

[en] We present a model for direct semi-parametric estimation of the state price density (SPD) implied by quoted option prices. We treat the observed prices as expected values of possible pay-offs at maturity, weighted by the unknown probability density function. We model the logarithm of the latter as a smooth function, using P-splines, while matching the expected values of the potential pay-offs with the observed prices. This leads to a special case of the penalized composite link model. Our estimates do not rely on any parametric assumption on the underlying asset price dynamics and are consistent with no-arbitrage conditions. The model shows excellent performance in simulations and in applications to real data.

Disciplines :

Quantitative methods in economics & management

Author, co-author :

Frasso, Gianluca ; Université de Liège - ULiège > Département des sciences sociales > Méthodes quantitatives en sciences sociales

Eilers, Paul H.C.; Erasmus University Medical Center, Rotterdam, Netherlands

Language :

English

Title :

Direct Semi-Parametric Estimation of the State Price Density Implied in Option Prices

Publication date :

2022

Journal title :

Journal of Business and Economic Statistics

ISSN :

0735-0015

eISSN :

1537-2707

Publisher :

American Statistical Association

Volume :

Issue :

Pages :

1179 - 1190

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.tandfonline.com/doi/pdf/10.1080/07350015.2021.1906686

Funding text :

We thank professor Oleg Bondarenko for sharing the code used to estimate risk-neutral densities with the positive convolution approximation. We also thank the two anonymous reviewers and the associated editor for their insightful comments and remarks.

Available on ORBi :

since 29 March 2023

Statistics

Number of views

41 (5 by ULiège)

Number of downloads

0 (0 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

Abken, P. A., Madan, D. B., and Ramamurtie, B. S. (1996), “Estimation of Risk-neutral and Statistical Densities by Hermite Polynomial Approximation: With an Application to Eurodollar Futures Options,” FRB Atlanta Working Paper 96–5, Federal Reserve Bank of Atlanta.
Aït-Sahalia, Y., and Duarte, J. (2003), “Nonparametric Option Pricing Under Shape Restrictions,” Journal of Econometrics, 116, 9–47. DOI: 10.1016/S0304-4076(03)00102-7.
Aït-Sahalia, Y. and Lo, A. W. (1998), “Nonparametric Estimation of State-price Densities Implicit in Financial Asset Prices,” Journal of Finance, 53, 499–547. DOI: 10.1111/0022-1082.215228.
Aït-Sahalia, Y. and Lo, A. W. (2000), “Nonparametric Risk Management and Implied Risk Aversion,” Journal of Econometrics, 94, 9–51.
Bahra, B. (1997), “Implied Risk-neutral Probability Density Functions From Option Prices: Theory and Application,” Bank of England Working Papers No. 66, Bank of England.
Barletta, A., Santucci de Magistris, P., and Violante, F. (2019), “A Non-structural Investigation of VIX Risk Neutral Density,” Journal of Banking & Finance, 99, 1–20.
Bates, D. S. (2000), “Post-’87 Crash Fears in the s&p 500 Futures Option Market,” Journal of Econometrics, 94, 181–238.
Berkowitz, J. (2001), “Testing Density Forecasts, With Applications to Risk Management,” Journal of Business & Economic Statistics, 19, 465–474.
Birru, J. and Figlewski, S. (2012), “Anatomy of a Meltdown: The Risk Neutral Density for the s&p 500 in the Fall of 2008,” Journal of Financial Markets, 15, 151–180. DOI: 10.1016/j.finmar.2011.09.001.
Black, F., and Scholes, M. (1973), “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637. DOI: 10.1086/260062.
Bliss, R. R., and Panigirtzoglou, N. (2002), “Testing the Stability of Implied Probability Density Functions,” Journal of Banking & Finance, 26, 381–422.
Bondarenko, O. (2003), “Estimation of Risk-neutral Densities Using Positive Convolution Approximation,” Journal of Econometrics, 116, 85–112. DOI: 10.1016/S0304-4076(03)00104-0.
Breeden, D. T. and Litzenberger, R. H. (1978), “Prices of State-contingent Claims Implicit in Option Prices,” The Journal of Business, 51, 621–51. DOI: 10.1086/296025.
Campa, J. M., Chang, P., and Reider, R. L. (1998), “Implied Exchange Rate Distributions: Evidence From OTC Option Markets,” Journal of International Money and Finance, 17, 117–160. DOI: 10.1016/S0261-5606(97)00054-5.
CBOE (2019), “Whitepaper: Cboe Volatility Index,” Technical report, Available at https://www.cboe.com/micro/vix/vixwhite.pdf.
Cox, J., Ross, S., and Rubinstein, M. (1979), “Option Pricing: A Simplified Approach,” Journal of Financial Economics, 7, 229–263. DOI: 10.1016/0304-405X(79)90015-1.
Cox, J. C., and Ross, S. A. (1976), “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics, 3, 145–166. DOI: 10.1016/0304-405X(76)90023-4.
Eilers, P. H. (2007), “Ill-posed Problems With Counts, the Composite Link Model and Penalized Likelihood,” Statistical Modelling, 7, 239–254. DOI: 10.1177/1471082X0700700302.
Eilers, P. H. C. (2005), “Unimodal Smoothing,” Journal of Chemometrics, 19, 317–328. DOI: 10.1002/cem.935.
Eilers, P. H. C., and Marx, B. D. (1996), “Flexible Smoothing With b-splines and Penalties,” Statistical Science, 11, 89–121. DOI: 10.1214/ss/1038425655.
Eilers, P. H. C., and Marx, B. D. (2003), “Multivariate Calibration With Temperature Interaction Using Two-dimensional Penalized Signal Regression,” Chemometrics and Intelligent Laboratory Systems, 66, 159–174.
Eilers, P. H. C., and Marx, B. D. (2010), “Splines, Knots, and Penalties,” Wiley Interdisciplinary Reviews: Computational Statistics, 2, 637–653.
Fengler, M. R. (2009), “Arbitrage-free Smoothing of the Implied Volatility Surface,” Quantitative Finance, 9, 417–428. DOI: 10.1080/14697680802595585.
Fengler, M. R., and Hin, L.-Y. (2015), “Semi-nonparametric Estimation of the Call-option Price Surface Under Strike and Time-to-expiry no-arbitrage Constraints,” Journal of Econometrics, 184, 242–261. DOI: 10.1016/j.jeconom.2014.09.003.
Figlewski, S. (2009), “ Estimating the Implied Risk Neutral Density for the U.S. Market Portfolio,” in Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle (eds.), eds. T. Bollerslev, J. R. Russell and M. Watson, Oxford: Oxford University Press.
Fleming, W. (1987), Functions of Several Variables. Undergraduate Texts in Mathematics. New York: Springer.
Green, P. J. (1984), “Iteratively Reweighted Least Squares for Maximum Likelihood Estimation, and Some Robust and Resistant Alternatives,” Journal of the Royal Statistical Society, Series B, 46, 149–192. DOI: 10.1111/j.2517-6161.1984.tb01288.x.
Härdle, W., and Hlávka, Z. (2009), “Dynamics of State Price Densities,” Journal of Econometrics, 150, 1–15. DOI: 10.1016/j.jeconom.2009.01.005.
Harrison, J., and Kreps, D. M. (1979), “Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory, 20, 381–408. DOI: 10.1016/0022-0531(79)90043-7.
Harrison, J. M. and Pliska, S. R. (1981), “Martingales and Stochastic Integrals in the Theory of Continuous Trading,” Stochastic Processes and their Applications, 11, 215–260. DOI: 10.1016/0304-4149(81)90026-0.
Hastie, T. J., and Tibshirani, R. J. (1990), Generalized Additive Models, London: Chapman & Hall.
Huynh, K. P., Kervella, P., and Zheng, J. T. (2002), “Estimating State-price Densities With Nonparametric Regression,” SFB 373 Discussion Papers 2002,40, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
Jackwerth, J., and Rubinstein, M. (1996), “Recovering Probability Distributions From Option Prices,” Journal of Finance, 51, 1611–32. DOI: 10.1111/j.1540-6261.1996.tb05219.x.
Jackwerth, J. C. (2000), “Recovering Risk Aversion From Option Prices and Realized Returns,” The Review of Financial Studies, 13, 433–451. DOI: 10.1093/rfs/13.2.433.
Jarrow, R., and Rudd, A. (1982), “Approximate Option Valuation for Arbitrary Stochastic Processes,” Journal of Financial Economics, 10, 347–369. DOI: 10.1016/0304-405X(82)90007-1.
Jondeau, E., and Rockinger, M. (2001), “Gram-Charlier Densities,” Journal of Economic Dynamics and Control, 25, 1457–1483. DOI: 10.1016/S0165-1889(99)00082-2.
Jondeau, E., Poon, S.-H., and Rockinger, M. (2007), Financial Modeling Under Non-Gaussian Distributions, London: Springer.
Krivobokova, T., and Kauermann, G. (2007), “A Note on Penalized Spline Smoothing With Correlated Errors,” Journal of the American Statistical Association, 102, 1328–1337. DOI: 10.1198/016214507000000978.
Lee, Y., Nelder, J., and Pawitan, Y. (2006), Generalized Linear Models with Random Effects: Unified Analysis Via H-likelihood, Chapman & Hall/CRC Monographs on Statistics & Applied Probability, New York: CRC Press.
Lipovetsky, S. (2009), “Linear Regression With Special Coefficient Features Attained Via Parameterization in Exponential, Logistic, and Multinomial–logit Forms,” Mathematical and Computer Modelling, 49, 1427–1435. DOI: 10.1016/j.mcm.2008.11.013.
Madan, D. B., and Milne, F. (1994), “Contingent Claims Valued and Hedged by Pricing and Investing in a Basis,” Mathematical Finance, 4, 223–245. DOI: 10.1111/j.1467-9965.1994.tb00093.x.
Melick, W. R., and Thomas, C. P. (1997), “Recovering an Asset’s Implied pdf From Option Prices: An Application to Crude Oil During the Gulf Crisis,” The Journal of Financial and Quantitative Analysis, 32, 91–115. DOI: 10.2307/2331318.
Ritchey, R. J. (1990), “Call Option Valuation for Discrete Normal Mixtures,” Journal of Financial Research, 13, 285–296. DOI: 10.1111/j.1475-6803.1990.tb00633.x.
Rosenberg, J. (1998), “Pricing Multivariate Contingent Claims Using Estimated Risk-neutral Density Functions,” Journal of International Money and Finance, 17, 229–247. DOI: 10.1016/S0261-5606(98)00001-1.
Ruppert, D., Wand, P., and Carroll, R. (2003), Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics, New York: Cambridge University Press.
Schall, R. (1991), “Estimation in Generalized Linear Models With Random Effects,” Biometrika, 78, 719–727. DOI: 10.1093/biomet/78.4.719.
Shimko, D. (1993), “Bounds of Probability,” RISK, 6, 33–47.
Song, Z., and Xiu, D. (2016), “A Tale of Two Option Markets: Pricing Kernels and Volatility Risk,” Journal of Econometrics, 190, 176–196. DOI: 10.1016/j.jeconom.2015.06.024.
Thompson, R., and Baker, R. J. (1981), “Composite Link Functions in Generalized Linear Models,” Journal of the Royal Statistical Society, Series C, 30, 125–131. DOI: 10.2307/2346381.
Wood, S. N., and Fasiolo, M. (2017), “A Generalized Fellner–Schall Method for Smoothing Parameter Optimization With Application to Tweedie Location, Scale and Shape Models,” Biometrics, 73, 1071–1081. DOI: 10.1111/biom.12666.
Yatchew, A., and Härdle, W. (2006), “Nonparametric State Price Density Estimation Using Constrained Least Squares and the Bootstrap,” Journal of Econometrics, 133, 579–599. DOI: 10.1016/j.jeconom.2005.06.031.
Yu, Y., and Ruppert, D. (2002), “Penalized Spline Estimation for Partially Linear Single-index Models,” Journal of the American Statistical Association, 97, 1042–1054. DOI: 10.1198/016214502388618861.