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
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.
scite shows how a scientific paper has been cited by providing the context of the citation, a classification describing whether it supports, mentions, or contrasts the cited claim, and a label indicating in which section the citation was made.
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.
Similar publications
Sorry the service is unavailable at the moment. Please try again later.
This website uses cookies to improve user experience. Read more
Save & Close
Accept all
Decline all
Show detailsHide details
Cookie declaration
About cookies
Strictly necessary
Performance
Strictly necessary cookies allow core website functionality such as user login and account management. The website cannot be used properly without strictly necessary cookies.
This cookie is used by Cookie-Script.com service to remember visitor cookie consent preferences. It is necessary for Cookie-Script.com cookie banner to work properly.
Performance cookies are used to see how visitors use the website, eg. analytics cookies. Those cookies cannot be used to directly identify a certain visitor.
Used to store the attribution information, the referrer initially used to visit the website
Cookies are small text files that are placed on your computer by websites that you visit. Websites use cookies to help users navigate efficiently and perform certain functions. Cookies that are required for the website to operate properly are allowed to be set without your permission. All other cookies need to be approved before they can be set in the browser.
You can change your consent to cookie usage at any time on our Privacy Policy page.
