Abstract
A new approach is considered to estimate risk-neutral densities (RND) within a kernel regression framework, through local cubic polynomial estimation using intraday data. There is a new strategy for the definition of a criterion function used in nonparametric regression that includes calls, puts, and weights in the optimization problem associated with parameters estimation. No-arbitrage constraints are incorporated into the problem through equality and bound constraints. The approach considered yields directly density functions of interest with minimum requirements needed. Within a simulation framework, it is demonstrated the robustness of proposed procedures. Additionally, RNDs are estimated through option prices associated with two indices, S&P500 and VIX.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aït-Sahalia, Y., & Duarte, J. (2003). Nonparametric option pricing under shape restrictions. Journal of Econometrics, 116(1), 9–47.
Aït-Sahalia, Y., & Lo, A. W. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53(2), 499–547.
Aït-Sahalia, Y., & Lo, A. W. (2000). Nonparametric risk management and implied risk aversion. Journal of Econometrics, 94(1), 9–51.
Aït-Sahalia, Y., Wang, Y., & Yared, F. (2001). Do option markets correctly price the probabilities of movement of the underlying asset? Journal of Econometrics, 102(1), 67–110.
Banz, R. W., & Miller, M. H. (1978). Prices for state-contingent claims: Some estimates and applications. Journal of Business, 51(4), 653–672.
Birke, M., & Pilz, K. F. (2008). Nonparametric option pricing with no-arbitrage constraints. Journal of Financial Econometrics, 7(2), 53–76.
Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of Business, 51(4), 621–651.
Cao-Abad, R. (1991). Rate of convergence for the wild bootstrap in nonparametric regression. The Annals of Statistics, 19(4), 2226–2231.
Cox, J. C., & Ross, S. A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(1–2), 145–166.
Dalderop, J. (2018). Nonparametric filtering of conditional state-price densities. Technical report.
DiCiccio, T. J., & Efron, B. (1996). Bootstrap confidence intervals. Statistical Science, 11(3), 189–212.
Efron, B. (1982). The jackknife, the bootstrap, and other resampling plans (Vol. 38). Philaphidia: SIAM.
Efron, B., & Tibshirani R.(1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, 1(1), 54–75.
Fan, J., & Gijbels, I. (1996). Local polynomial modelling and its applications (Vol. 66). London: CRC Press.
Fengler, M. R., & 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(2), 242–261.
Grith, M., Härdle, W. K., & Schienle, M. (2012). Nonparametric estimation of risk-neutral densities. In Handbook of computational finance, pp. 277–305. London: Springer.
Hall, P. (1988). Theoretical comparison of bootstrap confidence intervals. The Annals of Statistics, 16(3), 927–953.
Hall, P. (2013). The bootstrap and Edgeworth expansion. NY: Springer.
Härdle, W., & Bowman, A. W. (1988). Bootstrapping in nonparametric regression: Local adaptive smoothing and confidence bands. Journal of the American Statistical Association, 83(401), 102–110.
Härdle, W. (1990). Applied nonparametric regression (Vol. 19). Cambridge: Cambridge University Press.
Härdle, W., & Marron, J. (1991). Bootstrap simultaneous error bars for nonparametric regression. The Annals of Statistics, 19(2), 778–796.
Härdle, W., & Hlávka, Z. (2009). Dynamics of state price densities. Journal of Econometrics, 150(1), 1–15.
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2), 327–343.
Jackwerth, J. C. (2000). Recovering risk aversion from option prices and realized returns. The Review of Financial Studies, 13(2), 433–451.
Li, Q., & Racine, J. S. (2007). Nonparametric econometrics: Theory and practice. Princeton: Princeton University Press.
Liu, R. Y. (1988). Bootstrap procedures under some non-iid models. The Annals of Statistics, 16(4), 1696–1708.
Monteiro, A. M., Tütüncü, R. H., & Vicente, L. N. (2008). Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity. European Journal of Operational Research, 187(2), 525–542.
Nadaraya, E. A. (1964). On estimating regression. Theory of Probability & Its Applications, 9(1), 141–142.
Rosenberg, J. V., & Engle, R. F. (2002). Empirical pricing kernels. Journal of Financial Economics, 64(3), 341–372.
Song, Z., & Xiu, D. (2016). A tale of two option markets: Pricing kernels and volatility risk. Journal of Econometrics, 190(1), 176–196.
Watson, G. S. (1964). Smooth regression analysis. Sankhyā: The Indian Journal of Statistics, Series A, 26(4), 359–372.
Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. The Annals of Statistics, 14(4), 1261–1295.
Yatchew, A. (2003). Semiparametric regression for the applied econometrician. Cambridge: Cambridge University Press.
Yatchew, A., & Härdle, W. (2006). Nonparametric state price density estimation using constrained least squares and the bootstrap. Journal of Econometrics, 133(2), 579–599.
Zhang, X., Brooks, R. D., & King, M. L. (2009). A bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation. Journal of Econometrics, 153(1), 21–32.
Acknowledgements
We are grateful to the Associate Editor and to an anonymous referee for many insightful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Monteiro, A.M., Santos, A.A.F. Conditional risk-neutral density from option prices by local polynomial kernel smoothing with no-arbitrage constraints. Rev Deriv Res 23, 41–61 (2020). https://doi.org/10.1007/s11147-019-09156-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11147-019-09156-x