Abstract
This chapter deals with nonparametric estimation of the risk neutral density. We present three different approaches which do not require parametric functional assumptions on the underlying asset price dynamics nor on the distributional form of the risk neutral density. The first estimator is a kernel smoother of the second derivative of call prices, while the second procedure applies kernel type smoothing in the implied volatility domain. In the conceptually different third approach we assume the existence of a stochastic discount factor (pricing kernel) which establishes the risk neutral density conditional on the physical measure of the underlying asset. Via direct series type estimation of the pricing kernel we can derive an estimate of the risk neutral density by solving a constrained optimization problem. The methods are compared using European call option prices. The focus of the presentation is on practical aspects such as appropriate choice of smoothing parameters in order to facilitate the application of the techniques.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
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. Journal of Finance, 53, 499–547.
Aït-Sahalia, Y., & Lo, A. W. (2000). Nonparametric risk management and implied risk aversion. Journal of Econometrics, 94, 9–51.
Arrow, K. J. (1964). The role of securities in the optimal allocation of risk-bearing. Review of Economic Studies, 31, 91–96.
Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9(1), 69–107.
Black, F., & Scholes, M. (1973). The Pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.
Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. The Journal of Business, 51(4), 621–651.
Brown, D. P., & Jackwerth, J. C. (2004). The pricing kernel puzzle: Reconciling index option data and economic theory, Working Paper, University of Konstanz/University of Wisconsin.
Campbell, J., Lo, A., & McKinlay, A. (1997). The econometrics of financial markets. NJ: Princeton University Press.
Craven, P., & Wahba, G. (1979). Smoothing noisy data with spline functions, Numerische Mathematik, 31, 377–403
Debreu, G. (1959). Theory of value: An axiomatic analysis of economic equilibrium. New Haven: Yale University Press.
Engle, R. F., & Rosenberg, J. V. (2002). Empirical pricing kernels. Journal of Financial Economics, 64, 341–372.
Fan, J., & Gijbels, I. (1996). Local polynomial modelling and its applications. London: Chapman and Hall.
Fengler, M. R. (2005). Semiparametric modeling of implied volatility. Berlin: Springer
Grith, M., Härdle, W., & Park, J. (2010). Shape invariant modelling pricing kernels and risk aversion. Resubmitted to Journal of Financial Econometrics on 17 December 2010
Härdle, W. (1990). Applied nonparametric regression. Econometric Society Monographs No. 19. London: Cambridge University Press
Härdle, W., & Hlavka, Z. (2009). Dynamics of state price densities. Journal of Econometrics, 150(1), 1–15.
Härdle, W., Müller, M., Sperlich, S., & Werwatz, A. (2004). Nonparametric and semiparametric models. Heidelberg: Springer.
Härdle, W., Okhrin, Y., & Wang, W. (2009). Uniform confidence for pricing kernels. SFB649DP2010-003. Econometric Theory (Submitted).
Heston, S. (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.
Ingersoll, J.E. (1987). Theory of financial decision making. Rowman & Littlefield
Jackwerth, J. C. (2000). Recovering risk aversion from option prices and realized returns. Review of Financial Studies, 13(2), 433–451.
Li, K. C. (1987). Asymptotic optimality for c p , c l , cross-validation and generalized cross-validation: Discrete index set. Annals of Statistics, 15, 958–975.
Li, Q., & Racine, J. S. (2007). Nonparametric econometrics: Theory and practice. NJ: Princeton University Press.
Linton, O., & Nielsen, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika, 82(1), 93–100.
Lucas, R. E. (1978). Asset prices in an exchange economy. Econometrica, 46, 1429–1445.
Mallows, C. L. (1973). Some comments on c p . Technometrics, 15, 661–675.
Mammen, E., Linton, O., & Nielsen, J. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Annals of Statistics, 27(5), 1443–1490.
Marron, J. S., & Nolan, D. (1988). Canonical kernels for density estimation. Statistics and Probability Letters, 7(3), 195–199.
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics, 4(1), 141–183.
R. Merton (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125–183.
Müller, H. G. (1988). Nonparametric regression analysis of longitudinal data. Lecture Notes in Statistics (Vol. 46). New York: Springer.
Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. Journal of Econometrics, 79(1), 147–168.
Rookley, C. (1997). Fully exploiting the information content of intra day option quotes: Applications in option pricing and risk management, Working paper, University of Arizona.
Rubinstein. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics, 7(2), 407–425.
Ruppert, D., & Wand, M. P. (1994). Multivariate locally weighted least squares regression. Annals of Statistics, 22(3), 1346–1370.
Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions. J. Roy. Statist. Soc. Ser. B, 36, 111–147.
Wahba, G. (1985). A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Annals of Statistics, 13(4), 1378–1402.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Grith, M., Härdle, W.K., Schienle, M. (2012). Nonparametric Estimation of Risk-Neutral Densities. In: Duan, JC., Härdle, W., Gentle, J. (eds) Handbook of Computational Finance. Springer Handbooks of Computational Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17254-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-17254-0_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17253-3
Online ISBN: 978-3-642-17254-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)