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Handbook of Computational Finance pp 277–305Cite as

Nonparametric Estimation of Risk-Neutral Densities

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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.

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References

  • Aït-Sahalia, Y., & Duarte, J. (2003). Nonparametric option pricing under shape restrictions. Journal of Econometrics, 116(1), 9–47.

    Article  MathSciNet  MATH  Google Scholar 

  • Aït-Sahalia, Y., & Lo, A. W. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. Journal of Finance, 53, 499–547.

    Article  Google Scholar 

  • Aït-Sahalia, Y., & Lo, A. W. (2000). Nonparametric risk management and implied risk aversion. Journal of Econometrics, 94, 9–51.

    Article  MATH  Google Scholar 

  • Arrow, K. J. (1964). The role of securities in the optimal allocation of risk-bearing. Review of Economic Studies, 31, 91–96.

    Article  Google Scholar 

  • Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9(1), 69–107.

    Article  Google Scholar 

  • Black, F., & Scholes, M. (1973). The Pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.

    Article  Google Scholar 

  • Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. The Journal of Business, 51(4), 621–651.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Campbell, J., Lo, A., & McKinlay, A. (1997). The econometrics of financial markets. NJ: Princeton University Press.

    MATH  Google Scholar 

  • Craven, P., & Wahba, G. (1979). Smoothing noisy data with spline functions, Numerische Mathematik, 31, 377–403

    Article  MathSciNet  MATH  Google Scholar 

  • Debreu, G. (1959). Theory of value: An axiomatic analysis of economic equilibrium. New Haven: Yale University Press.

    MATH  Google Scholar 

  • Engle, R. F., & Rosenberg, J. V. (2002). Empirical pricing kernels. Journal of Financial Economics, 64, 341–372.

    Article  Google Scholar 

  • Fan, J., & Gijbels, I. (1996). Local polynomial modelling and its applications. London: Chapman and Hall.

    MATH  Google Scholar 

  • Fengler, M. R. (2005). Semiparametric modeling of implied volatility. Berlin: Springer

    MATH  Google Scholar 

  • 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

    Google Scholar 

  • Härdle, W. (1990). Applied nonparametric regression. Econometric Society Monographs No. 19. London: Cambridge University Press

    Google Scholar 

  • Härdle, W., & Hlavka, Z. (2009). Dynamics of state price densities. Journal of Econometrics, 150(1), 1–15.

    Article  MathSciNet  Google Scholar 

  • Härdle, W., Müller, M., Sperlich, S., & Werwatz, A. (2004). Nonparametric and semiparametric models. Heidelberg: Springer.

    Book  MATH  Google Scholar 

  • Härdle, W., Okhrin, Y., & Wang, W. (2009). Uniform confidence for pricing kernels. SFB649DP2010-003. Econometric Theory (Submitted).

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Ingersoll, J.E. (1987). Theory of financial decision making. Rowman & Littlefield

    Google Scholar 

  • Jackwerth, J. C. (2000). Recovering risk aversion from option prices and realized returns. Review of Financial Studies, 13(2), 433–451.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, Q., & Racine, J. S. (2007). Nonparametric econometrics: Theory and practice. NJ: Princeton University Press.

    MATH  Google Scholar 

  • Linton, O., & Nielsen, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika, 82(1), 93–100.

    Article  MathSciNet  MATH  Google Scholar 

  • Lucas, R. E. (1978). Asset prices in an exchange economy. Econometrica, 46, 1429–1445.

    Article  MathSciNet  MATH  Google Scholar 

  • Mallows, C. L. (1973). Some comments on c p . Technometrics, 15, 661–675.

    MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Marron, J. S., & Nolan, D. (1988). Canonical kernels for density estimation. Statistics and Probability Letters, 7(3), 195–199.

    Article  MathSciNet  MATH  Google Scholar 

  • Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics, 4(1), 141–183.

    Article  MathSciNet  Google Scholar 

  • R. Merton (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125–183.

    Article  MATH  Google Scholar 

  • Müller, H. G. (1988). Nonparametric regression analysis of longitudinal data. Lecture Notes in Statistics (Vol. 46). New York: Springer.

    Google Scholar 

  • Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. Journal of Econometrics, 79(1), 147–168.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • Rubinstein. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics, 7(2), 407–425.

    Google Scholar 

  • Ruppert, D., & Wand, M. P. (1994). Multivariate locally weighted least squares regression. Annals of Statistics, 22(3), 1346–1370.

    Article  MathSciNet  MATH  Google Scholar 

  • Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions. J. Roy. Statist. Soc. Ser. B, 36, 111–147.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Ladislaus von Bortkiewicz Chair of Statistics, Humboldt-Universität zu Berlin, Spandauer Straße 1, 10178, Berlin, Germany

    Maria Grith

  2. Ladislaus von Bortkiewicz Chair of Statistics and CASE – Center for Applied Statistics and Economics, Humboldt-Universität zu Berlin, Spandauer Straße 1, 10178, Berlin, Germany

    Wolfgang Karl Härdle

  3. Graduate Institute of Statistics, CDA – Centre for Complex Data Analysis, National Central University, No. 300, Jhongda Rd., Jhongli City, Taoyuan County, 32001, Taiwan (R.O.C.)

    Wolfgang Karl Härdle

  4. Chair of Econometrics and CASE – Center for Applied Statistics and Economics, Humboldt-Universität zu Berlin, Spandauer Straße 1, 10178, Berlin, Germany

    Melanie Schienle

Authors
  1. Maria Grith
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  2. Wolfgang Karl Härdle
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  3. Melanie Schienle
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Corresponding author

Correspondence to Maria Grith .

Editor information

Editors and Affiliations

  1. , Risk Management Institute, National University of Singapore, 21 Heng Mui Keng Terrace, Level 4, Singapore, 119613, Singapore

    Jin-Chuan Duan

  2. L.v.Bortkiewicz Chair of Statistics, C.A.S.E. Centre f. Appl. Stat. & Econ., Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin, 10099, Germany

    Wolfgang Karl Härdle

  3. , Department of Computational and Data Sci, George Mason University, University Drive 4400, Fairfax, 22030, Virginia, USA

    James E. Gentle

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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

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