Statistical Science

A Selective Overview of Nonparametric Methods in Financial Econometrics

Jianqing Fan

Full-text: Open access


This paper gives a brief overview of the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inference for instantaneous returns and volatility functions of time-homogeneous and time-dependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline the main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.

Article information

Statist. Sci., Volume 20, Number 4 (2005), 317-337.

First available in Project Euclid: 12 January 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Asset pricing diffusion drift GLR tests simulations state price density time-inhomogeneous model transition density volatility


Fan, Jianqing. A Selective Overview of Nonparametric Methods in Financial Econometrics. Statist. Sci. 20 (2005), no. 4, 317--337. doi:10.1214/088342305000000412.

Export citation


  • Ahn, D. H. and Gao, B. (1999). A parametric nonlinear model of term structure dynamics. Review of Financial Studies 12 721--762.
  • Aït-Sahalia, Y. (1996). Nonparametric pricing of interest rate derivative securities. Econometrica 64 527--560.
  • Aït-Sahalia, Y. (1996). Testing continuous-time models of the spot interest rate. Review of Financial Studies 9 385--426.
  • Aït-Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions. J. Finance 54 1361--1395.
  • Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 223--262.
  • Aït-Sahalia, Y. and Duarte, J. (2003). Nonparametric option pricing under shape restrictions. J. Econometrics 116 9--47.
  • Aït-Sahalia, Y. and Lo, A. W. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. J. Finance 53 499--547.
  • Aït-Sahalia, Y. and Lo, A. W. (2000). Nonparametric risk management and implied risk aversion. J. Econometrics 94 9--51.
  • Aït-Sahalia, Y. and Mykland, P. (2003). The effects of random and discrete sampling when estimating continuous-time diffusions. Econometrica 71 483--549.
  • Aït-Sahalia, Y. and Mykland, P. (2004). Estimators of diffusions with randomly spaced discrete observations: A general theory. Ann. Statist. 32 2186--2222.
  • Arapis, M. and Gao, J. (2004). Nonparametric kernel estimation and testing in continuous-time financial econometrics. Unpublished manuscript.
  • Arfi, M. (1998). Non-parametric variance estimation from ergodic samples. Scand. J. Statist. 25 225--234.
  • Bandi, F. (2002). Short-term interest rate dynamics: A spatial approach. J. Financial Economics 65 73--110.
  • Bandi, F. and Nguyen, T. (1999). Fully nonparametric estimators for diffusions: A small sample analysis. Unpublished manuscript.
  • Bandi, F. and Phillips, P. C. B. (2003). Fully nonparametric estimation of scalar diffusion models. Econometrica 71 241--283.
  • Banon, G. (1977). Estimation non paramétrique de densité de probabilité pour les processus de Markov. Thése, Univ. Paul Sabatier de Toulouse, France.
  • Banon, G. (1978). Nonparametric identification for diffusion processes. SIAM J. Control Optim. 16 380--395.
  • Banon, G. and Nguyen, H. T. (1981). Recursive estimation in diffusion models. SIAM J. Control Optim. 19 676--685.
  • Barndoff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein--Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167--241.
  • Bingham, N. H. and Kiesel, R. (1998). Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives. Springer, New York.
  • Black, F., Derman, E. and Toy, W. (1990). A one-factor model of interest rates and its application to Treasury bond options. Financial Analysts Journal 46(1) 33--39.
  • Black, F. and Karasinski, P. (1991). Bond and option pricing when short rates are lognormal. Financial Analysts Journal 47(4) 52--59.
  • Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Economy 81 637--654.
  • Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71 353--360.
  • Breeden, D. and Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. J. Business 51 621--651.
  • Cai, Z., Fan, J. and Yao, Q. (2000). Functional-coefficient regression models for nonlinear time series. J. Amer. Statist. Assoc. 95 941--956.
  • Cai, Z. and Hong, Y. (2003). Nonparametric methods in continuous-time finance: A selective review. In Recent Advances and Trends in Nonparametric Statistics (M. G. Akritas and D. N. Politis, eds.) 283--302. North-Holland, Amsterdam.
  • Campbell, J. Y., Lo, A. W. and MacKinlay, A. C. (1997). The Econometrics of Financial Markets. Princeton Univ. Press.
  • Chan, K. C., Karolyi, G. A., Longstaff, F. A. and Sanders, A. B. (1992). An empirical comparison of alternative models of the short-term interest rate. J. Finance 47 1209--1227.
  • Chapman, D. A. and Pearson, N. D. (2000). Is the short rate drift actually nonlinear? J. Finance 55 355--388.
  • Chen, R. and Tsay, R. S. (1993). Functional-coefficient autoregressive models. J. Amer. Statist. Assoc. 88 298--308.
  • Chen, S. X. (2005). Nonparametric estimation of expected shortfall. Econometric Theory. To appear.
  • Chen, S. X. and Tang, C. Y. (2005). Nonparametric inference of value-at-risk for dependent financial returns. J. Financial Econometrics 3 227--255.
  • Chen, X. and Ludvigson, S. (2003). Land of Addicts? An empirical investigation of habit-based asset pricing model. Unpublished manuscript.
  • Claeskens, G. and Hall, P. (2002). Effect of dependence on stochastic measures of accuracy of density estimators. Ann. Statist. 30 431--454.
  • Cochrane, J. H. (2001). Asset Pricing. Princeton Univ. Press.
  • Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53 385--407.
  • Cox, J. C. and Ross, S. (1976). The valuation of options for alternative stochastic processes. J. Financial Economics 3 145--166.
  • Dacunha-Castelle, D. and Florens, D. (1986). Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19 263--284.
  • Dalalyan, A. S. and Kutoyants, Y. A. (2002). Asymptotically efficient trend coefficient estimation for ergodic diffusion. Math. Methods Statist. 11 402--427.
  • Dalalyan, A. S. and Kutoyants, Y. A. (2003). Asymptotically efficient estimation of the derivative of the invariant density. Stat. Inference Stoch. Process. 6 89--107.
  • Duffie, D. (2001). Dynamic Asset Pricing Theory, 3rd ed. Princeton Univ. Press.
  • Egorov, A. V., Li, H. and Xu, Y. (2003). Maximum likelihood estimation of time-inhomogeneous diffusions. J. Econometrics 114 107--139.
  • Engel, K.-J. and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin.
  • Engelbert, H. J. and Schmidt, W. (1984). On one-dimensional stochastic differential equations with generalized drift. Stochastic Differential Systems. Lecture Notes in Control and Inform. Sci. 69 143--155. Springer, Berlin.
  • Fan, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87 998--1004.
  • Fan, J. and Gu, J. (2003). Semiparametric estimation of value-at-risk. Econom. J. 6 261--290.
  • Fan, J., Jiang, J., Zhang, C. and Zhou, Z. (2003). Time-dependent diffusion models for term structure dynamics. Statist. Sinica 13 965--992.
  • Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645--660.
  • Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • Fan, J., Yao, Q. and Tong, H. (1996). Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83 189--206.
  • Fan, J. and Yim, T. H. (2004). A crossvalidation method for estimating conditional densities. Biometrika 91 819--834.
  • Fan, J. and Zhang, C. (2003). A re-examination of diffusion estimators with applications to financial model validation. J. Amer. Statist. Assoc. 98 118--134.
  • Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153--193.
  • Florens-Zmirou, D. (1993). On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30 790--804.
  • Gao, J. and King, M. (2004). Adaptive testing in continuous-time diffusion models. Econometric Theory 20 844--882.
  • Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 29 119--151.
  • Gobet, E. (2002). LAN property for ergodic diffusions with discrete observations. Ann. Inst. H. Poincaré Probab. Statist. 38 711--737.
  • Gobet, E., Hoffmann, M. and Reiss, M. (2004). Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Statist. 32 2223--2253.
  • Gouriéroux, C. and Jasiak, J. (2001). Financial Econometrics: Problems, Models, and Methods. Princeton Univ. Press.
  • Gouriéroux, C., Monfort, A. and Renault, E. (1993). Indirect inference. J. Appl. Econometrics 8 suppl. S85--S118.
  • Hall, P. and Carroll, R. J. (1989). Variance function estimation in regression: The effect of estimating the mean. J. Roy. Statist. Soc. Ser. B 51 3--14.
  • Hall, P., Racine, J. and Li, Q. (2004). Cross-validation and the estimation of conditional probability densities. J. Amer. Statist. Assoc. 99 1015--1026.
  • Hall, P., Wolff, R. C. L. and Yao, Q. (1999). Methods for estimating a conditional distribution function. J. Amer. Statist. Assoc. 94 154--163.
  • Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50 1029--1054.
  • Hansen, L. P. and Scheinkman, J. A. (1995). Back to the future: Generating moment implications for continuous-time Markov processes. Econometrica 63 767--804.
  • Hansen, L. P., Scheinkman, J. A. and Touzi, N. (1998). Spectral methods for identifying scalar diffusions. J. Econometrics 86 1--32.
  • Härdle, W., Herwartz, H. and Spokoiny, V. (2003). Time inhomogeneous multiple volatility modelling. J. Financial Econometrics 1 55--95.
  • Härdle, W. and Tsybakov, A. B. (1997). Local polynomial estimators of the volatility function in nonparametric autoregression. J. Econometrics 81 223--242.
  • Härdle, W. and Yatchew, A. (2002). Dynamic nonparametric state price density estimation using constrained least-squares and the bootstrap. Discussion paper 16, Quantification and Simulation of Economics Processes, Humboldt-Universität zu Berlin.
  • Harrison, J. M. and Kreps, D. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 2 381--408.
  • Hart, J. D. (1996). Some automated methods of smoothing time-dependent data. Nonparametr. Statist. 6 115--142.
  • Hart, J. D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer, New York.
  • Hastie, T. J. and Tibshirani, R. J. (1993). Varying-coefficient models (with discussion). J. Roy. Statist. Soc. Ser. B. 55 757--796.
  • Ho, T. S. Y. and Lee, S.-B. (1986). Term structure movements and pricing interest rate contingent claims. J. Finance 41 1011--1029.
  • Hong, Y. and Lee, T.-H. (2003). Inference on predictability of foreign exchange rates via generalized spectrum and nonlinear time series models. Review of Economics and Statistics 85 1048--1062.
  • Hong, Y. and Li, H. (2005). Nonparametric specification testing for continuous-time models with applications to term structure of interest rates. Review of Financial Studies 18 37--84.
  • Hull, J. (2003). Options, Futures, and Other Derivatives, 5th ed. Prentice Hall, Upper Saddle River, NJ.
  • Hull, J. and White, A. (1990). Pricing interest-rate-derivative securities. Review of Financial Studies 3 573--592.
  • Itô, K. (1942). Differential equations determining Markov processes. Zenkoku Shijo Sugaku Danwakai 244 1352--1400. (In Japanese.)
  • Itô, K. (1946). On a stochastic integral equation. Proc. Japan Acad. 22 32--35.
  • Jorion, P. (2000). Value at Risk: The New Benchmark for Managing Financial Risk, 2nd ed. McGraw--Hill, New York.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist. 24 211--229.
  • Kessler, M. and Sørensen, M. (1999). Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli 5 299--314.
  • Kloeden, P. E., Platen, E., Schurz, H. and Sørensen, M. (1996). On effects of discretization on estimators of drift parameters for diffusion processes. J. Appl. Probab. 33 1061--1076.
  • Kutoyants, Y. A. (1998). Efficient density estimation for ergodic diffusion processes. Stat. Inference Stoch. Process. 1 131--155.
  • Mercurio, D. and Spokoiny, V. (2004). Statistical inference for time-inhomogeneous volatility models. Ann. Statist. 32 577--602.
  • Merton, R. (1973). Theory of rational option pricing. Bell J. Econom. Management Sci. 4 141--183.
  • Müller, H.-G. and Stadtmüller, U. (1987). Estimation of heteroscedasticity in regression analysis. Ann. Statist. 15 610--625.
  • Osborne, M. F. M. (1959). Brownian motion in the stock market. Operations Res. 7 145--173.
  • Pham, D. T. (1981). Nonparametric estimation of the drift coefficient in the diffusion equation. Math. Operationsforsch. Statist. Ser. Statist. 12 61--73.
  • Prakasa Rao, B. L. S. (1985). Estimation of the drift for diffusion process. Statistics 16 263--275.
  • Rosenblatt, M. (1970). Density estimates and Markov sequences. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 199--213. Cambridge Univ. Press.
  • Roussas, G. G. (1969). Nonparametric estimation in Markov processes. Ann. Inst. Statist. Math. 21 73--87.
  • Roussas, G. G. (1969). Nonparametric estimation of the transition distribution function of a Markov process. Ann. Math. Statist. 40 1386--1400.
  • Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scand. J. Statist. 9 65--78.
  • Ruppert, D., Wand, M. P., Holst, U. and Hössjer, O. (1997). Local polynomial variance function estimation. Technometrics 39 262--273.
  • Schurz, H. (2000). Numerical analysis of stochastic differential equations without tears. In Handbook of Stochastic Analysis and Applications (D. Kannan and V. Lakshmikantham, eds.) 237--359. Dekker, New York.
  • Shephard, N., ed. (2005). Stochastic Volatility: Selected Readings. Oxford Univ. Press.
  • Simonoff, J. S. (1996). Smoothing Methods in Statistics. Springer, New York.
  • Spokoiny, V. (2000). Adaptive drift estimation for nonparametric diffusion model. Ann. Statist. 28 815--836.
  • Stanton, R. (1997). A nonparametric model of term structure dynamics and the market price of interest rate risk. J. Finance 52 1973--2002.
  • Steele, J. M. (2001). Stochastic Calculus and Financial Applications. Springer, New York.
  • Vasicek, O. A. (1977). An equilibrium characterization of the term structure. J. Financial Economics 5 177--188.
  • Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.
  • Wang, Y. (2002). Asymptotic nonequivalence of GARCH models and diffusions. Ann. Statist. 30 754--783.
  • Yoshida, N. (1992). Estimation for diffusion processes from discrete observations. J. Multivariate Anal. 41 220--242.