Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed‐form Approximation Approach

When a continuous-time diffusion is observed only at discrete dates, in most cases the transition distribution and hence the likelihood function of the observations is not explicitly computable. Using Hermite polynomials, I construct an explicit sequence of closed-form functions and show that it converges to the true (but unknown) likelihood function. I document that the approximation is very accurate and prove that maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and shares its asymptotic properties. Monte Carlo evidence reveals that this method outperforms other approximation schemes in situations relevant for financial models.

[1]  Developments in Hermite Polynomials , 1927 .

[2]  H. McKean.  Elementary solutions for certain parabolic partial differential equations , 1956 .

[3]  John S. White.  THE LIMITING DISTRIBUTION OF THE SERIAL CORRELATION COEFFICIENT IN THE EXPLOSIVE CASE , 1958 .

[4]  T. W. Anderson.  On Asymptotic Distributions of Estimates of Parameters of Stochastic Difference Equations , 1959 .

[5]  Patrick Billingsley,et al.  Statistical inference for Markov processes , 1961 .

[6]  E. K. Wong.  The Construction of a Class of Stationary Markoff Processes , 1964 .

[7]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[8]  E. Renshaw,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS , 1974 .

[9]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .

[10]  R. C. Merton,et al.  On Estimating the Expected Return on the Market: An Exploratory Investigation , 1980 .

[11]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[12]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[13]  Jdh.  Statistical Inference for Stochastic Processes. By I. V. Basawa and B. L. S. Prakasa Rao. Pp. 435. Academic Press, London. £26.80. , 1981 .

[14]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[15]  Ishwar V. Basawa,et al.  Asymptotic optimal inference for non-ergodic models , 1983 .

[16]  C. Gouriéroux,et al.  PSEUDO MAXIMUM LIKELIHOOD METHODS: THEORY , 1984 .

[17]  L. Rogers.  Smooth Transition Densities for One‐Dimensional Diffusions , 1985 .

[18]  D. Florens-zmirou,et al.  Estimation of the coefficients of a diffusion from discrete observations , 1986 .

[19]  A. Gallant,et al.  Semi-nonparametric Maximum Likelihood Estimation , 1987 .

[20]  A. Lo.  Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data , 1986, Econometric Theory.

[21]  L. Hansen,et al.  Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes , 1993 .

[22]  C. Sims,et al.  Estimation of continuous-time models in finance , 1994 .

[23]  Anders Martin-Löf.  On some classes of series used in mathematical statistics , 1994 .

[24]  L. Hansen,et al.  Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes , 1993 .

[25]  Yacine Aït-Sahalia.  Nonparametric Pricing of Interest Rate Derivative Securities , 1995 .

[26]  A. Gallant,et al.  Which Moments to Match? , 1995, Econometric Theory.

[27]  Yacine Ait-Sahalia.  Testing Continuous-Time Models of the Spot Interest Rate , 1995 .

[28]  P. Jeganathan.  Some Aspects of Asymptotic Theory with Applications to Time Series Models , 1995, Econometric Theory.

[29]  A. Pedersen.  A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations , 1995 .

[30]  J. Cox.  The Constant Elasticity of Variance Option Pricing Model , 1996 .

[31]  Richard Stanton.  A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk , 1997 .

[32]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[33]  L. Hansen,et al.  Spectral methods for identifying scalar diffusions , 1998 .

[34]  Danielle Florens,et al.  Estimation of the Diffusion Coefficient from Crossings , 1998 .

[35]  Michael Sørensen,et al.  Estimating equations based on eigenfunctions for a discretely observed diffusion process , 1999 .

[36]  Yacine Aït-Sahalia.  Transition Densities for Interest Rate and Other Nonlinear Diffusions , 1999 .

[37]  Bjørn Eraker.  MCMC Analysis of Diffusion Models With Application to Finance , 2001 .

[38]  P. Glynn,et al.  Estimation of Continuous-Time Markov Processes Sampled at Random Time Intervals , 2004 .

[39]  T. Alderweireld,et al.  A Theory for the Term Structure of Interest Rates , 2004, cond-mat/0405293.

[40]  Christopher J. BISHOPAbstra,et al.  Orthogonal Functions , 2022 .