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Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approach

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  • Yacine Ait-Sahalia

Abstract

When a continuous-time diffusion is observed only at discrete dates, not necessarily close together, the likelihood function of the observations is in most cases not explicitly computable. Researchers have relied on simulations of sample paths in between the observations points, or numerical solutions of partial differential equations, to obtain estimates of the function to be maximized. By contrast, we construct a sequence of fully explicit functions which we show converge under very general conditions, including non-ergodicity, to the true (but unknown) likelihood function of the discretely-sampled diffusion. We document that the rate of convergence of the sequence is extremely fast for a number of examples relevant in finance. We then show that maximizing the sequence instead of the true function results in an estimator which converges to the true maximum-likelihood estimator and shares its asymptotic properties of consistency, asymptotic normality and efficiency. Applications to the valuation of derivative securities are also discussed.

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Bibliographic Info

Paper provided by National Bureau of Economic Research, Inc in its series NBER Technical Working Papers with number 0222.

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Date of creation: Feb 1998
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Publication status: published as "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approach", Econometrica, Vol. 70, pp. 223-262, (2002).
Handle: RePEc:nbr:nberte:0222

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  9. Yacine Ait-Sahalia, 1995. "Nonparametric Pricing of Interest Rate Derivative Securities," NBER Working Papers 5345, National Bureau of Economic Research, Inc.
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Cited by:
  1. Siddhartha Chib & Michael K Pitt & Neil Shephard, 2004. "Likelihood based inference for diffusion driven models," OFRC Working Papers Series 2004fe17, Oxford Financial Research Centre.
  2. Pascal St-Amour, 2005. "Direct Preference for Wealth in Aggregate Household Portfolio," Cahiers de Recherches Economiques du Département d'Econométrie et d'Economie politique (DEEP) 05.04, Université de Lausanne, Faculté des HEC, DEEP.
  3. Eric Ghysels & Jean-Pierre Florens & Mikhail Chernov & Marine Carrasco, 2003. "Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions," CIRANO Working Papers 2003s-02, CIRANO.
  4. Nour Meddahi & Éric Renault, 2000. "Temporal Aggregation of Volatility Models," CIRANO Working Papers 2000s-22, CIRANO.
  5. Pascal St-Amour, 2004. "Ratchet vs Blasé Investors and Asset Markets," CIRANO Working Papers 2004s-11, CIRANO.
  6. Das, Sanjiv Ranjan & Uppal, Raman, 2002. "Systemic Risk and International Portfolio Choice," CEPR Discussion Papers 3305, C.E.P.R. Discussion Papers.
  7. Bollerslev, Tim & Zhou, Hao, 2002. "Estimating stochastic volatility diffusion using conditional moments of integrated volatility," Journal of Econometrics, Elsevier, vol. 109(1), pages 33-65, July.
  8. Sergio Pastorello & Valentin Patilea & Éric Renault, 2003. "Iterative and Recursive Estimation in Structural Non-Adaptive Models," CIRANO Working Papers 2003s-08, CIRANO.
  9. Paulo M. M. Rodrigues & Antonio Rubia, 2004. "On the Small Sample Properties of Dickey Fuller and Maximum Likelihood Unit Root Tests on Discrete-Sampled Short-Term Interest Rates," Econometrics 0405004, EconWPA.
  10. Alexander David & Pietro Veronesi, 1998. "Option Prices with Uncertain Fundamentals: Theory and Evidence on the Dynamics of Implied Volatilities," CRSP working papers 485, Center for Research in Security Prices, Graduate School of Business, University of Chicago.
  11. Alexander David & Pietro Varonesi, 1999. "Option prices with uncertain fundamentals theory and evidence on the dynamics of implied volatilities," Finance and Economics Discussion Series 1999-47, Board of Governors of the Federal Reserve System (U.S.).
  12. Peter C.B. Phillips & Joon Y. Park, 1998. "Nonstationary Density Estimation and Kernel Autoregression," Cowles Foundation Discussion Papers 1181, Cowles Foundation for Research in Economics, Yale University.
  13. Michael W. Brandt & Pedro Santa-Clara, 2001. "Simulated Likelihood Estimation of Diffusions with an Application to Exchange Rate Dynamics in Incomplete Markets," NBER Technical Working Papers 0274, National Bureau of Economic Research, Inc.

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