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

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Yacine Ait-Sahalia

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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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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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Handle: RePEc:nbr:nberte:0222

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Find related papers by JEL classification:
G12 - Financial Economics - - General Financial Markets - - - Asset Pricing
C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Estimation

References listed on IDEAS
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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. 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. [Downloadable!]
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  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. [Downloadable!]
  3. Peter C.B. Phillips & Joon Y. Park, 1998. "Nonstationary Density Estimation and Kernel Autoregression," Cowles Foundation Discussion Papers 1181, Cowles Foundation, Yale University. [Downloadable!]
  4. Carrasco, Marine & Chernov, Mikhaël & Florens, Jean-Pierre & Ghysels, Eric, 2000. "Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions," IDEI Working Papers 116, Institut d'Économie Industrielle (IDEI), Toulouse, revised 2002. [Downloadable!]
    Other versions:
  5. Siddhartha Chib & Michael K Pitt & Neil Shephard, 2004. "Likelihood based inference for diffusion driven models," Economics Papers 2004-W20, Economics Group, Nuffield College, University of Oxford. [Downloadable!]
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  6. Pascal St-Amour, 2004. "Ratchet vs Blasé Investors and Asset Markets," CIRANO Working Papers 2004s-11, CIRANO. [Downloadable!]
  7. Nour Meddahi & Éric Renault, 2000. "Temporal Aggregation of Volatility Models," CIRANO Working Papers 2000s-22, CIRANO. [Downloadable!]
  8. 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.). [Downloadable!]
  9. 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. [Downloadable!] (restricted)
  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. [Downloadable!]
  11. Sergio Pastorello & Valentin Patilea & Éric Renault, 2003. "Iterative and Recursive Estimation in Structural Non-Adaptive Models," CIRANO Working Papers 2003s-08, CIRANO. [Downloadable!]
  12. Tim Bollerslev & Hao Zhou, 2001. "Estimating stochastic volatility diffusion using conditional moments of integrated volatility," Finance and Economics Discussion Series 2001-49, Board of Governors of the Federal Reserve System (U.S.). [Downloadable!]
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  13. Das, Sanjiv Ranjan & Uppal, Raman, 2002. "Systemic Risk and International Portfolio Choice," CEPR Discussion Papers 3305, C.E.P.R. Discussion Papers. [Downloadable!] (restricted)
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