Maximum Likelihood Estimation Of Discretely Sampled Diffusions
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Author | : Yacine Ait-Sahalia |
Publisher | : |
Total Pages | : 0 |
Release | : 2002 |
Genre | : |
ISBN | : |
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.
Author | : Yacine Ait-Sahalia |
Publisher | : |
Total Pages | : 52 |
Release | : 2000 |
Genre | : |
ISBN | : |
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 observation 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.
Author | : Yacine Aït-Sahalia |
Publisher | : |
Total Pages | : 64 |
Release | : 1998 |
Genre | : Diffusion processes |
ISBN | : |
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.
Author | : Gurdip Bakshi |
Publisher | : |
Total Pages | : 18 |
Release | : 2004 |
Genre | : |
ISBN | : |
This paper provides a closed-form density approximation when the underlying state variable is a one-dimensional diffusion. Building on Ait-Sahalia (2002), we show that our refinement is applicable under a wide class of drift and diffusion functions. In addition, it facilitates the maximum likelihood estimation of discretely sampled diffusion models of short interest-rate or stock volatility with unknown conditional densities. Our interest-rate examples demonstrate that the analytical approximation is accurate.
Author | : Rolf Poulsen |
Publisher | : |
Total Pages | : 34 |
Release | : 1999 |
Genre | : |
ISBN | : |
Author | : Yacine Aït-Sahalia |
Publisher | : |
Total Pages | : 80 |
Release | : 2001 |
Genre | : Economics |
ISBN | : |
High-frequency financial data are not only discretely sampled in time but the time separating successive observations is often random. We analyze the consequences of this dual feature of the data when estimating a continuous-time model. In particular, we measure the additional effects of the randomness of the sampling intervals over and beyond those due to the discreteness of the data. We also examine the effect of simply ignoring the sampling randomness. We find that in many situations the randomness of the sampling has a larger impact than the discreteness of the data.
Author | : Peter Honoré |
Publisher | : |
Total Pages | : |
Release | : 1997 |
Genre | : |
ISBN | : |
Author | : Michael W. Brandt |
Publisher | : |
Total Pages | : 55 |
Release | : 2001 |
Genre | : Diffusion process |
ISBN | : |
Abstract: We present an econometric method for estimating the parameters of a diffusion model from discretely sampled data. The estimator is transparent, adaptive, and inherits the asymptotic properties of the generally unattainable maximum likelihood estimator. We use this method to estimate a new continuous-time model of the Joint dynamics of interest rates in two countries and the exchange rate between the two currencies. The model allows financial markets to be incomplete and specifies the degree of incompleteness as a stochastic process. Our empirical results offer several new insights into the dynamics of exchange rates
Author | : William Greene |
Publisher | : Emerald Group Publishing |
Total Pages | : 371 |
Release | : 2010-12-03 |
Genre | : Business & Economics |
ISBN | : 0857241508 |
This collection of methodological developments and applications of simulation-based methods were presented at a workshop at Louisiana State University in November, 2009. Topics include: extensions of the GHK simulator; maximum-simulated likelihood; composite marginal likelihood; and modelling and forecasting volatility in a bayesian approach.
Author | : Christiane Fuchs |
Publisher | : Springer Science & Business Media |
Total Pages | : 439 |
Release | : 2013-01-18 |
Genre | : Mathematics |
ISBN | : 3642259693 |
Diffusion processes are a promising instrument for realistically modelling the time-continuous evolution of phenomena not only in the natural sciences but also in finance and economics. Their mathematical theory, however, is challenging, and hence diffusion modelling is often carried out incorrectly, and the according statistical inference is considered almost exclusively by theoreticians. This book explains both topics in an illustrative way which also addresses practitioners. It provides a complete overview of the current state of research and presents important, novel insights. The theory is demonstrated using real data applications.