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Accurate parameter estimation for coupled stochastic dynamics

Pages: 44 - 53, Issue Special, September 2009

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Robert Azencott - Department of Mathematics, University of Houston and Ecole Normale Sup. France, United States (email)
Yutheeka Gadhyan - Department of Mathematics, University of Houston, United States (email)

Abstract: We develop and implement an efficient algorithm to estimate the 5 parameters of Heston's model from arbitrary given series of joint observations for the stock price and volatility. We consider the time interval T separating two observations to be unknown and estimate it from the data, thereby estimating 6 parameters with a clear gain in fit accuracy. We compare the maximum likelihood parameter estimates based on an Euler discretization scheme to analogous estimates derived from the more accurate Milstein discretization scheme; we derive explicit conditions under which the two set of estimates are asymptotically equivalent, and we compute the asymptotic distribution of the difference of the two set of estimates. We show that parameter estimates derived from the Euler scheme by constrained optimization of the approximate maximum likelihood are consistent, and we compute their asymptotic variances. Numerically, our estimation algorithms are easy to implement,and require only very moderate amounts of CPU. We have performed extensive simulations which show that for standard range of the process parameters, the empirical variances of our parameter estimates are correctly approximated by their theoretical asymptotic variances.

Keywords:  Stochastic differential equations, parameter estimation, maximum likelihood estimators, Heston model, asset price dynamics, volatility dynamics
Mathematics Subject Classification:  Primary: 60H10, 60H35; Secondary: 62F12, 91B28

Received: August 2008;      Revised: March 2009;      Published: September 2009.