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differential equations
Accurate parameter estimation for coupled stochastic dynamics
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.