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Accuracy of maximum likelihood parameter estimators for Heston stochastic volatility SDE. (English) Zbl 1328.82037

Summary: We study approximate maximum likelihood estimators (MLEs) for the parameters of the widely used Heston Stock price and volatility stochastic differential equations (SDEs). We compute explicit closed form estimators maximizing the discretized log-likelihood of \(N\) observations recorded at times \(T,2T,\dots, NT\). We compute the asymptotic biases of these parameter estimators for \(T\) fixed and \(N \to \infty\), as well as the rate at which these biases vanish when \(T \to 0\). We determine asymptotically consistent explicit modifications of these MLEs. For the Heston volatility SDE, we identify a canonical form determined by two canonical parameters \(0 < \omega < 1\) and \(\zeta > 1/2\) which are explicit functions of the original SDE parameters. We analyze theoretically the asymptotic distribution of the MLEs and of their consistent modifications, and we outline their concrete speeds of convergence by numerical simulations. We clarify in terms of \(\zeta\) the precise dichotomy between asymptotic normality and attraction by stable like distributions with heavy tails. We illustrate numerical model fitting for Heston SDEs by two concrete examples, one for daily data and one for intraday data, both with moderate values of \(N\).

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35R60 PDEs with randomness, stochastic partial differential equations
91B24 Microeconomic theory (price theory and economic markets)
91G60 Numerical methods (including Monte Carlo methods)
35B45 A priori estimates in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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