Estimation of the Stochastic Volatility of a Diffusion Process I. Comparison of Haar basis Estimator and some Kernel Estimators

Pierre Bertrand 1
1 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : Let ($X_t$) be a stochastic process satisfying $dX_t= b(t, X_t) ¸dt + \theta (t) ¸dW_t$, with a stochastic volatility $\theta (t)$ (thus few regular). We have a discretized observation at sampling times $t_i=i\Dr $i=1,...,N $. a We want to estimate the diffusion coefficient $\theta(t)$, called volatility in financial applications. We compare three families of non-parametric Estimators: Wavelets Estimator in the Haar basis, Moving Average Estimator and Centered Moving Average Estimator (CMAE). We emphasis dependence of the Estimators on the size of Window A. This is a new point of view. We prove punctual convergence of the three Estimators at the same rate. Then, we study Mean Integrated Square Error (MISE) as a function of Window A, we show it is smaller for Centered Moving Average Estimator (CMAE) than for Haar Basis Estimator in most circumstances. Furthermore, MISE(A) is a hardly oscillating function for Wavelets Estimators and not for Centerd Moving Average Estimator which should be considered more robust. We prove a Central Limit Theorem for Integrated Square Error (ISE) in the deterministic case. We conclude by numerical simulations which illustrate our theorical results. AMS Classifications. 62M 05, 60G 35.
Type de document :
Rapport
[Research Report] RR-2739, INRIA. 1996, pp.42
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Dernière modification le : samedi 27 janvier 2018 - 01:31:36
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Pierre Bertrand. Estimation of the Stochastic Volatility of a Diffusion Process I. Comparison of Haar basis Estimator and some Kernel Estimators. [Research Report] RR-2739, INRIA. 1996, pp.42. 〈inria-00073954〉

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