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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.
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Submitted on : Wednesday, May 24, 2006 - 2:09:50 PM
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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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