Estimation of the Stochastic Volatility of a Diffusion Process I. Comparison of Haar basis Estimator and some Kernel Estimators - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Reports (Research Report) Year : 1996

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

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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Dates and versions

inria-00073954 , version 1 (24-05-2006)

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  • HAL Id : inria-00073954 , version 1

Cite

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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