Abstract
We consider nonparametric spot volatility estimation for diffusion models with discrete high frequency observations. Our estimator is carried out in two steps. First, using the local average of the range-based variance, we propose a crude estimator of the spot volatility. Second, we use usual nonparametric kernel smoothing to reconstruct the volatility function from the crude estimator. By inference, we find such a double smoothing operation can effectively reduce the estimation error.
Highlights
IntroductionStochastic differential equations have been widely studied by many researchers and some interesting results have appeared in the literature (see [1,2,3,4,5] and references therein)
In recent years, stochastic differential equations have been widely studied by many researchers and some interesting results have appeared in the literature
The double smoothing method can replicate the actual features of the volatility function more
Summary
Stochastic differential equations have been widely studied by many researchers and some interesting results have appeared in the literature (see [1,2,3,4,5] and references therein). Constraining a kernel function and choosing appropriate bandwidth parameters, Kristensen [11] proposed a filtered kernel-based spot volatility estimator for time-dependent diffusion models: β2τ. Under the coexistence of market microstructure noise and multiple transactions, with high frequency data, Liu et al [18] gave a spot volatility estimation procedure by taking the sum from the local range increments and discussed the estimator’s consistency and asymptotic normality. Bandi and Phillips [28] used double smoothing technique to estimate drift and diffusion function and proposed their robust and reliable spot volatility estimator in which they took both infill asymptotics and long span one into account. The double smoothing technique is used to estimate the spot volatility of time-dependent diffusion models, and the consistency and asymptotic normality of the estimator are obtained. With high frequency data at hand, the range-based estimator is more precise than the return-based one
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