Some New Estimators of Integrated Volatility

We develop higher order accurate estimators of integrated volatility in a stochastic volatility models by using kernel smoothing method and using different weights to kernels. The weights have some relationship to moment problem. As the bandwidth of the kernel vanishes, an estimator of the instantaneous stochastic volatility is obtained. We also develop some new estimators based on smoothing splines.


Introduction
These days high frequency intradaily data of asset returns are available.Hence realized volatility which is a measure of the integrated volatility has received considerable interest in recent days' empirical finance.The realized volatility is defined as the sum of squared increments of returns.In order to improve the realized volatility, we estimate the integrated volatility by kernel method and spline method.We obtain higher order nonparametric estimator of kernel smooth integrated volatility.We simply take a kernel weighted average of the squared increments of return.The method to choose weight has relation to moment problem.

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This model has been studied in [1].The integrated volatility is defined as In stochastic volatility model, calculation of conditional cummulants of the integrated volatility conditioned on the initial value is enough to be able to compute European style options.
When the Levy process is an inverse Gaussian process with parameters   ,   , the cummulant functions of IGOU process are given by We assume that the parameters of the Levy process are known.We study estimation of integrated volatility by kernel method.Observe that the realized volatility estimator is a histogram estimator of the integrated volatility where is the binwidth.Here we extend the realized volatility to include kernel weights.We take kernel weighted average of the squared increments of the observations.Our estimator includes as a special case the rolling window estimator of [2] and [3], the kernel can be chosen to satisfy the weighting schemes proposed there while the bandwidth determines the laglength.The paper also generalizes [4,5] to include weighting.The weighting scheme is jointly determined by the choice of > 0 h K and .With a two-sided kernel, kernel volatility (KV) takes a weighted average of the instantaneous volatility h over the whole sample period.We will choose one-sided kernel.
For fixed , KV gives a weighted measure of the integrated volatility.As , we recover the instantaneous volatility at any point of continuity of We have the following assumptions about the kernel.Consider a continuously differentiable kernel with shrinking bandwidth .Let For example consider the Epanechikov kernel and the kernel suggested in [6]     We consider kernel weighted average of the quadratic variation.The kernel estimators converge to the integrated variance as the bandwith vanishes.In order to improve the rate of convergence of kernel estimators, we consider its relation to a moment problem.h For simplicity of notation, we will denote Integrated volatility has to be estimated on the basis of discrete observations of the process   t Y at times The realized volatility is defined as The following theorem is well known in the literature, see [1].
Theorem 2.1 In order to improve the realized volatility with faster rate of convergence we follow the following path.The ideas are used in [7] for parametric drift estimation in diffusion processes.Define a weighted sum of squares where is a weight function.
General weighted kernel volatility (KV) is defined as and obtained the rate of convergence along with asymptotic distribution of the estimator .

 , , n T F
Our plan is to improve the rate of convergence by using appropriate weights for the kernel.With , the simple symmetric KV is defined as With the weight function the weighted symmetric KV is defined as Note that estimator (2.16) is analogous to the trapezoidal rule in numerical integration.One can instead use the midpoint rule to define another estimator We can use the Simpson's rule to define another estimator which is a convex combination of the midpoint estimator and the trapezoidal estimator where is a probability mass function of a discrete random variable on with Denote the -th moment of the random variable as .
If one chooses the probability distribution as uniform distribution for which the moments are a harmonic sequence there is no change in rate of convergence than second order.If one can construct a probability distribution for which the harmonic sequence is truncated at a point, then there is a rate of convergence improvement at the point of truncation.Given a positive integer , construct a probability mass function on such that Neither the probabilities j p nor the atoms, j s , of the distribution are specified in advance.
This problem is related to the truncated Hausdorff moment problem.I obtain examples of such probability distributions and use them to get higher order accurate (up to sixth order) KVs.
The order of approximation error (rate of convergence) The moment based estimators of integrated volatility which are given by (2.24)I construct probability distributions satisfying these moment conditions and obtain KVs of the rate of convergence up to order 6.
Theorem 2.2 Assume that the kernel is sufficiently smooth, continuously differentiable of order 6. 3

Probabilities  
This gives (d).
Probabilities   for which   gives (f).
Probabilities   for which   Copyright © 2011 SciRes.OJS for which  
Probabilities   produce the symmetric KV given by

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e realized volatility estimator of integrated volatility, we use an altern method of splines, see [9], [10] and [1 ste Since these are based on analysis of variance for diffusion models, we call it DAN DANOVA stands for ANOVA for Diffusions.
In the stochastic volatility model, the log-price

Spline Estimators
In order to improve th ative method, the 1].This is the first p towards the use of splines for volatility estimation.
OVA models.* = log y S with S being the asset price, follows where  and  are assumed to be independent of the standard Brownian motion an. Suppose one is interested in estim g the actual called the actual me atin using intra-observations.A natural e rea vol given by m lized h atility Thus the realized volatility is given by , realized volatility converges in to the integ atility.We consider th fixed The realized variance is a quadratic form.
that the realized volatility is based on first order nce.We introduce some new estimators: The above estimator is based on second order difference.

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The geometric mean based sym KV (which is based on ideas of partial autoco- To improve this error bound, we introduce the lagestimator k of the intercept  . 20)