1. 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.
2. Weighted Kernel Estimators
Consider the stochastic volatility model with asset price process
and volatility process
satisfying the stochastic differential equations
(2.1)
(2.2)
where
is a standard Brownian,
, a subordinator, that is, a Levy process with only positive jumps, and
are the parameters. If
the model can express leverage-effect. We denote by
, supported by
, the Levy measure of
and assume that ![](https://www.scirp.org/html/4-1240011\899a5328-8d7d-4bca-a191-9a45b1e3baea.jpg)
This model has been studied in [1]. The integrated volatility is defined as
(2.3)
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
![](https://www.scirp.org/html/4-1240011\93744ad8-3597-4aed-8346-ddb62654343d.jpg)
![](https://www.scirp.org/html/4-1240011\f885fc40-5c61-47d4-98f7-2f91914d30fd.jpg)
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
and
. With a two-sided kernel, kernel volatility (KV) takes a weighted average of the instantaneous volatility 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
(2.4)
where
is a kernel which normalize to
(2.5)
For example consider the Epanechikov kernel
(2.6)
and the kernel suggested in [6]
(2.7)
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.
For simplicity of notation, we will denote
and
.
Integrated volatility has to be estimated on the basis of discrete observations of the process
at times
with
. Denote
(2.8)
The realized volatility is defined as
(2.9)
The following theorem is well known in the literature, see [1].
Theorem 2.1 ![](https://www.scirp.org/html/4-1240011\7ac85fd0-e261-44ab-acc4-22ee4da7507f.jpg)
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
(2.10)
where
is a weight function.
Denote
(2.11)
(2.12)
General weighted kernel volatility (KV) is defined as
(2.13)
With
, we obtain the forward KV as
(2.14)
With
, we obtain the backward KV as
(2.15)
[8] studied asymptotics of the estimator
and obtained the rate of convergence along with asymptotic distribution of the estimator
.
Our plan is to improve the rate of convergence by using appropriate weights for the kernel. With
, the simple symmetric KV is defined as
(2.16)
With the weight function
![](https://www.scirp.org/html/4-1240011\c9c993b2-4e4b-46da-88dc-99cc37e074a0.jpg)
the weighted symmetric KV is defined as
(2.17)
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
(2.18)
We can use the Simpson’s rule to define another estimator which is a convex combination of the midpoint estimator and the trapezoidal estimator
(2.19)
In general, one can generalize Simpson’s rule as
(2.20)
for any
.
The case
produces the estimator (2.18). The case
produces the estimator (2.17). The case
produces the estimator (2.19).
I propose a very general form of the quadrature based KV as
(2.21)
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
then 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
(2.22)
(2.23)
Neither the probabilities
nor the atoms,
, 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) of a KV is
where
(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. The moment based estimators of integrated volatility which are given by
![](https://www.scirp.org/html/4-1240011\93062091-fa99-4058-9f44-82de91755c48.jpg)
![](https://www.scirp.org/html/4-1240011\17c7b926-d65d-4a2c-9aeb-a2765a1a31b4.jpg)
![](https://www.scirp.org/html/4-1240011\a4e22d54-9828-4fe8-bdb7-fa58bcbd8c97.jpg)
![](https://www.scirp.org/html/4-1240011\158df0c3-b2b9-4832-88c8-62cf0b01eea1.jpg)
![](https://www.scirp.org/html/4-1240011\06542a27-b4a1-4de2-9199-d403b6397acb.jpg)
![](https://www.scirp.org/html/4-1240011\816e5a76-6675-4c8e-83e9-ad7ae67ef4c9.jpg)
![](https://www.scirp.org/html/4-1240011\f2e56598-116a-4ba6-aec4-193a28af9a14.jpg)
![](https://www.scirp.org/html/4-1240011\14524853-0348-473b-93d2-b97ea57da1d3.jpg)
![](https://www.scirp.org/html/4-1240011\9ec4638f-1cc4-4ba9-83ba-7a3c0e6e6659.jpg)
![](https://www.scirp.org/html/4-1240011\9c4b0639-3550-448a-9463-8f23b19999f3.jpg)
![](https://www.scirp.org/html/4-1240011\382a3dea-43c8-4d5b-ac9a-9864a89c77d2.jpg)
satisfy
![](https://www.scirp.org/html/4-1240011\dce1710f-9153-4420-be73-9fd2822289a1.jpg)
![](https://www.scirp.org/html/4-1240011\d337e14f-4495-44f7-ae95-5adcf3fd9163.jpg)
![](https://www.scirp.org/html/4-1240011\c943f848-13e1-4215-80e4-edbaafe922fa.jpg)
![](https://www.scirp.org/html/4-1240011\5ae79955-83ec-4067-81db-26b6c789ad75.jpg)
![](https://www.scirp.org/html/4-1240011\f2064eca-d559-4d72-986e-3c23009c1b74.jpg)
![](https://www.scirp.org/html/4-1240011\ee01975a-f3bc-4863-9876-a215d174d9be.jpg)
![](https://www.scirp.org/html/4-1240011\9474750f-021a-479e-bbd0-5cd9ff463df7.jpg)
![](https://www.scirp.org/html/4-1240011\67cccdde-c96b-4239-afd7-60e360e40633.jpg)
![](https://www.scirp.org/html/4-1240011\749e5d03-0420-47f1-83fb-51cee7d621e4.jpg)
![](https://www.scirp.org/html/4-1240011\8b769006-eab2-46aa-9bde-d59324ac762a.jpg)
![](https://www.scirp.org/html/4-1240011\6fb27237-e69e-4ad8-8a52-f91b2256913d.jpg)
Proof We use (2.22)-(2.24). Probability
at the point
gives the KV (2.11) for which
.
Note that
. Thus
. This is gives (a).
Probability
at the point
gives the KV(2.12) for which
. Note that
. Thus
. This gives (b).
Probabilities
at the respective points
produces the KV
for which
. Thus
. This gives (c).
Probability
at the point
produce the KV
for which
. Thus
. This gives (d).
Probabilities
at the respective points
produce the asymmetric KV
(2.25)
for which
. Thus
. This gives (e).
Probabilities
at the respective points
produce asymmetric KV
(2.26)
for which
. Thus
. This gives (f).
Probabilities
at the respective points
produce the KV
for which
. Thus
.
This gives (g).
Probabilities
at the respective points
produce the symmetric KV
(2.27)
for which
. Thus
.
This gives (h).
Probabilities
at the respective points ![](https://www.scirp.org/html/4-1240011\ee8f3a3b-2956-4be5-8bb4-434862efa3b4.jpg)
produce the asymmetric KV
(2.28)
for which
. Thus
. This gives (i).
Probabilities
at the respective points ![](https://www.scirp.org/html/4-1240011\f6b0ad79-4cf3-479a-a8f5-58649610874d.jpg)
produce the symmetric KV
given by
(2.29)
for which
.
Thus
. This gives (j).
Probabilities
![](https://www.scirp.org/html/4-1240011\b291c810-37b6-47ae-8e89-a0d6009b0749.jpg)
at the respective points
produce symmetric KV
(2.30)
for which
. Thus
. This gives (k).
The KV
is based on the arithmetic mean of
and
. One can use geometric mean and harmonic mean instead.
Theorem 2.4 The geometric mean based symmetric KV (which is based on the ideas of partial autocorrelation) is given by
(2.31)
The harmonic mean based symmetric KV is given by
(2.32)
3. Spline Estimators
In order to improve the realized volatility estimator of integrated volatility, we use an alternative method, the method of splines, see [9], [10] and [11]. This is the first step towards the use of splines for volatility estimation. Since these are based on analysis of variance for diffusion models, we call it DANOVA models. DANOVA stands for ANOVA for Diffusions.
In the stochastic volatility model, the log-price
with
being the asset price, follows
![](https://www.scirp.org/html/4-1240011\deef94e4-cdf6-4f28-888e-394f518b41a3.jpg)
![](https://www.scirp.org/html/4-1240011\da1e5b75-0717-4932-b36b-c264231c075e.jpg)
where
and
are assumed to be independent of the standard Brownian motion
. The process
is called the instantaneous volatility or spot volatility and
is called the mean process and the Brownian motions
and
are allowed to be correlated. A simple example of this is
![](https://www.scirp.org/html/4-1240011\cf89edf7-9ec6-4798-8b09-01311a1fef66.jpg)
in which case
is called the risk premium and
is called the integrated variance.
Over an interval of time length
, returns are defined as
![](https://www.scirp.org/html/4-1240011\bf2d00b9-a990-4bb8-b7c2-bd3632d214c7.jpg)
which implies that
![](https://www.scirp.org/html/4-1240011\c8fb72b3-96fb-42f1-8f5b-22dd091f03fd.jpg)
where
![](https://www.scirp.org/html/4-1240011\b82373ac-9df2-4be8-9e06-39e823052c3b.jpg)
and
![](https://www.scirp.org/html/4-1240011\04919c0c-8544-4c2d-9e34-37c4e80baf26.jpg)
Here
is called the actual variance and
is called the actual mean.
Suppose one is interested in estimating the actual volatility
using
intra-
observations. A natural candidate is the realized volatility given by
![](https://www.scirp.org/html/4-1240011\28836255-25ec-4d88-9853-99a4f656deda.jpg)
where
![](https://www.scirp.org/html/4-1240011\28a60b5d-f614-4c24-a64f-57016926ab8d.jpg)
Denote
![](https://www.scirp.org/html/4-1240011\0c72737d-7fbd-4895-a285-b46d0fbd7def.jpg)
and
![](https://www.scirp.org/html/4-1240011\bb6ebc2a-58de-4702-983c-bede0762adbe.jpg)
Thus the realized volatility is given by
![](https://www.scirp.org/html/4-1240011\5a434ea5-812a-4d5a-ab77-374ec4fbfea7.jpg)
When
, realized volatility converges in
to the integrated volatility. We consider the fixed
case. The realized variance is a quadratic form.
Note that the realized volatility is based on first order difference.
We introduce some new estimators:
![](https://www.scirp.org/html/4-1240011\54bb70ee-2ebb-4b04-93f6-fede23c18be2.jpg)
The above estimator is based on second order difference.
![](https://www.scirp.org/html/4-1240011\96006960-f326-4506-aada-bd86c7cfb714.jpg)
where
and
are non-negative integers,
is called the order, and the difference sequence
satisfies
and ![](https://www.scirp.org/html/4-1240011\b37ec7e8-eb5c-4519-b4fd-200f2de1f727.jpg)
Note that for difference based estimators
![](https://www.scirp.org/html/4-1240011\c0d72f5c-67e8-497b-b8a5-28c2bdfdf8df.jpg)
To improve this error bound, we introduce the lag-
estimator
![](https://www.scirp.org/html/4-1240011\67c2b1f0-8860-4497-bcd5-9193b548cf4f.jpg)
In practice, the choice of the order
and an appropriate difference sequence which minimizes the finite sample MSE is difficult.
Theorem 3.1 The spline estimator of integrated volatility is given by
![](https://www.scirp.org/html/4-1240011\24b8d243-e578-4e4a-980f-16262b199528.jpg)
where
![](https://www.scirp.org/html/4-1240011\a19513e7-fb55-460c-a1ea-509385cd6b23.jpg)
![](https://www.scirp.org/html/4-1240011\f5efa346-2e7d-48ab-b58d-6d726efeba9a.jpg)
and
![](https://www.scirp.org/html/4-1240011\9c4e4bc4-8390-49e1-9913-7d0b688171ae.jpg)
Proof We fit the following regression model:
![](https://www.scirp.org/html/4-1240011\408240f9-7197-4830-8d8c-76e1745be220.jpg)
using the weighted least squares estimate
![](https://www.scirp.org/html/4-1240011\e769ffda-1c22-4f89-a1e2-e84810dd83f8.jpg)
where
is a sequence of i.i.d. random variables.
Let
![](https://www.scirp.org/html/4-1240011\ea21fbfd-eaec-4493-acf7-01cc2741ffb8.jpg)
and
![](https://www.scirp.org/html/4-1240011\ba147ac4-761e-4dfa-8e6a-49facd7d9960.jpg)
Then
![](https://www.scirp.org/html/4-1240011\d58abd19-5bdf-4fd9-a68f-637e11490a7b.jpg)
where
![](https://www.scirp.org/html/4-1240011\f7760cc2-9c10-4264-947b-50d2e164be20.jpg)
is the estimate of the intercept
.![](https://www.scirp.org/html/4-1240011\561b5924-f589-4674-b395-dee35f254615.jpg)
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