Volatility estimation from a view point of entropy

Statistics of continuous-time stochastic processes with stochastic calculus, as seen in the textbooks by Prof. Albert N. Shiryaev, with Prof. Robert S. Liptser [15, 16], and with Prof. Jean Jacod [6], has been the basis of the more recent topics including so-called high-frequency statistics (HFS, for short). The HFS is also based on, instead of stationarity, the semi-martingale principle

for the data of the semi-martingale $X$ sampled at $0=t_{0}<t_{1}<\cdots<t_{n}=t$.

The HFS has been intensively studied in view of applications in finance because convergence (1.1) has some kind of reality in financial markets that allows high-frequency trading. The reality, however, leaves us the problem of so-called microstructure noise; the convergence (1.1) is not really achieved but only with some modification, which can be well explained by supposing that we observe with noise whose distribution is rather irrelevant to the frequency (see, e.g. [1]).

There has been a great deal of literature on the construction of an estimator of $[X]_{t}$ under the microstructure noise. In the present paper, we introduce an approach that relies on a likelihood function along the line of the one proposed in [8] (which we call the Kunitomo-Sato type likelihood function), which is somewhat different from those found in Section 7.3 of [1]. The approach actually unifies the volatility estimator proposed by N. Kunitomo and S. Sato [8] and the one proposed by P. Malliavin and M. E. Mancino [17, 18]. Further, we will point out that both fail to be a consistent estimator if there is also a micro-structure noise in the initial observation. We then propose an alternative estimator which overcomes the difficulty, based on the likelihood function approach.

The rest of the present paper is organized as follows. In section 2, we study the Kunitomo–Sato approach, which is referred to as the separating information maximum likelihood method (SIML method, for short). In Section 3, we discuss how the Malliavin–Mancino estimator can be understood as a separating information estimator. In Section 4, we introduce an alternative estimator which could be consistent under the initial noise. Section 5 concludes the paper.

In the present section, we describe our setting and the problem, which is typical in HFS. Let $J\in\mathbb{N}$. Consider an Itô process

for $j=1,2,\dots,J$ and $t\in[0,1]$, where $b^{j}$ and $\sigma^{j}_{r}$ are integrable and square integrable adapted processes, respectively, for all $j=1,2,\dots,J$ and $r=1,2,\dots,d$, $\mathbf{W}\equiv(W^{1},W^{2},\cdots,W^{d})$ is a Wiener process, all of which are defined on a filtered probability space $(\Omega,\mathcal{F},\mathbf{P},\{\mathcal{F}_{t}\})$.

We suppose that the observations $Y$ are with some additive noise $v$, often called the micro-structure noise. Namely,

for $k=0,1,\dots,n^{j}$ and $j=1,2,\dots,J$, where $\{v^{j}_{k}\}_{j,k}$ is a family of random variables whose distribution will be specified later, and $n^{j}$ is an integer.

We are interested in constructing an estimator $(V^{j,j^{\prime},h})_{j,j^{\prime}=1,2,\dots,J}$ of $h$- integrated volatility matrix defined by

for $j,j^{\prime}=1,2,\dots,J$, where $h$ is a given function or out of the observations, which is consistent in the sense that each $V^{j,j^{\prime}}$ converges to $\int_{0}^{t}h(s)\Sigma^{j,j^{\prime}}(s)\,{\rm d}s$ in probability as $n\to\infty$, under the condition that

as $n\to\infty$.

Let us put

for $k,l=1,2,\dots,n$, $n\in\mathbf{N}$. Note that $\mathbf{P}_{n}=(p^{n}_{k,l})$ forms an orthogonal matrix. The separating information maximum likelihood (SIML) estimator, introduced by Prof. Naoto Kunitomo and Prof. Seisho Sato in [8] (see also [14, Chapters 3 & 5]), is given by

where $m_{n}(\ll n)$ is an integer, and we understand $\Delta$ to be the difference operator given by $(\Delta a)_{k}=a_{k}-a_{k-1}$ for a sequence $\{a_{k}\}_{k}$. We then write

They have proved the following two properties:

1. (a)

   (the consistency): the convergence in probability of $V^{j,j^{\prime}}_{n,m_{n}}$ to $\int_{0}^{1}\Sigma^{j,j^{\prime}}(s)\,\mathrm{d}s$ as $n\to\infty$ is attained, provided that $m_{n}=o(n^{1/2})$, and

2. (b)

   (the asymptotic normality of the error): the stable convergence of

   $$\begin{split}&\sqrt{m_{n}}\left(V^{j,j^{\prime}}_{n,m_{n}}-\int_{0}^{1}\Sigma^{j,j^{\prime}}(s)\,\mathrm{d}s\right)\\ &\to N\left(0,\int_{0}^{1}\left(\Sigma^{j,j}(s)\Sigma^{j^{\prime},j^{\prime}}(s)+(\Sigma^{j,j^{\prime}}(s))^{2}\right)\mathrm{d}s\right)\end{split}$$

   holds true as $n\to\infty$ if $m_{n}=o(n^{2/5})$.

when the following hold (see [14] for details).

1. (i)

   the observations are equally spaced, that is, $t^{j}_{k}\equiv k/n$,

2. (ii)

   $\{v^{j}_{k}\}_{k=1}^{n}$ are independently and identically distributed, and the common distribution is independent of the choices of $n$ and $j$,

3. (iii)

   $v_{0}^{j}\equiv 0$,

4. (iv)

   $\sigma$ is deterministic and $\mu\equiv 0$.

When $v^{j}_{0}\neq 0$, we emphasize that the consistency of the SIML estimator may fail. Let us see this when $J=1$. We start with the decomposition of $V_{n,m_{n}}\equiv V_{n,m_{n}}^{1,1}$ as

Under the general sampling scheme for which consistency is established in [3], we know that

converges to zero in $L^{2}$, and so its mean also converges to zero. On the other hand, we have clearly

The following observation claims that we cannot establish the consistency of the estimator with $m_{n}=o(1)$ as $n\to\infty$.

Let us introduce a prototype of Kunitomo–Sato (KS, for short) likelihood function. To focus on the study of estimations under micro-structure noise, we work only on the one-dimensional Itô process case for simplicity. We set

where $W$ is a one-dimensional Wiener process, $\sigma$ and $\mu$ are square- and absolute- integrable processes adapted to the filtration generated by $W$, respectively. Observations are set to be

Even though the process $Y$ is not a Gaussian process since $\sigma$ is not deterministic in general, it turned out to be beneficial to consider the likelihood function of $\mathbf{\Delta Y}:=\{\Delta Y_{k}=Y_{(k+1)/n}-Y_{k/n},k=0,1,\dots,n-1\}$ pretending that $\sigma$ is a deterministic constant (we write $\sigma^{2}=:c$), $\mu\equiv 0$ and $v_{k}\sim N(0,\nu)$, $k=1,2,\dots,n$, are zero-mean Gaussian random variables.

Under these specifications, we have

where $I_{n}$ is the $n\times n$ identity matrix, and the $n\times n$ matrix $J_{n}$ is defined by

It is noteworthy that the matrix $\mathbf{P}_{n}$ introduced in Section 2.2 does diagonalize $J_{n}$. Moreover, the diagonal is decreasing. To be precise, we have

where we denote by $\mathrm{diag}[b_{1},b_{2},\dots,b_{n}]$ the diagonal matrix whose components are $b_{1},b_{2},\dots,b_{n}$, and $\mathbf{P}_{n}^{\top}$ means the transpose of $\mathbf{P}_{n}$.

We can then write down the log-likelihood function of $\mathbf{z}=n^{-1/2}\mathbf{P}_{n}\mathbf{\Delta Y}_{n}$ as

where we put

Kunitomo and Sato [8] introduced the following decomposition of $L_{n}$:

where we put

which we call the prototype of the KS likelihood function,

and $L_{n}^{(r)}:=2L_{n}-L_{n}^{(1)}-L_{n}^{(2)}$. Note that the decomposition depends on the choice of $(m,l)$. The term separating information comes from this decomposition.

Now we see that the SIML estimator (2.4) maximize $L_{n}^{(1)}$, and

maximize $L_{n}^{(2)}$. With a proper choice of $(m_{n},l_{n})$, we may have

so that the estimators are asymptotically optimal (see [14, Chapter 3] for a discussion).

The Malliavin-Mancino’s Fourier method (MM method, for short), introduced in [17, 4, 18] (see also [19], which is a good review of the literature), is an estimation method for the spot volatility $\Sigma(s)$ in Section 2.1, by constructing an estimator of the Fourier series of $\Sigma$. The series consists of estimators of Fourier coefficients given by

for $j,j^{\prime}=1,2,\dots,J$ and $q\in\mathbf{Z}$. We note that $\widehat{\Sigma^{j,j^{\prime}}_{n,m_{n}}}(0)$ is a variant of the SIML estimator (2.4). In fact, we have

and so, if we put

for $k=1,2,\cdots,n_{j}$, we have

Now we see the similarity between the SIML estimator (2.4) and the MM estimator (3.2). In the following section, we show that the difference of the two estimators can be explained in terms of the KS likelihood function.

From now on, we consider the case where $d=1$, so that the dependence on the coordinate $j$ vanishes, where $n$ is odd, so that we rewrite it as $2n+1$, and where $t_{k}=k/(2n+1)$ for $k=0,1,\dots,2n+1$. Then, we obtain a $(2n+1)\times(2n+1)$ matrix $\mathbf{Q}_{n}:=(q^{2n+1}_{k,l})_{0\leq k,l\leq 2n}$ with

The above lemma insists that the KS likelihood function associated with the MM estimator $\widehat{\Sigma^{j,j^{\prime}}_{n,m_{n}}}(0)$ of (3.2) is the one obtained by replacing $J_{n}$ of (2.6) with (3.3).

Looking back the procedure for constructing KS likelihood function from the observation structure in Section 2.4, we notice that the matrix $\widetilde{J}_{n}$ might come from the equation $v_{n}=v_{0}$, which is of course unrealistic. The following proposition may reflect this structure.

As we have seen, the KS estimator (2.4) cannot be consistent if $v_{0}\neq 0$. In this section, we propose an adjusted estimator by replacing $J_{n}$ of (2.6) with

which reflects $v_{0}\sim N(0,\nu)$ in defining the KS likelihood function.

Given (4.1), we may consider an estimator of the integrated volatility $\int_{0}^{1}\Sigma^{j,j^{\prime}}(s)\,\mathrm{d}s$ gievn by

instead of (2.4). The good news is that, in contrast with Proposition 2.3, we have