In this paper we derive lower bounds in minimax sense for estimation of the instantaneous volatility if the diffusion type part cannot be observed directly but under some additional Gaussian noise. Three different models are considered. Our technique is based on a general inequality for Kullback-Leibler divergence of multivariate normal random variables and spectral analysis of the processes. The derived lower bounds are indeed optimal. Upper bounds can be found in Munk and Schmidt-Hieber [18]. Our major finding is that the Gaussian microstructure noise introduces an additional degree of ill-posedness for each model, respectively.
In this paper, we develop econometric tools to analyze the integrated volatility of the efficient price and the dynamic properties of microstructure noise in high-frequency data under general dependent noise. We first develop consistent estimators of the variance and autocovariances of noise using a variant of realized volatility. Next, we employ these estimators to adapt the pre-averaging method and derive a consistent estimator of the integrated volatility, which converges stably to a mixed Gaussian distribution at the optimal rate $n^{1/4}$. To refine the finite sample performance, we propo…
In this article we consider the volatility inference in the presence of both market microstructure noise and endogenous time. Estimators of the integrated volatility in such a setting are proposed, and their asymptotic properties are studied. Our proposed estimator is compared with the existing popular volatility estimators via numerical studies. The results show that our estimator can have substantially better performance when time endogeneity exists.
We propose a new concept of modulated bipower variation for diffusion models with microstructure noise. We show that this method provides simple estimates for such important quantities as integrated volatility or integrated quarticity. Under mild conditions the consistency of modulated bipower variation is proven. Under further assumptions we prove stable convergence of our estimates with the optimal rate $n^{-{1}/{4}}$. Moreover, we construct estimates which are robust to finite activity jumps.
We consider the problem of testing the parametric form of the volatility for high frequency data. It is demonstrated that in the presence of microstructure noise commonly used tests do not keep the preassigned level and are inconsistent. The concept of preaveraging is used to construct new tests, which do not suffer from these drawbacks. These tests are based on a Kolmogorov-Smirnov or Cramer-von-Mises functional of an integrated stochastic process, for which weak convergence to a (conditional) Gaussian process is established. The finite sample properties of a bootstrap version of the test are…
The basic model for high-frequency data in finance is considered, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $σ$. As an application, simple rate-optimal estimators of the volatility and efficient estimators of the integrated volatility are constructed.
We first revisit the problem of estimating the spot volatility of an Itô semimartingale using a kernel estimator. We prove a Central Limit Theorem with optimal convergence rate for a general two-sided kernel. Next, we introduce a new pre-averaging/kernel estimator for spot volatility to handle the microstructure noise of ultra high-frequency observations. We prove a Central Limit Theorem for the estimation error with an optimal rate and study the optimal selection of the bandwidth and kernel functions. We show that the pre-averaging/kernel estimator's asymptotic variance is minimal for exponen…
Volatility estimation based on high-frequency data is key to accurately measure and control the risk of financial assets. A Lévy process with infinite jump activity and microstructure noise is considered one of the simplest, yet accurate enough, models for financial data at high-frequency. Utilizing this model, we propose a "purposely misspecified" posterior of the volatility obtained by ignoring the jump-component of the process. The misspecified posterior is further corrected by a simple estimate of the location shift and re-scaling of the log likelihood. Our main result establishes a Bernst…
In this paper, we present a test for the maximal rank of the volatility process in continuous diffusion models observed with noise. Such models are typically applied in mathematical finance, where latent price processes are corrupted by microstructure noise at ultra high frequencies. Using high frequency observations we construct a test statistic for the maximal rank of the time varying stochastic volatility process. Our methodology is based upon a combination of a matrix perturbation approach and pre-averaging. We will show the asymptotic mixed normality of the test statistic and obtain a con…
With the availability of high frequency financial data, nonparametric estimation of volatility of an asset return process becomes feasible. A major problem is how to estimate the volatility consistently and efficiently, when the observed asset returns contain error or noise, for example, in the form of microstructure noise. The former (consistency) has been addressed heavily in the recent literature, however, the resulting estimator is not quite efficient. In Zhang, Mykland and Ait-Sahalia (2003), the best estimator converges to the true volatility only at the rate of n^{-1/6}. In this paper, …
We consider discrete-time observations of a continuous martingale under measurement error. This serves as a fundamental model for high-frequency data in finance, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $σ$ and a nonstandard noise level. As an application, new rate-optimal estimators of the volatility function and simple efficient estimators of the integrated volatility are constructed.
We consider microstructure as an arbitrary contamination of the underlying latent securities price, through a Markov kernel $Q$. Special cases include additive error, rounding and combinations thereof. Our main result is that, subject to smoothness conditions, the two scales realized volatility is robust to the form of contamination $Q$. To push the limits of our result, we show what happens for some models that involve rounding (which is not, of course, smooth) and see in this situation how the robustness deteriorates with decreasing smoothness. Our conclusion is that under reasonable smoothn…
We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the multivariate density of the finite dimensional distributions of which we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A multivariate Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the multivariate volatility density. An expansion of the bias and a bound on the variance are derived. Key words: stochastic…
We study the non-parametric estimation of an unknown stationary density fV of an unobserved strictly stationary volatility process $(\bm V_t)_{t\geq 0}$ on $\IRp^2 := (0,\infty)^2$ based on discrete-time observations in a stochastic volatility model. We identify the underlying multiplicative measurement error model and build an estimator based on the estimation of the Mellin transform of the scaled, integrated volatility process and a spectral cut-off regularisation of the inverse of the Mellin transform. We prove that the proposed estimator leads to a consistent estimation strategy. A fully d…
In this paper, we are concerned with nonparametric inference on the volatility of volatility process in stochastic volatility models. We construct several estimators for its integrated version in a high-frequency setting, all based on increments of spot volatility estimators. Some of those are positive by construction, others are bias corrected in order to attain the optimal rate $n^{-1/4}$. Associated central limit theorems are proven which can be widely used in practice, as they are the key to essentially all tools in model validation for stochastic volatility models. As an illustration we g…
We consider discrete time models for asset prices with a stationary volatility process. We aim at estimating the multivariate density of this process at a set of consecutive time instants. A Fourier type deconvolution kernel density estimator based on the logarithm of the squared process is proposed to estimate the volatility density. Expansions of the bias and bounds on the variance are derived.
The partially observed linear Gaussian system of stochastic differential equations with low noise in observations is considered. A kernel-type estimators are used for estimation of the quadratic variation of the derivative of the limit of the observed process. Then this estimator is used for nonparametric estimation of the integral of the square of volatility of unobservable component. This estimator is also used for construction of substitution estimators in the case where the drift in observable component and the volatility of the state component depend on some unknown parameter. Then this s…
We consider a microstructure model for a financial asset, allowing for price discreteness and for a diffusive behavior at large sampling scale. This model, introduced by Delattre and Jacod, consists in the observation at the high frequency $n$, with round-off error $α_n$, of a diffusion on a finite interval. We give from this sample estimators for different forms of the integrated volatility of the asset. Our method is based on variational properties of the process associated with wavelet techniques. We prove that the accuracy of our estimation procedures is $α_n\vee n^{-1/2}$. Using compensat…
We consider two kinds of stochastic volatility models. Both kinds of models contain a stationary volatility process, the density of which, at a fixed instant in time, we aim to estimate. We discuss discrete time models where for instance a log price process is modeled as the product of a volatility process and i.i.d. noise. We also consider samples of certain continuous time diffusion processes. The sampled time instants will be be equidistant with vanishing distance. A Fourier type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate t…
In this paper, we present the double smoothed nonparametric approach for infinitesimal conditional volatility of jump-diffusion model based on high frequency data. Under certain minimal conditions, we obtain the strong consistency and asymptotic normality for the estimator as the time span $T \rightarrow \infty$ and the sample interval $Δ_{n} \rightarrow 0.$ The procedure and asymptotic behavior can be applied for both null Harris recurrent and positive Harris recurrent processes.
We consider Stochastic Volatility processes with heavy tails and possible long memory in volatility. We study the limiting conditional distribution of future events given that some present or past event was extreme (i.e. above a level which tends to infinity). Even though extremes of stochastic volatility processes are asymptotically independent (in the sense of extreme value theory), these limiting conditional distributions differ from the i.i.d. case. We introduce estimators of these limiting conditional distributions and study their asymptotic properties. If volatility has long memory, then…
This paper provides a semiparametric model of estimating states of the volatility defined as the squared diffusion coefficient of a stochastic differential equation. Without assuming any functional form of the volatility function, we consider the volatility state as an unobservable state in a state space model and estimate it by filtering. By using the estimates, we can draw the information on what kind of functions the volatility has behind the observable state of the process. We also prove the consistency of the model in the sense that estimated states converges to the true ones as the obser…
An extensive empirical literature documents a generally negative correlation, named the "leverage effect," between asset returns and changes of volatility. It is more challenging to establish such a return-volatility relationship for jumps in high-frequency data. We propose new nonparametric methods to assess and test for a discontinuous leverage effect --- i.e. a relation between contemporaneous jumps in prices and volatility. The methods are robust to market microstructure noise and build on a newly developed price-jump localization and estimation procedure. Our empirical investigation of si…
The usage of a spot volatility estimate based on a volatility decomposition in a time-changed price-model according to the trading times is investigated. In this model clock-time volatility splits up into the product of tick-time volatility and trading intensity, which both can be estimated from data and contain valuable information. By inspecting these two curves individually we gain more insight into the cause and structure of volatility. Several examples are provided where the tick-time volatility curve is much smoother than the clock-time volatility curve meaning that the major part of flu…
High-frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro-structure noise, and realized volatility-based methods are often used to estimate integrated volatility. For problems involving a large number of assets, the estimation objects we face are volatility matrices of large size. The existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large. In fact, they are inconsistent when both the number, $p$, of the assets and the average sample size, $n$, of the price data …
We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis. The co-volatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the co-volatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in proba…
The problem of integrated volatility estimation for the solution X of a stochastic differential equation with L{é}vy-type jumps is considered under discrete high-frequency observations in both short and long time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results the condition $β$ > 1 2(2--$α$) on $β$ (that is such that (1/n) $β$ is…
We statistically analyse a multivariate HJM diffusion model with stochastic volatility. The volatility process of the first factor is left totally unspecified while the volatility of the second factor is the product of an unknown process and an exponential function of time to maturity. This exponential term includes some real parameter measuring the rate of increase of the second factor as time goes to maturity. From historical data, we efficiently estimate the time to maturity parameter in the sense of constructing an estimator that achieves an optimal information bound in a semiparametric se…
We study the estimation of the high-dimensional covariance matrix andits eigenvalues under dynamic volatility models. Data under such modelshave nonlinear dependency both cross-sectionally and temporally. We firstinvestigate the empirical spectral distribution (ESD) of the sample covariancematrix under scalar BEKK models and establish conditions under which thelimiting spectral distribution (LSD) is either the same as or different fromthe i.i.d. case. We then propose a time-variation adjusted (TV-adj) sample co-variance matrix and prove that its LSD follows the same Marcenko-Pasturlaw as the i…
We consider a multidimensional Ito semimartingale regularly sampled on [0,t] at high frequency $1/Δ_n$, with $Δ_n$ going to zero. The goal of this paper is to provide an estimator for the integral over [0,t] of a given function of the volatility matrix. To approximate the integral, we simply use a Riemann sum based on local estimators of the pointwise volatility. We show that although the accuracy of the pointwise estimation is at most $Δ_n^{1/4}$, this procedure reaches the parametric rate $Δ_n^{1/2}$, as it is usually the case in integrated functionals estimation. After a suitable bias corre…
Stochastic processes are often used to model complex scientific problems in fields ranging from biology and finance to engineering and physical science. This paper investigates rate-optimal estimation of the volatility matrix of a high-dimensional Itô process observed with measurement errors at discrete time points. The minimax rate of convergence is established for estimating sparse volatility matrices. By combining the multi-scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate. The minimax lower bound is derived by considering a su…
In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on $μ$ and volatility coefficient depends on $σ$, two unknown parameters. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $Δ$n = sup i=0,...,n--1 (t n i+1 -- t n i) $\rightarrow$ 0. We introduce an estimator of $θ$ := ($μ$, $σ$), based on a contrast function, which is asymptotically gaussian without requiring any conditions on the rate at which $Δ$n $\rightarrow$ 0, assuming a finite jump activity. This extends earlier results where a condition on the step …
We consider the models Y_{i,n}=\int_0^{i/n} σ(s)dW_s+τ(i/n)ε_{i,n}, and \tilde Y_{i,n}=σ(i/n)W_{i/n}+τ(i/n)ε_{i,n}, i=1,...,n, where W_t denotes a standard Brownian motion and ε_{i,n} are centered i.i.d. random variables with E(ε_{i,n}^2)=1 and finite fourth moment. Furthermore, σand τare unknown deterministic functions and W_t and (ε_{1,n},...,ε_{n,n}) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for σ^2 and τ^2 and investigate their rate of convergence of the MISE in dependence of their smoothness. To this…
We propose new nonparametric estimators of the integrated volatility of an Itô semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally "stable" Lévy processes, that is, processes whose Lévy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characte…
After a quick review of superpositions of OU (supOU) processes, integrated sup\-OU processes and the supOU stochastic volatility model we estimate these processes by using the generalized method of moments (GMM). We show that the GMM approach yields consistent estimators and that it works very well in practice. Moreover, we discuss the influence of long memory effects.
We study the asymptotic normality of two feasible estimators of the integrated volatility of volatility based on the Fourier methodology, which does not require the pre-estimation of the spot volatility. We show that the bias-corrected estimator reaches the optimal rate $n^{1/4}$, while the estimator without bias-correction has a slower convergence rate and a smaller asymptotic variance. Additionally, we provide simulation results that support the theoretical asymptotic distribution of the rate-efficient estimator and show the accuracy of the latter in comparison with a rate-optimal estimator …
The optimal rate of convergence of estimators of the integrated volatility, for a discontinuous Itô semimartingale sampled at regularly spaced times and over a fixed time interval, has been a long-standing problem, at least when the jumps are not summable. In this paper, we study this optimal rate, in the minimax sense and for appropriate "bounded" nonparametric classes of semimartingales. We show that, if the $r$th powers of the jumps are summable for some $r\in[0,2)$, the minimax rate is equal to $\min(\sqrt{n},(n\log n)^{(2-r)/2})$, where $n$ is the number of observations.
Time- and state-domain methods are two common approaches for nonparametric prediction. The former predominantly uses the data in the recent history while the latter mainly relies on historical information. The question of combining these two pieces of valuable information is an interesting challenge in statistics. We surmount this problem via dynamically integrating information from both the time and the state domains. The estimators from both domains are optimally combined based on a data driven weighting strategy, which provides a more efficient estimator of volatility. Asymptotic normality …
Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuous time processes and discrete time models will be discussed. The key insight for the analysis is a transformation of the volatility density estimation problem to a deconvolution model for which standard methods exist. Three type of nonparametric density estimators are reviewed: …
Kernel Estimation is one of the most widely used estimation methods in non-parametric Statistics, having a wide-range of applications, including spot volatility estimation of stochastic processes. The selection of bandwidth and kernel function is of great importance, especially for the finite sample settings commonly encountered in econometric applications. In the context of spot volatility estimation, most of the proposed selection methods are either largely heuristic or just formally stated without any feasible implementation. In this work, an objective method of bandwidth and kernel selecti…
We propose a two stage procedure for the estimation of the parameters of a fairly general, continuous-time stochastic volatility. An important ingredient of the proposed method is the Cuchiero-Teichmann volatility estimator, which is based on Fourier transforms and provides a continuous time estimate of the latent process. This estimate is then used to construct an approximate likelihood for the parameters of interest, whose restrictions are taken into account through prior distributions. The procedure is shown to be highly successful for constructing the posterior distribution of the paramete…
This paper is concerned with the estimation of the volatility process in a stochastic volatility model of the following form: $dX_t=a_tdt+σ_tdW_t$, where $X$ denotes the log-price and $σ$ is a càdlàg semi-martingale. In the spirit of a series of recent works on the estimation of the cumulated volatility, we here focus on the instantaneous volatility for which we study estimators built as finite differences of the \textit{power variations} of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of $σ$. In particular, these theorems yield some con…
For a semi-martingale $X_t$, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation $\langle X, X \rangle_t$ is constructed based on observations in the vicinity of $X_t$. The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion areas. We derive $n^{-1/3}$ as optimal convergence rate in a high-frequency framework with $n$ observations (in mean). We discuss a potential application for the estimation of the integrated squared volatility of an efficient price process $X_t$ from intra-…
We propose a new estimator for the integrated covariance of two Ito semimartingales observed at a high-frequency. This new estimator, which we call the pre-averaged truncated Hayashi-Yoshida estimator, enables us to separate the sum of the co-jumps from the total quadratic covariation even in the case that the sampling schemes of two processes are nonsynchronous and the observation data is polluted by some noise. It is the first estimator which can simultaneously handle these three issues, which are fundamental to empirical studies of high-frequency financial data. We also show the asymptotic …
We develop a general class of noise-robust estimators based on the existing estimators in the non-noisy high-frequency data literature. The microstructure noise is a parametric function of the limit order book. The noise-robust estimators are constructed as plug-in versions of their counterparts, where we replace the efficient price, which is non-observable, by an estimator based on the raw price and limit order book data. We show that the technology can be applied to five leading examples where, depending on the problem, price possibly includes infinite jump activity and sampling times encomp…
In this paper, we provide non-parametric statistical tools to test stationarity of microstructure noise in general hidden Ito semimartingales, and discuss how to measure liquidity risk using high frequency financial data. In particular, we investigate the impact of non-stationary microstructure noise on some volatility estimators, and design three complementary tests by exploiting edge effects, information aggregation of local estimates and high-frequency asymptotic approximation. The asymptotic distributions of these tests are available under both stationary and non-stationary assumptions, th…
This paper deals with the filtering problem for a class of discrete time stochastic volatility models in which the disturbances have rational probability density functions. This includes the Cauchy distributions and Student t-distributions with odd number of degrees of freedom. Using state space realizations to represent the rational probability density functions we are able to solve the filtering problem exactly. However the size of the involved state space matrices grows exponentially with each time step of the filter. Therefore we use stochastically balanced truncation techniques to approxi…
Consider discrete time observations (X_{\ellδ})_{1\leq \ell \leq n+1}$ of the process $X$ satisfying $dX_t= \sqrt{V_t} dB_t$, with $V_t$ a one-dimensional positive diffusion process independent of the Brownian motion $B$. For both the drift and the diffusion coefficient of the unobserved diffusion $V$, we propose nonparametric least square estimators, and provide bounds for theirrisk. Estimators are chosen among a collection of functions belonging to a finite dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method wo…
We consider a multidimensional Itô process $Y=(Y_t)_{t\in[0,T]}$ with some unknown drift coefficient process $b_t$ and volatility coefficient $σ(X_t,θ)$ with covariate process $X=(X_t)_{t\in[0,T]}$, the function $σ(x,θ)$ being known up to $θ\inΘ$. For this model we consider a change point problem for the parameter $θ$ in the volatility component. The change is supposed to occur at some point $t^*\in (0,T)$. Given discrete time observations from the process $(X,Y)$, we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimat…
