# High-Frequency Financial Econometrics

Yacine Aït-Sahalia
Jean Jacod
Pages: 688
https://www.jstor.org/stable/j.ctt6wq07x

1. Front Matter
(pp. i-vi)
(pp. vii-xvi)
3. Preface
(pp. xvii-xxii)
Yacine Aït-Sahalia and Jean Jacod
4. Notation
(pp. xxiii-xxiv)
5. I Preliminary Material
• Chapter 1 From Diffusions to Semimartingales
(pp. 3-56)

This chapter is a quick review of the theory of semimartingales, these processes being those for which statistical methods are considered in this book.

Aprocessis a collectionX= (Xt) of random variables with values in the Euclidean space${{\mathbb{R}}^{d}}$for some integerd≥ 1, and indexed on the half line${{\mathbb{R}}_{+}}=\left[ 0,\infty \right)$, or a subinterval of${{\mathbb{R}}_{+}}$, typically [0,T] for some realT> 0. The distinctive feature however is that all these variables are defined onthe same probability space$\Omega,{\mathcal F},{\mathbb{P}}$. Therefore, for any outcomeω∈ Ω one can consider thepath(or...

• Chapter 2 Data Considerations
(pp. 57-78)

Semimartingales of the type described in the previous chapter are used as modeling tools in a number of applications including the study of Internet packet flow, turbulence and other meteorological studies, genome analysis, physiology and other biological studies, particle modeling and finance. We will focus in this chapter on the specific characteristics of high-frequency financial data that distinguish them from other, more standard, time series data. The statistical methods we will describe in this book rely on statistics of the processXsampled at timesiΔnfori= 0, … , [Tn], and sometimes at unevenly spaced times, possibly...

6. II Asymptotic Concepts
• Chapter 3 Introduction to Asymptotic Theory: Volatility Estimation for a Continuous Process
(pp. 83-108)

In this chapter, we consider, as an introduction to asymptotic theory, the very simple situation of a one-dimensional continuous martingale of the form${{X}_{t}}={{X}_{0}}+\int_{0}^{t}{{{\sigma }_{s}}d{{W}_{s}}}.\caption {(3.1)}$

Our objective is to “estimate” its integrated volatility, which is${{C}_{t}}=\int_{0}^{t}{{{c}_{s}}ds},\quad \text{where}\ \ {{c}_{s}}=\sigma _{s}^{2},$on the basis of discrete observations ofXover the time interval [0,T], with the horizonTfixed. The sampling is the simplest possible: equidistant observations, without market microstructure noise. The observation times areiΔnfori= 0, 1, … , [Tn], where [Tn] is the biggest integer less than or equal toTn, and the time interval Δneventually goes to 0:...

• Chapter 4 With Jumps: An Introduction to Power Variations
(pp. 109-130)

As seen at the end of Chapter 3, the situation is indeed quite different when the observed process is continuous and when it is not. This is why, in this chapter, we study the simplest possible process having both a non-trivial continuous part and jumps, that is,${{X}_{t}}={{X}_{0}}+\sigma {{W}_{t}}+{{Y}_{t}},\caption {(4.1)}$whereYtis a compound Poisson process (see Example 1.5), the volatilityσ> 0 is a constant parameter, andWis a standard Brownian motion. The processXis again observed, without microstructure noise, at regularly spaced timesiΔnfori= 0, 1, … , [Tn] for some fixed time horizon...

• Chapter 5 High-Frequency Observations: Identifiability and Asymptotic Efficiency
(pp. 131-164)

This chapter starts with a brief reminder about a number of concepts and results which pertain to classical statistical models, without specific reference to stochastic processes (although the examples are always stochastic process models). This should help the reader make the connection between classical statistics and the specific statistical situations encountered in this book.

Next, we introduce a general notion of identifiability for a parameter, in a semi-parametric setting. A parameter can be a number (or a vector), as in classical statistics; it can also be a random variable, such as the integrated volatility, as already seen in Chapter 3....

7. III Volatility
• Chapter 6 Estimating Integrated Volatility: The Base Case with No Noise and Equidistant Observations
(pp. 169-208)

This chapter covers the various problems arising in the estimation of the integrated volatility, in the idealized situation where the process is observed without error (no microstructure noise) and along a regular observation scheme. In this case the situation is quite well understood, although not totally straightforward when the process has jumps.

The setting is as follows: the underlying processXis ad-dimensional Itô semimartingale (often withd= 1), defined on a filtered space$\left (\Omega,\mathcal{F},{{({{\mathcal{F}}_{t}})}_{t\ge 0}},\mathbb{P} \right )$, and we recall the Grigelionis representation of (1.74):${{X}_{t}}={{X}_{0}}+\int_{0}^{t}{{{b}_{s}}ds}+\int_{0}^{t}{{{\sigma }_{s}}d{{W}_{s}}}+(\delta {{1}_{\{\left\| \delta \right\|\le 1\}}})\star{{(\underline{p}-\underline{q})}_{t}}+(\delta {{1}_{\{\left\| \delta \right\|>1\}}})\star{{\underline{p}}_{t}}.\caption {(6.1)}$

HereWis ad′-dimensional Brownian motion and${\underline p}$is a Poisson measure...

• Chapter 7 Volatility and Microstructure Noise
(pp. 209-258)

In the previous chapter it is assumed that the observations are perfect, in the sense that the valueXiΔnof the process of interest at any observation timeiΔnis observedwithout error. However, ifXtrepresents the value at timetof some physical parameter, it is typically observed with a measurement error, which is often assumed to be a “white noise.” In finance, as discussed in Chapter 2, things are more complicated, and we can consider at least three types of error, or “noise,” in asserting the value of, for example, a log-priceXtat some observation time...

• Chapter 8 Estimating Spot Volatility
(pp. 259-298)

The estimation of integrated volatility, studied in Chapter 6, has been the first object of interest in financial econometrics, because of its relevance in finance, and also because it is the simplest object which can be estimated. However, under mild assumptions, such as the volatility being right-continuous or left-continuous, not only is the integrated volatility identifiable, but the volatility process as a whole is identifiable as well, and is potentially quite useful for financial applications. This question of estimating the “spot volatility,” that is, the value of the volatility at a given time, deterministic, or possibly random, time, is the...

• Chapter 9 Volatility and Irregularly Spaced Observations
(pp. 299-324)

Up to now, we only considered regularly spaced observation times. This is pertinent in many cases, like in physical or biological sciences where measurement devices are set to record data every minute, or second, or millisecond. In finance, this is also pertinent for indices, which are computed often, but typically at regular times.

For asset prices, things are different. For example transaction prices are recorded at actual transaction times, these times being also recorded (in principle, although there might be some delays or approximations, see Chapter 2). So, at least theoretically, we observe the values ofXat successive times...

8. IV Jumps
• [IV Introduction]
(pp. 327-328)

So far, we have been interested in volatility, integrated or spot, hence in the second characteristicCtof the log-price processX, which describes its continuous martingale part. Jumps ofXwere not excluded, but were viewed as a nuisance for the purpose of estimating the volatility component of the model and we tried to eliminate them in one way or another.

In this part, jumps become our main interest. We first try to answer the fundamental question of whether there are jumps at all. When the answer is positive, our aim is to estimate as many features of these...

• Chapter 10 Testing for Jumps
(pp. 329-392)

This chapter is devoted to the most basic question about jumps: are they present at all? As seen in Chapter 5, this question can be answered unambiguously when the full path of the underlying processXis observed over the time interval of interest [0,T]. However, we suppose thatXis discretely observed along a regular scheme with lag Δn, so no jump can actually be exactly observed, since observing a large discrete increment$\Delta _{i}^{n}X$may be suggestive that a jump took place, but provides no certitude. We wish to derive testing procedures which are at least consistent (that...

• Chapter 11 Finer Analysis of Jumps: The Degree of Jump Activity
(pp. 393-428)

After having developed tests for deciding whether the underlying processXhas jumps, we go further in the statistical analysis of the jumps. This of course makes sense only if we believe that jumps exist, for example because one of the tests of the previous chapter rejects the “continuous” hypothesis.

The previous analysis can be extended in two different directions. One direction amounts to studying the same kind of testing problems in more complex situations, such as finding whether two components of a multidimensional process have jumps at the same times (always, sometimes, never?), or whether a process and its...

• Chapter 12 Finite or Infinite Activity for Jumps?
(pp. 429-440)

The previous chapter was concerned with the estimation of the degree of activity of the jumps of a processX, that is, of its Blumenthal-Getoor index. If the resulting confidence interval does not contain the value 0, one may conclude that the genuine activity index is positive, and thus the jumps haveinfinite activity(infinitely many jumps inside the observation interval [0,T]). Otherwise the true BG index may be equal to 0, and then a natural question to ask oneself is whether or not the observed path has infinitely many jumps on [0,T]. As we know, if this...

• Chapter 13 Is Brownian Motion Really Necessary?
(pp. 441-452)

So far we have explained how to test the presence of jumps and, when the answer to this question is positive, how to test whether the jumps have finite activity or not, and how to estimate the activity index (or BG index) of the jumps.

Now suppose that these procedures end up with the answer that jumps have infinite activity, and perhaps, even, that the BG index is “high” (on its scale: between 0 and 2). In the latter case, high-activity (compensated) jumps look pretty much like a Brownian motion plus occasional large jumps. It is thus legitimate to ask...

• Chapter 14 Co-jumps
(pp. 453-476)

So far, and except for Chapter 6 about volatility estimation, we have mainly considered one-dimensional processes, or at least addressed one-dimensional questions. The real (financial) world, however, is multidimensional. This is why, in this chapter, we study some questions which only make sense in a multivariate setting.

We will be concerned with two problems: one is about a multidimensional underlying processX, and we want to decide whether two particular components ofXjump at the same time: this can happen always, or never, or for some but not all jump times. The second problem is again about a one-dimensional...

9. Appendix A Asymptotic Results for Power Variations
(pp. 477-506)
10. Appendix B Miscellaneous Proofs
(pp. 507-632)
11. Bibliography
(pp. 633-656)
12. Index
(pp. 657-659)