Notes: Ait-Sahalia and Jacod (2010)

1. Introduction

In this post, I summarize the econometric method introduced in Analyzing the Spectrum of Asset Returns (JEL 2011) by Yacine Ait-Sahalia and Jean Jacod. From an economic perspective, Delbaen and Schachermayer (1994) (Theorem $1.1$ ) tells us that a (log) stock price follows a semi-martingale if and only if there is no arbitrage where a semi-martingale is defined as follows:

Definition (Semi-Martingale):
A real valued process $\{ X_t \}_{t \geq 0}$ defined on the filtered probability space $(\Omega, \{\mathcal{F}_t \}_{t \geq 0}, \mu)$ is called a semi-martingale if it can be decomposed:

(1) $\begin{align*} X_t &= M_t + A_t \end{align*}$

where $\{M_t\}_{t \geq 0}$ is a local martingale and $\{A_t\}_{t \geq 0}$ is an adapted process with locally bounded variation.

What’s more, from a purely statistical perspective, we know that a semi-martingale $\{ X_t \}_{t \geq 0}$ can be decomposed into the sum of a drift component, a Brownian component and small and large jump components:

(2) $\begin{align*} X_t &= X_0 + \underbrace{\int_0^t b_s \cdot ds}_{\text{``Drift''}} + \underbrace{\int_0^t \sigma_s \cdot dW_s}_{\text{``Brownian''}} + \underbrace{\int_0^t \int_{|x| \leq \epsilon} x \cdot (\mu - \nu)[ds,dx]}_{\text{\tiny ``Jump''}} + \underbrace{\int_0^t \int_{|x| > \epsilon} x \cdot \mu[ds,dx]}_{\text{\Large ``Jump''}} \end{align*}$

In the characterization above, $\epsilon > 0$ which marks the cutoff between large and small jumps is arbitrary, but must be fixed. In this paper, the authors develop a suite of statistical tools based on spectrographic analysis in order to examine a time series properties of high frequency stock data over the time interval $[0,T]$ and determine whether or not the data has:

A Brownian component,
A jump component, and
(if yes…) A finite number of jumps.

The authors ignore questions about the drift component since it is invisible for all intents and purposes at high frequencies.

2. Rough Idea

The authors’ basic unit of observation is the change in the log stock price $X$ over the interval $[0,T]$ sampled at a frequency of $\Delta_n$ seconds. For, notational convenience, the authors write the change in $X$ from observation $(i - 1)$ to observation $i$ when sampling at a frequency of $\Delta_n$ as:

(3) $\begin{align*} \Delta_i^n X &= X_{i \cdot \Delta_n} - X_{(i - 1) \cdot \Delta_n} \end{align*}$

Since the authors are investigating the volatility and jump behavior of the process $X$ , a natural starting point is to look at the sum of the powers of the incremental changes $\Delta_i^n X$ . For instance, if $p=2$ then we’d have $\mathtt{Var}[\Delta^n X] = (1/ \lfloor T/\Delta_n \rfloor) \cdot \sum_{i=1}^{ \lfloor T/\Delta_n \rfloor} \left| \Delta_i^n X \right|^2$ .¹ Thus, the authors use the extended concept of variation $\mathtt{B}(p,u_n,\Delta_n)$ defined below to decompose movements in $X$ into smooth Brownian components and large and small jump components:

(4) $\begin{align*} \mathtt{B}(p,u_n,\Delta_n) &= \sum_{i=1}^{\lfloor T/\Delta_n \rfloor} \left| \Delta_i^n X \right|^p \cdot 1_{\{| \Delta_i^n X| \leq u_n \}} \end{align*}$

The power variation function has $3$ free parameters which can be adjusted by the econometrician:

Power: $p$ . As $p \searrow 0$ the smaller $\Delta_i^n X$ s get more and more weight in the summation $\mathtt{B}(p,u_n,\Delta_n)$ ; whereas, when $p \nearrow \infty$ the larger $\Delta_i^n X$ s get more and more weight. Thus, for $p < 2$ the power variation estimator $\mathtt{B}(p,u_n,\Delta_n)$ is tuned to the continuous variation in $X$ , while for $p > 2$ the estimator is tuned to pick up jumps in $X$ .
Truncation Level: $u_n$ . Truncating large increments will eliminate large jumps from the series. $X$ will always contain a finite number of big jumps but may have an infinite number of small jumps.
Sampling Frequency: $\Delta_n$ . By sampling at slightly different frequencies $k \cdot \Delta_n$ and $\Delta_n$ , we can identify the asymptotic behavior of the power variation as $\Delta_n \searrow 0$ . For instance, if $\lim_{\Delta_n \searrow 0} \mathtt{B}(p,u_n,k \cdot \Delta_n)/ \mathtt{B}(p,u_n,k \cdot \Delta_n) < 1$ , then the power variation will diverge to $\infty$ for a particular power level $p$ and truncation level $u_n$ . Alternatively, if this limit is $1$ or $>1$ the power variation with converge to a finite value or to $0$ respectively.

The power parameter controls whether the power variation emphasizes small or large movements in X.

Varying the sampling frequency allows identifies of convergence of the power variation of X at different power and truncation level parameters.

3. Examples

To get a feel for how to use the power variation of a series $X$ to decompose its movements into Brownian and jump components, I now walk through $2$ examples:

Example (Brownian Component Exists): Consider the null hypothesis that the Brownian motion component $\int_0^t \sigma_s \cdot dW_s$ exists in the stochastic process $X$ . If we consider powers $p < 2$ , then if the log price process $X$ contains a Brownian component the many tiny incremental changes due to Brownian motion will dominate the power variation $\mathtt{B}(p,u_n,\Delta_n)$ and it will diverge to $\infty$ . Conversely, without a Brownian component, for some cutoff $\beta \in (0,2)$ the power variation of $X$ will converge to $0$ at exactly the same rate for $k \cdot \Delta_n$ and $\Delta_n$ for all $p > \beta$ since only the size of the jumps not their frequency will matter. This intuition yields a test statistic $\mathtt{S}_{\scriptscriptstyle \exists dW}$ :

(5) $\begin{align*} \mathtt{S}_{\scriptscriptstyle \exists dW}(p, u_n, k, \Delta_n) &= \frac{\mathtt{B}(p,u_n,\Delta_n)}{\mathtt{B}(p,u_n,k \cdot \Delta_n)} \end{align*}$

The asymptotic results for this test as $\Delta_n \searrow 0$ are:

(6) $\begin{align*} \mathtt{S}_{\scriptscriptstyle \exists dW}(p, u_n, k, \Delta_n) &\overset{\mathtt{P}}{\longrightarrow} \begin{cases} k^{1 - p/2} &\text{ if Brownian component exists} \\ 1 &\text{ else} \end{cases} \end{align*}$

Example (Jump Component Exists): Consider the null hypothesis that at least $1$ of the jump components exists in the log price process $X$ . Then, if we set the power parameter $p > 2$ , then as we sample at higher and higher frequencies only the jump components of $X$ should matter for the value of the power variation. This yields a test statistic $\mathtt{S}_{\scriptscriptstyle \exists J}$ defined below with $p > 2$ :

(7) $\begin{align*} \mathtt{S}_{\scriptscriptstyle \exists J}(p,k,\Delta_t) &= \frac{\mathtt{B}(p,\infty, k \cdot \Delta_n)}{\mathtt{B}(p, \infty, \Delta_n)} \end{align*}$

The asymptotic results for this test as $\Delta_n \searrow 0$ are:

(8) $\begin{align*} \mathtt{S}_{\scriptscriptstyle \exists J}(p,k,\Delta_t) &\overset{\mathtt{P}}{\longrightarrow} \begin{cases} 1 &\text{ if either jump component exists} \\ k^{p/2 - 1} &\text{ else} \end{cases} \end{align*}$

Here, the $\lfloor \cdot \rfloor$ operator computes the nearest integer less than or equal to its argument. So for instance, $\lfloor 4.123 \rfloor = 4$ . ↩