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## Realized Variance Versus Number of Trades

It is widely known that, in a good approximation, the variance over some period of time is proportional to the number of trades during that time period (see e.g. [28]). Figure 2.1 illustrates this property on Bund data. On 15 minutes intraday intervals, averaging on every single day available, we look at (dashed curve) the average intraday rate of trading (i.e., the average number of trades per second) and (solid curve) the average (15 minutes) realized variance (estimated summing on 1mn-squared returns (pt+1mnpt)2). We see that the so-called intraday seasonality of the variance is highly correlated with the intraday seasonality of the trading rate [28].

In order to have more insights, we look at some daily statistics : Figure 2.2 shows a scatter plot in which each point corresponds to a given day k whose abscissa is the number of trades within this day, i.e., Nk, and the ordinate is the daily variance (estimated summing over 5-mn quadratic returns) of the same day k. It shows that, despite some dispersion, the points are mainly distributed around a mean linear trend confirming again the idea shown in Figure 2.1 that, to a good approximation, the variance is proportional to the number of trades. In that respect, trading time models (Eq. (2.1)) should capture most of the return variance fluctuations through the dynamics of the transaction rate. However, in Figure 2.2, the points with high abscissa values (i.e., days with a lot of activity) tend to be located above the linear line, whereas the ones with low abscissa (low activity) cluster below the linear line, suggesting that the variance per trade is dependant on the daily intensity of trading. Before moving on, we need to define a few quantities. Let t be an intraday time scale and let N be a number of trades. We define V (t;N) as the estimated price variance over the scale t conditioned by the fact that N trades occurred. Using notations, 2.2.1 (vii) and (viii), from a computational point of view, when t = t0 is fixed and N is varying, V (t = t0;N) is estimated as: V (t = t0;N) = Et (pt+t0 pt)2 j NT[t; t + t0] 2 [N N;N + N] : (2.5) where N is some bin size. And, along the same line, when we study V (t;N) for a fixed N = N0 value over a range of different values of t, one defines a temporal bin size t and computes V (t;N = N0) as1 V (t;N = N0) = Ei

### Single Trade Impact on the Midpoint Price

In this section, we will mainly focus on the impact of a trade i, and more specifically on the influence of its arrival time ti on the return rti = pti+1 pti . In order to do so, it is natural to consider the return rti conditioned by ti ti1, the time elapsed since previous transaction. We want to be able to answer questions such as : how do compare the impacts of the ith trade depending on the fact that it arrived right after or long after the previous trade ? Of course, in the framework of trading time models this question has a very simple answer : the impacts are the same ! Let us first study the conditional variance of the returns.

#### Impact on the Return Variance

In order to test the last assertion, we are naturally lead to use Eqs (2.6) and (2.7) for N0 = 1, i.e, v(t;N = 1) = Ei r2 ti j ti ti1 2 [t t;t + t] : (2.8) Let us illustrate our purpose on the DAX and the SP futures. They trade on two different exchanges, (EUREX and CME) and have very different daily statistics (e.g., DAX has the smallest perceived tick and SP the largest as one can see in Table 2.1). Figure 2.4 shows for both assets v(t;N = 1) (expressed in squared tick) as a function of t (in milliseconds). We notice that both curves present a peak for very small t and stabilize around an asymptotic constant value for larger t. This value is close to 0:7 ticks2 for the DAX and to 0:005 ticks2 for the SP. The peak reaches 0:95 (35% above the asymptote) for the DAX, and 0:1 (2000% above the asymptote).

**Large Scale Conditional Variance and Impact**

One of the key issue associated to our single trade study is the understanding of the consequences of our findings to large scale return behavior. This question implies the study of (conditional) correlations between successive trades, which is out of the scope of this paper and will be addressed in a forthcoming work.

However one can check whether the impact or the variance averaged locally over a large number of trades still display a dependence with respect to the tradingrate. Indeed, in Figure 2.3 we have already seen that this feature seems to persist when one studies returns over a fixed time (e.g., 5 min) period conditioned by the mean intertrade duration over this period. Along the same line, one can fix a large N = N0 value and compute v(t;N = N0) and I(t;N = N0) as functions of t. Note that v(t;N = N0) is defined in Eq. (2.7) while the aggregated impact can be defined similarly as: I(t;N = N0) = Ei i(pti+N0 pti) j ti1+N0 ti1 2 [t t;t + t] : (2.13) and the return impact I(t;N = 200) as functions of t. One sees that at these coarse scales, the increasing of these two quantities as the activity increases is clear (except maybe for the variance of the EURO). As compared to single trade curves, the threshold-like behavior are smoothed out and both variance and return impacts go continuously from small to large values as the trading rate increases.

**Liquidity Decreases When Trading Rate Increases**

One possible interpretation of these results would be that when the trading rate gets greater and greater, the liquidity tends to decrease, i.e., the orderbook tends to empty.

In [87], the authors mention that the spread is an indicator of the thinness of the book and that the distance from the best bid or ask to the next level of the orderbook is in fact equivalent to the spread. Moreover, they bring empirical evidence and theoretical no-arbitrage arguments suggesting that the spread and the variance per trade are strongly correlated. Accordingly, we define the mean spread over N trades .

**Table of contents :**

**1 Introduction **

1.1 Motivation, Topics and Thesis Axis

1.2 Continuous Auction and Electronic Markets

1.2.1 Continuous Auction

1.2.2 The Orderbook

1.3 Tick Size

1.4 Background and Models

1.4.1 Transaction Impact Models

1.4.2 Orderbook Models

1.4.3 Relationship to Our Work

1.5 Data Presentation

1.6 Thesis Results

1.6.1 Intensity of Trading and the Orderbook

1.6.2 Discrete Microstructure Noise

1.6.3 Non Parametric Estimation of Hawkes Decay Kernels

**2 The Nature of Price Returns During Periods of High Market Activity **

2.1 Introduction

2.2 Data Description

2.3 Realized Variance Versus Number of Trades

2.4 Single Trade Impact on the Midpoint Price

2.4.1 Impact on the Return Variance

2.4.2 Impact on the Return

2.5 From Fine to Coarse

2.5.1 Large Scale Conditional Variance and Impact

2.5.2 Liquidity Decreases When Trading Rate Increases

2.6 Concluding Remarks

**3 Discrete Microstructure Noise **

3.1 Introduction

3.1.1 The Signature Plot Effect

3.1.2 Our Approach

3.2 General Framework Model

3.2.1 Realized Variance

3.2.2 Autocovariance

3.3 Data Presentation and Tick Size

3.4 Application to Financial Data

3.4.1 Parameter Fitting

3.4.2 Application to the Bund Futures

3.4.3 Results for Other Assets

3.5 Supplemental material: a point process approach

**Appendices **

3.A Tables

3.B Figures

3.B.1 Trade Prices

3.B.2 Last Bid Prices

3.B.3 Midpoint Prices

**4 Hawkes Kernel Estimation **

4.1 Introduction

4.2 Multivariate Hawkes Processes

4.2.1 Notations and Definitions

4.2.2 Martingale Representation of t

4.3 The Covariance Operator of Hawkes Processes

4.3.1 The Infinitesimal Covariance Operator

4.3.2 The Covariance Operator

4.4 Non-parametric Estimation of the Kernel t

4.4.1 The Estimation Principle

4.4.2 Further Hypothesis on the Kernel t

4.4.3 The Estimator

4.4.4 Some Particular Cases

4.5 Numerical Illustrations

4.5.1 The Case n = 1

4.5.2 The Case n = 2

4.6 Application to High Frequency Market Data

4.6.1 1D Model

4.6.2 2D model

**Appendices **

4.A Example of a 2D Mutually Exciting Hawkes Process

4.A.1 A Mutually Exciting Process

4.A.2 Simulation

4.A.3 Covariance Operator

4.A.4 Convergence of the Covariance Operator

**Bibliography**