Get Complete Project Material File(s) Now! »

## Relationship to Our Work

We find the modeling approach presented above particularly appealing. First, we relish the idea of having models that are built from the ground up as they are in good position to replicate market behavior over multiple observation scales, from fine scales to coarse scales in particular. In this framework, the modeling frontier is pushed to the most elementary level by modeling the arrival of different orders and their interactions. The price therefore becomes a byproduct of the inner workings of the model.

We also appreciate the fact that these models are conceptually simple and imitate realistically the workings of the market and do not make any assumptions about what is not observable, like the traders’ strategies or behavior. Hence, their input parameters can be directly estimated from observing the market. The results of the models, when compared against real data, would be interpreted in light of assumed participants’ behavior, but the behavior is never a parameter of the model.

In this work, we pretend to bring additional elements that contribute to the understanding of a fine to coarse model. For instance, if we look at the models described in section 1.4.1 equation (2.1), we notice that physical time does not play any role in the way market prices vary from trade to trade. This implies notably that the variance per trade (or per unit of volume traded) is constant and therefore tha the volatility over a fixed physical time scale, is only dependent on the number of trades. In chapter 2 of the thesis, we critically examine this underlying assumption and provide empirical evidence portraying in what way this is assumption is broken We follow in chapter 3 with a simple model that mimics the random arrivals of trades whose prices present a mean reverting behavior at the fine scale, that vanishes at coarse scales. Tested against real data, the model was able to replicate multiscale empirical observations on the variance and the autocorrelation of trade prices. This also provides a new way to look at the tick size as defined in section Finally, chapter 4 studies a statistical method to estimate a powerful class of point processes. Inspired by previous work on modeling mutually and self exciting processes [9] and convinced by their potential in modeling the arrival of orders, we provide a non parametric moment method to estimate the decay kernel of these processes.

We introduce the works in section 1.6 of this introduction and present them in details in parts 2, 3 and 4. But now let’s introduce the data we worked with.The database we use is extensive, covering several world markets and different Futures asset classes that we describe in detail in section 1.5.

### Data Presentation

The large scope of the data is necessary to show the universality of our findings, in electronic double auctions at least. Thus, we restrict our analysis to assets that trade in markets that match the framework of the electronic double auction described above (section 1.2).

In particular, we use level 15 data6 of 10 futures contracts on assets of different classes that trade in different exchanges. The data has millisecond accuracy and was recorded over the past three years. In chapter 2 we have a date range going from 2008/08 till 2010/03, in chapter 3 we mainly use data from June 2009, and in chapter 4 we use data from October 2009. On the EUREX exchange, localized in Zürich, Switzerland, we use futures on the DAX index (DAX) and on the EURO-STOXX 50 index (ESX), and on three interest rates based on German government debt: 10-years Euro-Bund (Bund), 5-years Euro-Bobl (Bobl) and the 2-years Euro-Schatz (Schatz). On the CBOT exchange, localized in Chicago, US, we use the futures on the Dow Jones index (DJ) and the 5-Year U.S. Treasury Note Futures (BUS5). On the CME, also in Chicago, US, we use the forex EUR/USD futures (EURO) and the the futures on the SP500 index (SP). Finally we use the Light Sweet Crude Oil Futures (CL) that trades on the NYMEX, localized in New-York, US. As for their asset classes, the DAX, ESX, DJ, and SP are equity futures, the Bobl, Schatz, Bund, and BUS5 are

fixed income futures, the EURO is a foreign exchange futures and finally the CL is an energy futures. This information is summarized in table 1.1. A futures contract is a contract allowing the purchase of an underlying physical good (be that a government bond, natural gas tankers, stocks or currency) on a future settlement date, at a price determined in the present [23]. It allows the traders of these contracts to secure the price of the good that they will physically deliver or receive at the settlement date. Therefore at any particular time, there may exist infinitely many contracts, one for every conceivable future settlement date. On an exchange however, settlement dates are standardized, one every three months (March, June, September and December) and generally three future settlement months are trading at the same time. We dealt with this issue by keeping, on each day, the contract that recorded the most number of trades and, discarding the other maturities.

We also remind that when one market order hits several limit orders it results in several trades being reported (section 1.2.2). We aggregate together all such transactions and consider them as one market order whose size is the the sum of the volumes of all the reported trades and whose price is the price of the last limit order filled.

We refer the reader to table 1.1 for a summary of the information presented above and some simple statistics about the assets, such as the daily trading session, the average number of trades per day, and the proxies for the tick size and P=. We can see in that table that the assets are not homogenous: they have different exchanges (CBOT, CME, EUREX and NYMEX), different classes (Energy, Equity, Foreign Exchange and Interest Rates), different tick values and currency (e.g. DAX: 12.5e, DJ: 5$), different daily trading sessions, and different trading behavior:

average trades per day (e.g. Schatz: 10000 trades per day, SP: 100000 trades per day) and different tick sizes (e.g. DAX: small, DJ: large). Throughout this work, we rely on this diversity in order to confirm the universality of our findings.

#### Intensity of Trading and the Orderbook

As we have pointed in section 1.4, we have seen in recent years the development of what is usually referred to as transaction time models. Early works on alternative time clocks [56, 26] show that an asset price can be modeled as a Brownian motion with an alternative time clock Xt = Bt where Bt is the Brownian motion and t a stochastic time clock independent of Bt [43]. Since then, a lot of research has been made in effort to find the best plausible time clock t ([84] and the references therein) and today, there is a general consensus that the cumulated number of transactions is a good choice for the time clock ([7, 28]). This transaction time clock is rightfully seen to account for the highly intermittent nature of the volatility, a feature that manifests itself at practically all time scales, from intradaily scales (where periods of intense variations are observed, for instance around publications of important news), to monthly scales ([23, 28]).

In [43], the authors critically examine this view about transaction time. They show that while the transaction time clock is indeed an adequate Brownian subordination, it does not account for the clustered nature of the volatility nor for large price movements. Our article sheds a different light on the matter by looking at what role the physical time still plays in the series of prices, even as we look at the data in transaction time. In fact, the transaction time subordinated Brownian motion model suggests that the physical time does not play any role in the way prices vary from trade to trade. This implies notably that the variance per trade is constant as a function of the physical time and therefore that the volatility over a fixed physical time scale, is only dependent on the number of 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**