High-Frequency Trading and Probability Theory

Zhaodong Wang, Weian Zheng

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814616539
• Topic: Business
• Subtopic: Finance
• Index: Business

This book is the first of its kind to treat high-frequency trading and technical analysis as accurate sciences. The authors reveal how to build trading algorithms of high-frequency trading and obtain stable statistical arbitrage from the financial market in detail. The authors' arguments are based on rigorous mathematical and statistical deductions and this will appeal to people who believe in the theoretical aspect of the topic.

Investors who believe in technical analysis will find out how to verify the efficiency of their technical arguments by ergodic theory of stationary stochastic processes, which form a mathematical background for technical analysis. The authors also discuss technical details of the IT system design for high-frequency trading.

Contents:

• Introduction
• Market Microstructure
• Some Basic HFT Strategies
• IT System
• Stationary Process and Ergodicity
• Stationarity and Technical Analysis
• HFT of a Single Asset
• Bid, Ask and Trade Prices
• Financial Engineering
• Debate and Future

Readership: Graduates and researchers interested in frequency trading; finance professionals. Key Features:

• Reveals algorithms and scientific background of high-frequency trading
• Reveals the IT system preparation for high-frequency trading
• Emphasizes that technical analysis is not pseudo-science

Chapter 1

Introduction

 

In recent years, high-frequency trading (HFT) became an extremely hot topic in capital markets around the world. Normally, HFT has the following attributes:

• It is a kind of automatic trading strategy, which means that the trading is done by computer and there is no instance of human decision making during the buying or selling of any instrument. Some HFT systems allow traders to have some influence on it, but they can only change a few parameters of the strategy, and must allow the system to apply these parameters to the strategy making the trades.
• The traders keep positions for a very short time. In most cases, only several seconds or minutes. Some people believe that traders should not take any position over night, while others think it is allowed. But most agree that, HFT should not keep the main parts of the position to the second day.
• High-frequency traders only trade via electronic trading systems, not over-the-counter (OTC) markets, or any market still using outcries.
• It uses a high-speed connection to connect to the market, so as to retrieve high-frequency market data and place orders. In most cases, HFT requires the fastest access methods for a specified market, for example, the securities exchange place.
• Low latency is always very important for HFT. Various technologies are used for this target, including software optimization, hardware accelerators and dedicated network equipments. The time difference between an input message and corresponding order insert action has been defined as the internal response time, which is a key benchmark of latency for HFT. Recent competition has reduced it to several microseconds.
• Unlike other low latency trading strategies, high-frequency traders normally calculate profit and Sharpe ratio on a daily basis, not just annually.

Automatic trading is a result of wide acceptance of electronic trading platforms. Compared with doing these strategies manually, automatic trading has a lot of advantages: accuracy, objectiveness, no emotion, and low cost. Therefore, it is becoming more and more popular. Any strategies that are without the feelings of traders can be converted to an automatic trading strategy. As the market grows, the trading costs become lower and lower, allowing for more high-frequency strategy to apply to the market. A trader cannot guarantee that his trades are all with positive return. However, if the strategy is correct and can be applied repeatedly in a statistically stationary way, then according to the strong ergodic theorem, the accumulated profit will be increasing at a stable rate. Therefore, all automatic traders may try to find some HFT algorithm for a large accumulated volume. So we can say that, HFT is the result of market and technological improvements.

At the end of the last century, electronic trading platforms have become the main trading media for the US and European exchanges. HFT was created by proprietary trading firms, hedge funds and investment banks. As HFT achieved higher profit rates and stable performances, more and more institutional investors started to use it. It became so popular that in 2009, 73% of the US equity trading volume was made by HFT firms, accounting for only 2% of 20,000 funds [15].

Like any new financial tools, the HFT comes with a considerable amount of controversy. Recall the discussion about the greedy usurer in the middle ages, and the consequence of South Sea Bubble on the London Stock Market, the cases are somewhat similar to what is happening for HFT.

In this book, we first discuss some classical program trading based on the Chinese futures market. The classical program trading is a trade of a basket of assets which is executed by a computer program (accurate to the level of microseconds) based on a predetermined algorithm. There have been essentially two reasons to use this type of program trading:

(a)	When one desires to trade several stocks or futures at the same time (for example, when a mutual fund receives an influx of money, it will use that money to increase its holdings in the multiple stocks on which the fund is based);
(b)	When one wishes to arbitrage temporary price discrepancies between related financial instruments, such as between an index and its constituent parts.

Probability theory and statistics are deeply involved in the two types of trading mentioned above, because of the existence of the time gap (although only relevant for a few seconds) between the exchange platform sending out the current bid-ask information and receiving traders’ order. The main difficulty is that a high-frequency trader may take a loss if his orders have been only partially executed. Therefore, a successful trader should design an algorithm such that his profits may be related to an ergodic stationary process with a positive mean so that the strong ergodic theorem assures the accumulated profitability increasing in a stable manner.

Generally speaking, the more assets in one’s basket to trade, the more risk that his order will be only partially executed. Therefore, it is wise to reduce the number of assets in the trading basket. A problem is raised naturally: can one trade only one asset repeatedly to take stable profits in HFT? In the second part of this book, we discuss some “new” (possibly known by some traders but unpublished due to its profitability) algorithms of HFT which is built on the ergodicity of stationary processes. We show that the technical indicators used in the market form a multidimensional stationary process, which may lead considerable statistical arbitrage in HFT. The technical indicators have a mathematical and statistical background. They work much better in HFT than in ordinary trading, which can be explained from the behavioral finance point of view (see Section 9.3).

Our method is also useful when one wants to trade a large quantity of assets. A direct trade order of a large quantity of an asset will affect the price by pushing it toward the undesired direction (for example, imagine what would happen if a trader bids several millions of shares of a stock—the price will jump). So if a trader knows the algorithms of HFT, he knows how to separate his orders to reduce the buying cost and get a better selling price.

Let us briefly describe here one asset HFT. The price P(t) at time t of certain asset is a stochastic process in t. That is, for each fixed t, P(t) is a random variable. Denote P(t) = log P(t) and ∆p(t) as its increment, i.e., ∆p(t) = P(t) − p(t − δ) for some fixed positive δ. We will mainly use per 0.5 second high-frequency data, so our time unit is 0.5 second and δ = 1 unit (0.5 second). {∆p(t)} is called as the logarithmic return in finance, which can be considered as a strongly stationary process (see Sections 5.2 and 6.2) for certain heavily traded securities during the main trading hours. That will be our major hypothesis throughout the whole book. That hypothesis is in the common core of most popular mathematical models for security price. As we know, the more sophisticated a stochastic model might be, the more difficult to be verified in applications. Therefore, we select the simplest one, which can be (partially) tested by statistics.

We show that the basic technical indicators can be considered as (or transformed into) a multidimensional function

X(t) = f(∆p(s), t − q ≤ s ≤ t),

which depends only on the logarithmic returns in the past time interval [t − q, t] of the length q. As a function of stationary process, {X(t)} is also stationary. When we use an algorithm based on {X(t)} to repeatedly trade one unit of the asset, then the instant logarithmic return will be {H(t −1)∆p(t)}, where H(t − 1) = 1 if we have the asset or H(t − 1) = 0 if we do not have the asset at time t − 1. These instant logarithmic returns form a stationary time series. Denote by M(T) the cumulated logarithmic return by time T after deduction of the trading costs. Then, we can use the strong ergodic theorem to show that the mean logarithmic return M(T)/T converge for large T. The limit will be a positive constant when the instant logarithmic return is ergodic and has positive mean after cost deduction. That is the mathematical background of HFT. An interesting remark is that when one uses 0.5 second as the time unit, four trading hours are equivalent to T = 28,800 (0.5 second), which is a quite large number in mathematical statistics.

The algorithm of HFT depends on the microstructure of the market. The trading regulations and transaction costs heavily affect the HFT algorithms. In this book, we use the Chinese futures market as an example. Nevertheless, most of our principles also apply to the other markets.

China has the fastest growing capital market in the world, especially after the release of the CSI 300 index futures contracts in 2010. Here is a figure on the growth of trading volume of CSI 300 futures in 2012 (based on public data from the China Financial Futures Exchange).

Here is a sub-table on top index futures contracts in 2012 from FIA [1], which illustrates the tremendous growth of Chinese index futures market.

The Chinese capital market has electronic trading platform, which has no obstacle for automatic trading. However, the high trading costs and low liquidity made HFT unavailable until 2005 in the futures market. We cannot find any official statistics on the percentage of the trading volume made by HFT in China, but it is widely believed that at least 20% of futures market trading volume comes from HFT. On the other hand, because of the T + 1 rule (the shares cannot be sold on the same day that they have been purchased) in the Chinese stock market, HFT can rarely be used in cash markets in China. HFT is fast growing, and has not yet reached the peak in current Chinese derivative markets.

We discuss in Chapter 2 the microstructure of the Chinese futures markets, including trading, clearing, market data, risk control, etc. If you are quite familiar with the market microstructure of the Western exchanges, you can skip most of the sections of this chapter, and just read the last section, which describes the differences between the Chinese futures market and the Western markets. Based on our market microstructure understanding and simple mathematical tools, we can analyze classical HFT strategies, and their applications in the Chinese market, which are the topics of Chapter 3. After having a suitable strategy, the next problem is to design an IT system to implement; this is described in Chapter 4. In order to reach more readers from disciplines other than mathematics, we use Chapter 5 to introduce the basic concepts of probability and statistics, especially the ergodic theorem for stationary process. Mathematicians may skip that chapter, except for the last example of HFT based on stationary process. We use Chapter 6 to show that the technical analysis of financial market has a statistical foundation. Technical indicators are associated with stationary processes. Therefore, a trader can calculate statistics based on these stationary processes and the strong ergodic theorem that guarantee the observed relative frequencies will converge to the corresponding probabilities. Chapter 7 is used to describe the mathematical foundation for HFT of a single asset and gives two examples. We will show how to use the bid-ask spread and the last price to insert trade orders in Chapter 8. HFT can be considered an important application of financial engineering. Chapter 9 is an overview of financial engineering including a few related fields such as computational finance, mathematical finance, statistical finance and behavioral finance. Chapter 10 is used to discuss the future of HFT.

Chapter 2

Market Microstructure

Algorithms of high-frequency trading (HFT) are heavily based on the market. A profitable algorithm in one market may not work in another market. That is the reason why we need to study the market microstructure on which our algorithms are based.

2.1 Trading Products

Currently, there are four futures exchanges in China, China Financial Futures Exchange (CFFEX), Shanghai Futures Exchange (SHFE), Dalian Commodity Exchange (DCE) and Zhengzhou Commodity Exchange (CZCE).

CFFEX is the only futures exchange focusing on financial products in China. It was set up in 2006, and launched its first financial product, CSI 300 futures (Symbol IF), in 2010. CFFEX is growing so fast that it has become the largest futures exchange in China according to trading tu...

Table of contents

• Cover Page
• Title
• Copyright
• Contents
• Foreword
• Preface
• About the Authors
• 1. Introduction
• 2. Market Microstructure
• 3. Some Basic HFT Strategies
• 4. IT System
• 5. Stationary Process and Ergodicity
• 6. Stationarity and Technical Analysis
• 7. HFT of a Single Asset
• 8. Bid, Ask and Trade Prices
• 9. Financial Engineering
• 10. Debateand Future
• References
• Index