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Gaussian Process-Based Algorithmic Trading

Strategy Identification

Steve Y. Yang†, Qifeng Qiao†, Peter A. Beling†,

William T. Scherer†, and Andrei A. Kirilenko‡ ‡

†The University of Virginia, 151 Engineer’s Way, P.O. Box 400747,

Charlottesville, VA 22904

‡ MIT Sloan School of Management, 50 Memorial Drive,

Cambridge, Massachusetts 02142

Corresponding author. Email: yy6a@viginia.edu; and ak67@mit.edu

The research presented in this paper was co-authored by Andrei Kirilenko, a former full-time CFTC employee, Peter Beling,

a former CFTC contractor who performed work under CFTC OCE contract (CFCE-11-CO-0233), William Scherer, a former

CFTC contractor who performed work under CFTC OCE contract (CFCE-11-CO-0234), Steve Yang, a former CFTC contractor

who performed work under CFTC DOE contract (CFEN-12-CO-0092), and Qifeng Qiao, a Ph.D. student at the University

of Virginia who had no official affiliation with the CFTC. The Office of the Chief Economist and CFTC economists produce

original research on a broad range of topics relevant to the CFTCs mandate to regulate commodity futures markets, commodity

options markets, and the expanded mandate to regulate the swaps markets pursuant to the Dodd-Frank Wall Street Reform and

Consumer Protection Act. These papers are often presented at conferences and many of these papers are later published by

peer-review and other scholarly outlets. The analyses and conclusions expressed in this paper are those of the authors and do

not reflect the views of other members of the Office of the Chief Economist, other Commission staff, or the Commission itself.
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Abstract

Many market participants now employ algorithmic trading, commonly defined as the use of computer algorithms to automatically make certain trading decisions, submit orders, and manage those

orders after submission. Identifying and understanding the impact of algorithmic trading on financial

markets has become a critical issue for market operators and regulators. Advanced data feed and audit

trail information from market operators now make the full observation of market participants’ actions

possible. A key question is the extent to which it is possible to understand and characterize the behavior

of individual participants from observations of trading actions.

In this paper, we consider the basic problems of categorizing and recognizing traders (or, equivalently, trading algorithms) on the basis of observed limit orders. These problems are of interest to

regulators engaged in strategy identification for the purposes of fraud detection and policy development.

Methods have been suggested in the literature for describing trader behavior using classification rules

defined over a feature space consisting of the summary trading statistics of volume and inventory and

derived variables that reflect consistency of buying or selling behavior. Our principal contribution is to

suggest an entirely different feature space that is constructed by inferring key parameters of a sequential

optimization model that we take as a surrogate for the decision making process of the traders. In

particular, we model trader behavior in terms of a Markov decision process (MDP). We infer the reward

(or objective) function for this process from observation of trading actions using a process from machine

learning known as inverse reinforcement learning (IRL). The reward functions learned through IRL then

constitute a feature space that can be the basis for supervised learning (for classification or recognition

of traders) or unsupervised learning (for categorization of traders). Making use of a real-world data

set from the E-Mini futures contract, we compare two principal IRL variants, linear IRL (LIRL) and

Gaussian Process IRL (GPIRL), against a method based on summary trading statistics. Results suggest

that IRL-based feature spaces support accurate classification and meaningful clustering. Further, we

argue that, because they attempt to learn traders’ underlying value propositions under different market

conditions, the IRL methods are more informative and robust than the summary statistic-based approach

and are well suited for discovering new behavior patterns of market participants.

Index Terms

Inverse Reinforcement Learning; Gaussian Process; High Frequency Trading; Algorithmic

Trading; Behavioral Finance; Markov Decision Process; Support Vector Machine

I. INTRODUCTION

Financial markets have changed dramatically over the past 10 years or so. These changes

reflect the culmination of a decade-long trend from a market structure with primarily manual floor
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trading to a market structure dominated by automated computer trading. This rapid transformation

has been driven by the evolution of technologies for generating, routing, and executing orders,

which have dramatically improved the speed, capacity, and sophistication of the trading functions

that are available to market participants.

High-quality trading markets promote capital formation and allocation by establishing prices

for securities and by enabling investors to enter and exit their positions in securities wherever

and whenever they wish to do so. The one important feature of all types of algorithmic trading

strategies is to discover the underlying persistent tradable phenomena and generate trading

opportunities. These trading opportunities include microsecond price movements that allow a

trader to benefit from market-making trades, several minute-long strategies that trade on momentum forecasted by market microstructure theories, and several hour-long market movements

that surround recurring events and deviations from statistical relationship (Aldridge (2010)).

Algorithmic traders then design their trading algorithms and systems with the aim of generating

signals that result in consistent positive outcomes under different market conditions. Different

strategies may target different frequencies, and the profitability of a trading strategy is often

measured by a certain return metric. The most commonly used measure is the Sharpe ratio, a

risk-adjusted return metric first proposed by Sharpe (Edwards (1966)).

In particular, there is a subgroup within the algorithmic trading strategies called High Frequency Trading (HFT) strategies that have attracted a lot of attention from investors, regulators,

policy makers, and academics broadly. According to the U.S. Securities and Exchange Commission, high-frequency traders are ”professional traders acting in a proprietary capacity that engage

in strategies that generate a large number trades on daily basis.” (The SEC Concept Release on

Equity Market Structure, 75 Fed. Reg. 3603, January 21, 2010). The SEC characterized HFT

as (1) the use of extraordinary high-speed and sophisticated computer programs for generating,

routing, and executing orders; (2) use of co-location services and individual data feeds offered

by exchanges and others to minimize network and other types of latencies; (3) very short timeframes for establishing and liquidating positions; (4) the submission of numerous orders that are

canceled shortly after submission; and (5) ending the trading day in as close to a flat position as

possible (that is, not carrying significant, unhedged positions over night). Although many HFT

strategies exist today and they are largely unknown to public, researchers have shed lights on

their general characteristics recently. Several illustrative HFT strategies include: (1) acting as an
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informal or formal market-maker, (2) high-frequency relative-value trading, and (3) directional

trading on news releases, order flow, or other high-frequency signals (Jones (2012)).

In the past few years, there have been a number of studies of HFT and algorithmic trading

more generally. Their primary objective is to understand the economic impact of these algorithmic

trading practices to the market quality including liquidity, price discovery process, trading costs,

etc. On the empirical side, some researchers have been able to identify a specific HFT in data,

and others are able to identify whether a trade is from algorithmic traders. Given the amount

of information provided by exchanges and data vendors, it is possible to describe patterns in

algorithmic order submission, order cancellation, and trading behavior. It is also possible to

see whether algorithmic or HFT activities are correlated with bid-ask spreads, temporary and/or

permanent volatility, trading volume, and other market activity and quality measures. Hendershott

et al. (2011) study the implementation of an automated quote at the New York Exchange. They

find that the implementation of auto-quote is associated with an increase in electronic message

traffic and an improvement in market quality including narrowed effective spreads, reduced

adverse selection, and increase price discovery. These effects are concentrated in large-cap firms,

and there is little effect in small-cap stocks. Menkveld (2012) studies the July 2007 entry of

a high-frequency market-maker into the trading of Dutch stocks. He argues that competition

between trading venues facilitated the arrival of this high-frequency market-maker and HFT

more generally, and he shows that high-frequency market-maker entry is associated with 23%

less adverse selection. The volatility measured using 20 minutes realized volatility is unaffected

by the entry of the high-frequency market-maker. Riordan et al. (2012) examine the effect

of a technological upgrade on the market quality of 98 actively traded German stocks. They

conclude that the ability to update quotes faster helps liquidity providers minimize their losses

to liquidity demanders, and there are more price discovery take place. Boehmer et al. (2012)

examine international evidence on electronic message traffic and market quality across 39 stock

exchanges over the 2001-2009 period. They find that co-location increases algorithmic trading

and HFT, and introduction of co-location improves liquidity and the information efficiency of

prices. However, they claim volatility does not decline as much as it would be based on the

observed narrower bid-ask spreads. Gai et al. (2012) study the effect of two recent 2010 Nasdaq

technology upgrades that reduce the minimum time between messages from 950 nanoseconds

to 200 nanoseconds. These technological changes lead to substantial increase in the number of
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canceled orders without much change in overall trading volume. There is so little change in

bid-ask spreads and depths. Overall, these studies have focused on empirical evidence that an

increase in algorithmic trading has positive influence on market quality in general.

On the theoretical side, there are a number of models developed to understand the economic

impact of these algorithmic trading practices. Biais et al. (2012) conclude HFT can trade on

new information more quickly, generating adverse selection costs, and they also find multiple

equilibrium in their model, and some which exhibit socially inefficient over investment in HFT.

The model from Jovanovic et al. (2010) shows that HFT can avoid some adverse selection,

and can provide some that benefit to uninformed investors who need to trade. Martinez et

al. (2012) conclude from their model that HFT obtains and trades on information an instant

before it is available to others, and it imposes adverse selection on market-makers. Therefore

liquidity is worse, and prices are no longer efficient. Martinez et al. (2012) focus on HFTs that

demand liquidity, and suggest that HFT makes market prices extremely efficient by incorporating

information as soon as it becomes available. Markets are not destabilized, as long as there is

a population of market-makers standing ready to provide liquidity at competitive prices. Other

related theoretical models include Pagnotta et al. (2012), who focus on the investment in speed

made by exchanges in order to attract trading volume from speed sensitive investors. Moallemi

et al. (2012) argue that a reduction in latency allows limit order submitters to update their

orders more quickly, thereby reducing the value of the trading option that a limit order grants

to a liquidity demander. The common theme in these models is that HFT may increase adverse

selection, and it is harmful for liquidity. However, the ability to intermediate traders who arrive

at different times is generally good for liquidity.

Moreover, there have been a number of studies focused on algorithmic traders’ behaviors.

These studies examine the trading activities of different types of traders and try to distinguish

their behavioral differences. Hendershott et al. (2012) use exchange classifications distinguish

algorithmic traders from orders managed by humans. They find that algorithmic traders concentrates in smaller trade sizes, while large block trades of 5,000 shares or more are predominantly

originated by human traders. Algorithmic traders consume liquidity when bid-ask spreads are

relatively narrow, they supply liquidity when bid-ask spreads are relatively wide. This suggests

that algorithmic traders provide a more consistent level of liquidity through time. Brogaard (2012)

and Hendershott et al. (2011) work with Nasdaq data that flag whether trades involves HFT.
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Hendershott et al. (2011) find that HFT accounts for about 42% of (double-counted) Nasdaq

volume in large-cap stocks but only about 17% of volume in small-cap stocks. They estimate a

state-space model that decomposes price changes into permanent and temporary components, and

measures the contribution of HFT and non-HFT liquidity supply and liquidity demand to each

of these price change components. They find that when HFTs initiate trades, they trade in the

opposite direction to the transitory component of prices. Thus, HFTs contribute to price discovery

and contribute to efficient stock prices. Brogaard (2012) similarly finds that 68% of trades have

an HFT on at least one side of the transaction, and he also finds that HFT participation rates

are higher for stocks with high share prices, large market caps, narrow bid-ask spreads, or low

stock-specific volatility. He estimates a vector autoregressive permanent price impact model and

finds that HFT liquidity suppliers face less adverse selection than non-HFT liquidity suppliers,

suggesting that they are somewhat judicious in supplying liquidity. Kirilenko et al. (2011) use

account-level tick-by-tick data on the E-Mini S&P 500 futures contract, and they classify traders

into various categories, including HFTs, opportunistic traders, fundamental traders and noise

traders. Benos et al. (2012) conduct a similar analysis using UK equity data. These different

datasets provide considerable insight into overall HFT trading behavior.

One of the important goals of learning traders trading strategies is to be able to categorize

and identify the market participants, and be able to further understand their influences related

to such important economic issues as multiple characterizations of price formation processes,

market liquidity, and order flow, etc. (Hasbrouchk et al. (2001), Gabaix et al. (2003), Gatheral

(2010), Hasbrouck (1991), and Jones et al. (1994)). We assert that enhanced understanding of

the economic implication of these different algorithmic trading strategies will yield quantitative

evidence of value to market policy makers and regulators seeking to maintain transparency,

fairness and overall health in the financial markets.

In particular, traders deploy different trading strategies where each strategy has a unique value

proposition under a particular market condition. In other words, we can cast this problem as a

sequential decision problem under different conditions. Traders aim to optimize their decisions

overtime and consequently maximize their reward under different market conditions. We can

theoretically use reward functions to represent the value system that are encapsulated in the

various different trading strategies. It is possible to derive new policies based on the reward

functions learned and apply them in a new environment to govern a new autonomous process.
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This process is defined as reward learning under the framework of inverse reinforcement learning

( Ng et al. (2000), Abbeel et al. (2004) and Ramachandran et al. (2007). For example, a

simple keep-or-cancel strategy for buying one unit, the trader has to decide when to place

the order and when to cancel the order based on the market condition which may likely be

characterized as a stochastic process. However the value proposition for the trader is to buy

one unit of the security at a lowest price possible. This could be realized in a number of

ways. It could be described as a reward function meaning when the system is in a particular

state, the trader is always looking for a fixed reward. This notion of value proposition drives

the trader to take corresponding actions according to the market conditions. This ultimately

constitutes trader’s policies or strategies. Therefore a strategy under certain value proposition can

be consistently programmed in algorithms to achieve its goal of buy-one-unit in an optimal way.

Consequently, strategies developed under certain value frameworks can be observed, learned and

even reproduced in a different environment (such as a simulated financial market where impact

of these strategies can be readily assessed). As documented by Yang et al. (2012), Hayes et al.

(2012) and Paddrik et al. (2012), the manipulative or disruptive algorithmic strategies can be

studied and monitored by market operators and regulators to prevent unfair trading practices.

Furthermore, new emerging algorithmic trading practices can be assessed and new regulations

and policies can be evaluated to maintain the overall health of the financial markets.

In this study, we model the trading behavior of different market participants by the solution

to the inverse Markov decision process (MDP). We try to describe how traders are able to take

actions in a highly uncertain environment to reach return goals on different horizons. This task

can be solved using dynamic programming (DP) and reinforcement learning (RL) based on MDP.

The model accounts for traders’ preferences and expectations of uncertain state variables. In a

general MDP modeling setting, we describe these variables in two spaces: the state space and

the action space. From the trading decision perspective, we can parameterize learning agents

using reward functions that depend on state and action. We consider the market dynamics in

view of the learning agents’ subjective beliefs. The agents perform DP/RL through a sense, trial

and learn cycle. First, the agents gain state information from sensory input. Based on the current

state, knowledge and goals, the agents find and choose the best action. Upon receiving new

feedback, the agents learn to update their knowledge with a goal of maximizing their cumulative

expected reward. In the discrete-valued state and action problem space, DP and RL methods use
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similar techniques involving policy iteration and value iteration algorithms (Bertsekas (1996),

and Sutton et al. (1998)) to solve MDP problems. Formalisms for solving forward problems

of RL are often divided into model-based and model-free approaches (Daw et al. (2005), and

Sutton et al. (1998)).

As framed by Abbeel et al. (2004) under the Inverse Reinforcement Learning (IRL) framework,

the entire field of reinforcement learning is founded on the presupposition that the reward

function, rather than policy, is the most succinct, robust, and transferable definition of the task.

However, the reward function is often difficult to know in advance for some real-world tasks,

so the following difficulties may arise: 1) We have no experience to tackle the problem; 2) We

have experience but can not interpret the reward function explicitly; 3) The problem we solve

may be interacting with the adversarial decision makers who make all their effort to keep the

reward function secret. Rather than accessing the true reward function, it is easier to observe the

behavior of some other agents (teacher/expert) to determine how to solve the problem. Hence, we

have motivation to learn from observations. Technical approaches to learning from observations

generally fall into two broad categories Ratliff et al. (2009). The first category, called imitation

learning, attempts to use supervised learning to predict actions directly from observations of

features of the environments, which is unstable and vulnerable to highly uncertain environment.

The second category is concerned with how to learn the reward function that characterizes the

agent’s objectives and preferences in MDP ( Ng et al. (2000)).

IRL was first introduced in machine learning literature by Ng et al. (2000) in formulating

it as an optimization problem to maximize the sum of differences between the quality of the

optimal action and the quality of the next-best action. Other algorithms have been developed or

integrated into apprenticeship learning based on this linear approximation of the reward function.

The principal idea of apprenticeship learning using IRL is to search mixed solutions in a space

of learned policies with the goal that the cumulative feature expectation is near that of the expert

(Abbeel et al. (2004) and Syed et al. (2008)).

Other algorithms have also been developed under the IRL framework. A game-theoretic

approach to apprenticeship learning using IRL was developed in the context of a two-player

zero-sum game in which the apprentice chooses a policy and the environment chooses a reward

function (Syed et al. (2007)). Another algorithm for IRL is policy matching, in which the

loss function penalizing deviations from the expert’s policy that is minimized by tuning the
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parameters of the reward functions (Neu et al. (2007)). The maximum entropy IRL is proposed

in the context of modeling real-world navigation and driving behaviors (Ziebart et al. (2008)).

The algorithms for apprenticeship learning using IRL do not actually aim to recover the reward

function but instead are only concerned with the optimal policy. Ramachandran and Amir

consider IRL from a Bayesian perspective without assuming the linear approximation of the

reward function (Ramachandran et al. (2007)). Their model interprets the observations from the

expert as the evidence that is used to obtain a posterior distribution over reward using Markov

Chain Monte Carlo simulation. Recent theoretical works on IRL such as the framework of the

linear-solvable MDP (Dvijotham et al. (2010)), bootstrap learning (Boularias et al. (2010)) and

feature construction (Levine et al. (2010)), have also improved the learning performance. IRL

has also been successfully applied to many real-world problems, such as the automatic control

of helicopter flight (Abbeel et al. (2010)) and the motion control of an animation system in

computer graphics (Lee et al. (2010)).

We apply a Gaussian process-based IRL (GPIRL) model proposed by Qiao and Beling (2011)

to learn the trading behaviors under different market conditions. In this GPIRL, a Gaussian prior

is assigned on the reward function and the reward function is treated as a Gaussian process. This

approach is similar to that of Ramachandran et al. (2007), who view the state-action samples

from agents as the evidence that will be used to update a prior value in the reward function,

under a Bayesian framework. The solution (Ramachandran et al. (2007)) depends on non-convex

optimization using Markov Chain Monte Carlo simulation. Moreover, the ill-posed nature of the

inverse learning problem also presents difficulties. Multiple reward functions may yield the

same optimal policy, and there may be multiple observations at a single state given the true

reward function. The GPIRL model aims to address the ill-posed nature of this problem by

applying Bayesian inference and preference graphs. Here we are faced with the challenge of

modeling traders’ action as non-deterministic policies. In general, agent’s policies range from

deterministic Markovian to randomized history dependent, depending on how traders incorporate

past information and how traders select actions. Due to the uncertainty of the environment and

the random error of the measurement in the observations, a deterministic policy could very likely

be perceived as a non-deterministic one. Modeling traders’ reward function using a Gaussian

process is well suited to address these issues. One of the main novel features of this approach

is that it not only represents a probabilistic view but is also computationally tractable.
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The dynamic nature of financial markets makes it possible to postulate a priori a relationship

between the market variables we observe and those we wish to predict. The main contributions

of this study can be summarized as follows:

1) We propose an entirely different feature space that is constructed by inferring key parameters

of a sequential optimization model that we take as a surrogate for the decision making

process of the traders. We infer the reward (or objective) function for this process from

observation of trading actions using a process from machine learning known as inverse

reinforcement learning (IRL).

2) We model traders’ reward functions using a Gaussian process. We also apply preference

graphs to address the non-deterministic nature of the observed trading behaviors, reducing

the uncertainty and computational burden caused by the ill-posed nature of the inverse

learning problem.

3) We suggest a quantitative behavioral approach to categorizing algorithmic trading strategies

using weighted scores over time in the reward space, and we conclude that it performs

consistently better than the existing summary statistic-based trader classification approach

(Kirilenko et al. (2011)).

The remainder of this paper is organized as follows: First we discuss summary statistics

approach to trader classification in section II We then discuss the framework of which we use

to model market dynamics and the traders’ decisions in sectionII We extend the MDP and

introduce IRL formulation in section IV We review the original linear IRL formulation and

provide a Bayesian probabilistic model to infer the reward function using Gaussian processes.

We apply the GPIRL algorithm to the E-Mini S&P 500 Futures market as experiments in section

V. We show that the GPIRL algorithm can accurately capture algorithmic trading behavior based

on observations of the high frequency data. We also compare our behavior-based classification

results with the results from Kirilenko et al. (2011), and show that our behavioral approach

represents a consistent improvement. Finally we provide concluding remarks about the GPIRL

and its applications in section VI

II. SUMMARY STATISTICS APPROACH TO TRADER CLASSIFICATION

Kirilenko et al. (2011) suggest an approach to classifying individual trading accounts based

on the summary statistics of trading volume and inventory and consistency of buying or selling
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behavior. Six categories are used to describe individual trading accounts:

1) High Frequency Traders – high volume and low inventory;

2) Intermediaries – low inventory;

3) Fundamental Buyers – consistent intraday net buyers;

4) Fundamental Sellers – consistent intraday net sellers;

5) Opportunistic Traders – all other traders not classified;

6) Small Traders – low volume.

In this section, we develop the details for classification rules corresponding to the six categories

from Kirilenko et al. (2011), and then apply these rules to a real-world futures contract data set.

A. E-Mini Market Data Description

The E-Mini S&P 500 is a stock market index of futures contracts traded on the Chicago

Mercantile Exchange’s (CME) Globex electronic trading platform. The notional value of one

contract is $50 times the value of the S&P 500 stock index. The tick size for the E-Mini S&P

500 is 0.25 index points or $12.50. For example, the S&P 500 Index futures contract is trading

at $1,400.00, then the value of one contract is $70,000. The advantages to trading E-mini S&P

500 contracts include liquidity, greater affordability for individual investors and around-the-clock

trading.

Trading takes place 24 hours a day with the exception of a short technical maintenance

shutdown period from 4:30 p.m. to 5:00 p.m. The E-Mini S&P 500 expiration months are March,

June, September, and December. On any given day, the contract with the nearest expiration date

is called the front-month contract. The E-Mini S&P 500 is cash-settled against the value of the

underlying index and the last trading day is the third Friday of the contract expiration month.

The initial margin for speculators and hedgers are $5,625 and $4,500, respectively. Maintenance

margins for both speculators and hedgers are $4,500. There is no limit on how many contracts

can be outstanding at any given time.

The CME Globex matching algorithm for the E-Mini S&P 500 offers strict price and time

priority. Specifically, limit orders that offer more favorable terms of trade (sell at lower prices

and buy at higher prices) are executed prior to pre-existing orders. Orders that arrived earlier

are matched against the orders from the other side of the book before other orders at the same

price. This market operates under complete price transparency. This straight forward matching
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algorithm allows us to reconstruct the order book using audit trail messages archived by the

exchanges and allows us to replay the market dynamics at any given moment.

In this paper, empirical work is based on a month of E-Mini order book audit trail data. The

audit trail data includes all the order book events timestamped at a millisecond time resolution,

and contains the following data fields: date, time (the time when the client submits the order to the

exchange), conf time (the time when the order is confirmed by the matching engine), customer

account, tag 50 (trader identification number), buy or sell flag, price, quantity, order ID, order

type (market or limit), and func code (message type, e.g. order, modification, cancellation, trade,

etc.).

B. Summary Statistic-based Classification of E-Mini Market Data

We apply the set of the statistics based trader classification rules documented by Kirilenko et

al. (2011) on our E-Mini dataset. For Fundamental traders, we calculate their end of the day net

position. If the end-of-the-day net position is more than 15% of their total trading volume on

that day, we categorize them either as Fundamental Buyers or Fundamental Sellers depending

on their trading directions. We also identify Small Traders as those accounts with a trading

volume of 9 contracts or less. We apply the criteria (Kirilenko et al. (2011)) for Intermediaries,

Opportunistic Traders and High Frequency Traders, and obtain consistent results based on the

one-month data. There are two steps involved in this process. First, we ensure that the account’s

net holdings fluctuate within 1.5% of its end of day level, and second, we ensure the account’s

end of the day net position is no more than 5% of its daily trading volume. Then if we define

HFTs as a subset of Intermediaries (top 7% in daily trading volume), we find that there is a

significant amount of overlap between HFTs and Opportunistic Traders. The problem is that

the first criterion is not well defined, as the fluctuation of net holdings is vaguely defined. Net

holdings could be measured in different ways.

In consultation with the authors of Kirilenko et al. (2011), we choose the standard deviation

of an account’s net position measured on the event clock as a measure of an account’s holding

fluctuation. With this definition, we find that a 1.5% fluctuation is too stringent for HFTs,

because many high trading volume accounts are classified as Opportunistic Traders, while in

reality their end of day positions are still very low compared with other Opportunistic Traders.

Therefore, it is adequate to relax the first criterion requiring that the standard deviation of
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the account’s net holdings throughout the day is less than its end of day holding level. We

find that the newly adjusted criteria classify most high volume trading accounts as HFTs, and

this classification rule is validated from the registration information we can acquire. Without

this adjustment, almost all the top trading accounts are incorrectly classified as Opportunistic

Traders. Table I summarizes the results after applying the new classification rule demonstrating

that the modified classification criteria identified more HFTs. On average, there are 38 HTF

accounts, 118 Intermediary accounts, 2,658 Opportunistic accounts, 906 Fundamental Buyer

accounts, 775 Fundamental Seller accounts, and 5,127 Small Trader accounts. Over the 4-week

period, only 36% of the 120 accounts that consistently classified as the same type of traders.

If we rank these accounts by their daily trading volume, we find that only 40% of the top 10

accounts are consistently classified as the same trader types. The variation occurs among the

HFTs, Intermediaries, and Opportunistic Traders.

III. MARKOV DECISION PROCESS MODEL OF MARKET DYNAMICS

In this section, we develop a Markov decision process (MDP) model of trader behavior. This

model will then serve as the basis for the inverse reinforcement learning process described in

section IV.

A. MDP Background and Notation

The primary aim of our trading behavior-based learning approach is to uncover decision

makers’ policies and reward functions through the observations of an expert whose decision

process is modeled as an MDP. In this paper, we restrict our attention to a finite countable MDP

for easy exposition, but our approach can be extended to continuous problems if desired. A

discounted finite MDP is defined as a tuple M = (S, A,P, γ, r), where

• S = {sn}

N

n=1 is a set of N states. Let N = {1, 2, · · · , N}.

• A = {am}

M

m=1 is a set of M actions. Let M = {1, 2, · · · , M}.

• P = {Pam}

M

m=1 is a set of state transition probabilities (here Pam is a N ×N matrix where

each row, denoted as Pam(sn, :), contains the transition probabilities upon taking action am

in state sn. The entry Pam(sn, sn0) is the probability of moving to state sn0, n0 ∈ N in the

next stage.).

• γ ∈ [0, 1] is a discount factor.
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TABLE I

THE E-MINI S&P 500 FUTURES MARKET DATA SUMMARY

Market Opportunistic Fundamental Fundamental Total Number Total Trading

Date HFTs Makers Traders Buyers Sellers of Accounts Volume

10/04/2012 39 193 2,833 940 818 10,425 3,261,85210/05/2012 38 162 2,598 1191 1055 11,495 3,875,23210/06/2012 38 167 2,401 895 712 9,065 2,852,24410/07/2012 39 196 2,726 919 747 9,841 3,424,76810/08/2012 32 162 2,511 847 812 9,210 3,096,80010/11/2012 21 118 1,428 636 573 6,230 1,765,25410/12/2012 38 186 2,687 896 745 9,771 3,236,90410/13/2012 38 187 2,582 1020 840 10,297 3,699,10810/14/2012 30 198 3,001 1070 795 10,591 4,057,82410/15/2012 46 210 3,109 890 773 9,918 4,437,82610/18/2012 37 173 2,126 869 724 8,735 2,458,51010/19/2012 52 216 3,651 1030 974 11,600 5,272,67210/20/2012 39 176 2,949 951 877 10,745 3,956,79010/21/2012 43 240 3,370 952 771 10,980 4,230,19410/22/2012 32 143 1,837 676 629 7,370 2,026,23410/25/2012 38 181 2,533 888 684 9,228 3,074,55810/26/2012 37 175 2,726 816 709 9,568 3,000,62810/27/2012 45 186 2,973 919 820 10,472 3,850,55610/28/2012 39 185 2,873 914 705 9,777 3,485,91010/29/2012 37 160 2,247 794 744 8,369 3,012,860
15

• r denotes the reward function, mapping from S × A to < with the property that

r(sn, am) ,

X

n0∈N

Pam(sn, sn0)r(sn, am, sn0)

where r(sn, am, sn0) denotes the function giving the reward of moving to the next state sn0 after

taking action am in current state sn. The reward function r(sn, am) may be further reduced to

r(sn), if we neglect the influence of the action. We use r for reward vector through out this

paper. If the reward only depends on state, we have r = (r(s1), . . . , r(sN )). If we let r be the

vector of the reward depending on both state and action. We have

r = (r1(s1), ..., r1(sN )

| {z }

, . . . , rM(s1), . . . , rM(sN )

| {z }

)

= ( r1, · · · , rM).

In an MDP, the agent selects an action at each sequential stage, and we define a policy

(behavior) as the way that the actions are selected by a decision maker/agent. Hence this process

can be described as a mapping between state and action, i.e., a random state-action sequence

(s

0

, a0, s1, a1, · · · s

t

, at, · · ·),

1 where st+1 is connected to (st

, at) by Pa

t (s

t

, st+1).

We also define rational agents as those that behave according to the optimal decision rule

where each action selected at any stage maximizes the value function. The value function for a

policy π evaluated at any state s

0

is given as V

π

(s

0

) = E[

P∞

t=0 γ

t

r(s

t

, at)|π]. This expectation is

over the distribution of the state sequence {s

0

, s1, ...} given the policy π = {μ

0

, μ1, · · · }, where

a

t = μt

(s

t

), μ

t

(s

t

) ∈ U(s

t

) and U(s

t

) ⊂ A. The objective at state s is to choose a policy that

maximizes the value of V

π

(s). The optimal policy is then V

∗

(s

0

) = supπ E[

P∞

t=0 γ

t

r(s

t

, at)|π].

Similarly, there is another function called the Q-function (or Q-factor) that judges how well an

action is performed in a given state. The notation Qπ(s, a) represents the expected return from

state s when action a is taken and thereafter policy π is followed.

In the infinite-horizon case, the stationary Markovian structure of the problem implies that the

only variable that affects the agent’s decision rule and the corresponding value function should

be time invariant. We then have the essential theory of MDPs (Bellman R. (1957)) as follows:

Theorem 1 (Bellman Equations): Given a stationary policy π, ∀n ∈ N , m ∈ M, V

π

(sn) and

1

Superscripts represent time indices. For example s

t

and a

t

, with the upper-index t ∈ {1, 2, · · · }, denote state and action at

the t-th horizon stage, while sn (or am) represents the n-th state (or m-th action) in S (or A).
16

Qπ(sn, am) satisfy

V

π

(sn) = r(sn, π(sn)) + γ

X

n0∈N

Pπ(sn)(sn, sn0)V

π

(sn0),

Q

π

(sn, am) = r(sn, am) + γ

X

n0∈N

Pam(sn, sn0)V

π

(sn0).

Theorem 2 (Bellman Optimality): π is optimal if and only if, ∀n ∈ N , π(sn) ∈ arg maxa∈A Qπ(sn, a).

Based on the above theorem of MDPs, we have the following equations to represent the

Q-function as a the reward function.

Q

π

(sn, am) = rm(sn) + γPam(sn, :)(I − γPπ)

−1

rm,

where Pπ represents the state transition probability matrix for following policy π at every state,

and rm represents the reward vector under action am.

B. Constructing an MDP Model from Order Book Data

Figure 1 shows the entire life-cycle of an order initiated by a client of an exchange. The order

book audit trail data contains these messages, and the entire order history (i.e. order creation,

order modifications, fills, cancellation, etc.) can be retrieved and analyzed. To construct an MDP

model of trader behavior, we first reconstruct the limit order book using the audit trail messages.

The order book then contains bid/ask prices, market depth, liquidity, etc. During this process

on the E-Mini data described in section 2.1, we processed billions of messages for each trading

date, and built price queues using the price and time priority rule.

Once we have the order book at any given event tick, we take the market depth at five

different levels as our base variables and then discretize these variables to generate an MDP

model state space. This study extends the MDP model documented by Yang et al. (2012) to

obtain five variables, i.e., order volume imbalance between the best bid and the best ask prices,

order volume imbalance between the 2nd best bid and the 2nd best ask prices, order volume

imbalance between the 3rd best bid and the 3rd best ask prices, the order book imbalance at

the 5th best bid and the 5th ask prices, and the inventory level/holding position (see Figure 2

(b)). Then we discretize the values of the five variables into three levels defined as high (above

μ + 1.96σ), neutral (μ ± 1.96), and low (below μ − 1.96σ). Based on our observation that the

first 3 best bid and ask prices change the most, we select the first 3 level order book imbalance
17

Fig. 1. CME Globex Order Lifecycle. T1: Trader submits a new order; T2: The state of an order is changed, if a stop is

activated; T3: A trader may choose to cancel an order, and the state of an order can be modified multiple times; T4: When

an order is partially filled, the quantity remaining decreases; T5: Order elimination is similar to order cancellation except it

is initiated by the trading engine; T7: An order may be filled completely; T6: Trades can be eliminated after the fact by the

exchanges.

variables in modeling the limit order book dynamics. As argued by Yang et al. (2012), these

volume-related variables reflect the market dynamics on which the traders/algorithms depend to

place their orders at different prices.

As the volume imbalance at the best bid/ask prices is the most sensitive indicator of the trading

behavior of HFTs, Intermediaries and some of the Opportunistic traders, we also hypothesize

that the volume imbalance at other prices close to the book prices will be useful in inferring

trader behavior. As demonstrated in previous work (Yang et al. (2012)), the private variable

of a trader’s inventory level provides critical information about trader’s behavior. Traders in

high frequency environments strive to control their inventory levels as a critical measure of

controlling the risk of their position (Kirilenko et al. (2011), Easley et al. (2010) and Brogaard

et al. (2010)). HFTs and Market Makers tend to turn over their inventory level five or more

times a day and to hold very small or even zero inventory positions at the end of the trading

session. These observations provide strong support for the introduction of a position variable
18

Fig. 2. Order Book MDP Model: This graph shows the state variables used in the MDP model.

to characterize trader behavior in our model. Therefore, together with the volume imbalance

variables, we propose a computational model with 3

5 = 243 states.

Next, we need to define the action space. In general, there are three types of actions: placing

a new order, canceling an existing order, or placing a market order. We divide the limit order

book into 10 buckets at any given point of time by the following price markers: the best bid

price, the 2nd best bid price, the 3rd best bid price, between the 4th and 5th bid prices, below

the 5th best bid price, the best ask price, the 2nd best ask price, the 3rd best ask price, between

the 4th and 5th ask prices, and above the 5th best ask price. Then, at any given point of time,

a trader can take 22 actions. The price markers used to define the price ranges are illustrated

in Figure (2). We use unit size for all the actions. Orders other than unit size are treated as

repeated actions without state transition.

IV. INVERSE REINFORCEMENT LEARNING

Given an MDP M = (S, A,P, γ, r), let us define the inverse Markov decision process (IMDP)

MI = (S, A,P, γ, O). The process MIincludes the states, actions, and dynamics of M, but
19

lacks a specification of the reward function, r. By way of compensation, MIincludes a set

of observations O that consists of state-action pairs generated through the observation of a

decision maker. We can define the inverse reinforcement learning (IRL) problem associated with

MI = (S, A,P, γ, O) to be that of finding a reward function such that the observations O could

have come from an optimal policy for M = (S, A,P, γ, r). The IRL problem is, in general, highly

under-specified, which has led researchers to consider various models for restricting the set of

reward functions under consideration. Ng et al. (2000), in a seminal consideration of IMDPs

and associated IRL problems, observed that, by the optimality equations, the only reward vectors

consistent with an optimal policy π are those that satisfy the set of inequalities

(Pπ − Pa)(I − γPπ)

−1

r ≥ 0, ∀a ∈ A, (1)

where Pπ is the transition probability matrix relating to observed policy π and Pa denotes the

transition probability matrix for other actions. Note that the trivial solution r = 0 satisfies the

constraints (1), which highlights the under-specified nature of the problem and the need for

reward selection mechanisms.

In the machine learning and artificial intelligence literature, a principal motivation for considering IRL problems is the idea of apprenticeship learning, in which observations of state-action

pairs are used to learn the policies followed by experts for the purpose of mimicking or cloning

behavior. By its nature, apprenticeship learning problems arise in situations where it is not

possible or desirable to observe all state-action pairs for the decision maker’s policy. The basic

idea of apprenticeship learning through IRL is to first use IRL techniques to learn the reward

function (vector) and then use that function to define an MDP problem, which can then be

solved for an optimal policy. Our process is quite different. We learn the reward function with

IRL and then directly use the rewards as features for classifying and clustering traders or trading

algorithms.

A. Linear IRL

Ng et al. (2000) advance the idea choosing the reward function to maximize the difference

between the optimal and suboptimal policies, which can be done using a linear programming

formulation.
20

Most of the existing IRL algorithms make some assumption about the form of the reward

function. Prominent examples include the model in Ng et al. (2000), which we term linear IRL

(LIRL) because of its linear nature. In LIRL, the reward function is written linearly in terms of

basis functions, and effort is made to maximize the quantity

X

s∈S

[Q

π

(s, a0) − max

a∈A\a

0

Q

π

(s, a)], ∀a ∈ A. (2)

The optimization problem in Ng et al. (2000) is equivalent to the following optimization

program:

maxr

P

s∈Sβ(s) − λ

P

s∈S |r(s)|

s.t.

(Pπ − Pa)(I − γPπ)

−1

r ≥ β(s), ∀a ∈ A, ∀s ∈ S

β(s) ≥ 0, ∀s ∈ S,

where λ is a regularization parameter included to encourage sparse solution vectors. Yang et al.

(2012) use this approach to find a feature space that can be used to classify and cluster simulated

trading agents.

B. Bayesian IRL Framework

Ramachandran et al. (2007) originally proposed a Bayesian Framework for IRL. The posterior

over reward is written as

p(r|O) = p(O|r)p(r) ∝

Y

(s,a)∈O

p(a|s, r).

Then, the IRL problem is written as maxr log p(O|r) + log p(r). For many problems, however,

the computation of p(r|O) may be complicated and some algorithms use Markov chain Monte

Carlo (MCMC) to sample the posterior probability. Below we adopt a different approach that

uses the idea of selecting reward on the basis of maximum a posteriori (MAP) estimate computed

using convex optimization.

C. Gaussian Process IRL

We now turn to an IRL problem that addresses observations from a decision making process

in which the reward function has been contaminated by Gaussian noise. In particular, we assume
21

(a) (b)

Fig. 3. Action Preference Graph Examples: (a). This graph shows an example action preference graph at state 158. (b). This

graph shows an example action preference graph at state 14.

that the reward vector can be modeled as r + N (0, σ2), where N (0, σ2) is Gaussian noise. In

the financial trading problem setting, we may observe certain trading behavior over a period

of time, but we may not observe the complete polices behind a particular trading strategy. As

discussed earlier, different trading strategies tend to look at different time horizons. Therefore,

the observation period becomes critical to the learning process. Furthermore, two types of errors

may be introduced into our observations: The first type of error may be introduced during our

modeling process. Resolution of these discrete models will introduce errors into our observations.

The second potential source of error is the strategy execution process. Execution errors will occur

due to the uncertainty of market movements and will eventually appear in our observations,

confounding our efforts to determine the true policy. Overall, there are two types of challenges

in this learning problem: the uncertainty about reward functions given the observation of decision

behavior and the ambiguity involved in observing multiple actions at a single state.

Qiao and Beling (2011) argue for two different modeling techniques in learning reward

functions. To lessen the ambiguity of observing multiple actions at a state, they argue that

Bayesian inference should be the basis for understanding the agent’s preferences over the action

space. This argument is reasonable because the goal of IRL is to learn the reward subjectively

perceived by the decision maker from whom we have collected the observation data. The intuition

is that decision makers will select some actions at a given state because they prefer these actions

to others. These preferences are among the countable actions that can be used to represent

multiple observations at one state.

Here we use two examples to demonstrate how the action preference relationships have been

constructed based on the MDP model and observed actions. Table II shows two example states
22

TABLE II

ACTION PREFERENCE GRAPH EXAMPLES

State Action Frequency Observed State Action Frequency Observed

14 1 0.23 158 1 0.3014 2 0.14 158 3 0.0714 7 0.06 158 7 0.1114 11 0.26 158 11 0.3014 12 0.09 158 17 0.0714 16 0.17 158 18 0.0714 26 0.06 158 20 0.07
23

(a) (b)

Fig. 4. Examples Preference Graphs: (a) An example of observing two actions at a state. (b) An example of a unique

observation at a state.

with multiple observed actions. We then sort the frequency in descending order and construct

a two-layer graph: the top layer has the most frequently observed actions and the bottom layer

holds all the other actions. Based on this preference observation, we can construct two preference

graphs as shown in Figure (3). The state transition matrix can be constructed for the entire market

for the observation period. In our MDP model, we have a 243x243 matrix for every single action.

In the following, we first introduce the preference theory for the IMDP model, and then we

formalize the idea of modeling the reward function as a Gaussian process under the Bayesian

inference framework.

1) Action Preference Learning: In this section, we first define the action preference relationship and the action preference graph. At state sn, ∀a, ˆ aˇ ∈ A, we define the action preference

relation as:

1) Action aˆ is weakly preferred to aˇ, denoted as aˆ sn aˇ, if Q(sn, aˆ) ≥ Q(sn, aˇ);

2) Action aˆ is strictly preferred to aˇ, denoted as aˆ sn aˇ, if Q(sn, aˆ) > Q(sn, aˇ);

3) Action aˆ is equivalent to aˇ, denoted as aˆ ∼sn aˇ, if and only if aˆ sn aˇ and aˇ sn aˆ.

An action preference graph is a simple directed graph showing preference relations among

the countable actions at a given state. At state sn, the action preference graph Gn = (Vn, En)

comprises a set Vn of nodes together with a set En of edges. For the nodes and edges in graph

Gn, let us define

1) Each node represents an action in A. Define a one-to-one mapping φ : Vn → A.

2) Each edge indicates a preference relation.

Furthermore, we make the following assumption as a rule to build the preference graph, and

then we show how to draw a preference graph at state sn:
24

At state sn, if action aˆ is observed, we have the following preference relations: aˆ sn a, ˇ ∀aˇ ∈

A \ {aˆ}.

It is therefore straightforward to show the following according to Bellman optimality. The

variable aˆ is observed if and only if aˆ ∈ arg maxa∈A Q(sn, a). Therefore, we have

Q(sn, aˆ) > Q(sn, aˇ), ∀aˇ ∈ A \ {aˆ}

According to the definition on preference relations, it follows that if Q(sn, aˆ) > Q(sn, aˇ), we

have aˆ sn aˇ. Hence, we can show that the preference relationship has the following properties:

1) If a, ˆ aˇ ∈ A, then at state sn either aˆ sn aˇ or aˇ sn aˆ.

2) If aˆ sn aˇ or aˇ sn a ̃, then aˆ sn a ̃.

At this point, we have a simple representation of the action preference graph that is constructed

by a two-layer directed graph. We may have either multiple actions at sn as in Figure (4) (a)

or a unique action at sn as in Figure (4) (b). In this two-layer directed graph, the top layer V

+

n

is a set of nodes representing the observed actions and the bottom layer V

−

n

contains the nodes

denoting the other actions. The edge in the edge set En can be represented by a formulation of

its beginning node u and ending node v. We write the k-th edge as (u → v)k if u ∈ V+

n

, v ∈ V−

n

,

or the l-th edge (u ↔ v)lif u ∈ V−

n

, v ∈ V−

n

. Recall the mapping between Vn and A, the

representation u → v indicates that action φ(u) is preferred over φ(v). Similarly, u ↔ v means

that action φ(u) is equivalent to φ(v).

In the context of financial trading decision process, we may observe multiple actions from

one particular trader under certain market conditions. That is to say that the observation data O

may represent multiple decision trajectories generated by non-deterministic policies. To address

IRL problems in those cases, Qiao and Beling (2011) propose to process O into the form of

pairs of state and preference graphs similar to the representation shown in Figure (5), and then

we apply Bayesian inference using the new formulation.

According to Qiao and Beling (2011), we can represent O as shown in Figure (5). At state

sn, its action preference graph is constructed by a two-layer directed graph: a set of nodes V

+

n

in the top layer and a set of nodes V

−

n

in the bottom layer. Under the non-deterministic policy

assumption, we adopt a reward structure depending on both state and action.
25

Fig. 5. Observation structure for MDP.

2) Gaussian Reward Process: Recall that the reward depends on both state and action, and

consider rm, the reward related to action am, as a Gaussian process. We denote by km(si, sj )

the function generating the value of entry (i, j) for covariance matrix Km, which leads to rm ∼

N(0, Km). Then the joint prior probability of the reward is a product of multivariate Gaussian,

namely p(r|S) = QM

m=1 p(rm|S) and r ∼ N(0, K). Note that r is completely specified by

the positive definite covariance matrix K, which is block diagonal in the covariance matrices

{K1, K2..., KM} based on the assumption that the reward latent processes are uncorrelated . In

practice, we use a squared exponential kernel function, written as:

km(si, sj ) = e

1

2

(si−sj )Tm(si−sj ) + σ2

mδ(si

, sj ),

where Tm = κmI and I is an identity matrix. The function δ(si, sj ) = 1, when si = sj; otherwise

δ(si, sj ) = 0. Under this definition the covariance is almost unity between variables whose inputs

are very close in the Euclidean space, and decreases as their distance increases.

Then, the GPIRL algorithm estimates the reward function by iteratively conducting the following two main steps:

1) Get estimation of rMAP by maximizing the posterior p(r|O), which is equal to minimize

− log p(O|r)−log p(r|θ), where θ denotes the vector of hyper-parameters including κm and

σm that control the Gaussian process.
26

2) Optimize the hyper-parameters by using gradient decent method to maximize log p(O|θ, rMAP ),

which is the Laplace approximation of p(θ|O).

3) Likelihood Function and MAP Optimization: GPIRL adopts the following likelihood functions to capture the strict preference and equivalent preference respectively.

p((ˆa sn aˇ)k|r) = Φ(Q(sn, aˆ) − Q(sn, aˇ)

√

2σ

) (3)

p((ˆa ∼sn aˆ

0

)l|r) ∝ e

− 1

2

(Q(sn,aˆ)−Q(sn,aˆ

0

))2

(4)

In Eq. 3, the function Φ(x) = R x

−∞ N(v|0, 1)dv, where N(v|0, 1) denotes a standard Gaussian

variable.

As we stated earlier, if we model the reward functions as being contaminated with Gaussian

noise that has a mean of zero and an unknown variance σ

2

, we can then define the likelihood

function for both the k-th strict preference relation and the l-th equivalent preference relation.

Finally, we can formulate the following proposition:

Proposition 3: The likelihood function, given the evidence of the observed data (O in the

form of pairs of state and action preference graph (G), is calculated by

p(O|r) ∝ p(G|S, r) = Y

N

n=1

p(Gn|sn, r) = Y

N

n=1

Ynn

k=1

p((ˆa sn aˇ)k|r)

Ymn

l=1

p((ˆa ∼sn aˆ

0

)l|r), (5)

where nn denotes the number of edges for strict preference and mn means the number of edges

for equivalent preference at state sn.

In conclusion, the probabilistic IRL model is controlled by the kernel parameters κm and σm

which compute the covariance matrix of reward realizations, and by σ which tunes the noise

level in the likelihood function. We put these parameters into the hyper-parameter vector θ =

(κm, σm, σ). More often than not, we do not have prior knowledge about the hyper-parameters.

And then we can apply maximum a posterior estimate to evaluate the hyper-parameters.

Essentially, we now have a hierarchical model. At the lowest level, we have reward function

values encoded as a parameter vector r. At the top level, we have hyper-parameters in θ

controlling the distribution of the parameters. Inference takes place one level at a time. At
27

the bottom level, the posterior over function values is given by Bayes’ rule:

p(r|S, G, θ) = p(G|S, θ, r)p(r|S, θ)

p(G|S, θ)

. (6)

The posterior combines the prior information with the data, reflecting the updated belief about

r after observing the decision behavior. We can calculate the denominator in Eq.6 by integrating

p(G|S, θ, r) over the function space with respect to r, which requires a high computational

capacity. Fortunately, we are able to maximize the non-normalized posterior density of r without

calculating the normalizing denominator, as the denominator p(G|S, θ) is independent of the

values of r. In practice, we obtain the maximum posterior by minimizing the negative log

posterior, which is written as

U(r) ,

1

2

X

M

m=1

r

T

mK−1m rm −

X

N

n=1

Xnn

k=1

ln Φ(Q(sn, aˆ) − Q(sn, aˇ)

√

2σ

)

+

X

N

n=1

Xmn

l=1

1

2

(Q(sn, aˆ) − Q(sn, aˆ

0

))2(7)

Qiao and Beling (2011) present a proof that Proposition (7) is a convex optimization problem.

At the minimum of U(r) we have

∂U(r)

∂rm

= 0 ⇒ ˆrm = Km(∇ log P(G|S, θ, ˆr)) (8)

where ˆr = (ˆr1, · · · , ˆram, · · · , ˆrm). In Eq.8, we can use Newton’s method to find the maximum

of U with the iteration,

ˆr

new

m = ˆrm − (

∂

2U(r)

∂rm∂rm

)

−1

∂U(r)

∂rm

V. EXPERIMENT WITH THE E-MINI S&P 500 EQUITY INDEX FUTURES MARKET

In this section, we conduct two experiments using the MDP model defined earlier to identify

algorithmic trading strategies. We consider the six trader classes defined by Kirilenko et al.

(2011), namely High Frequency Traders, Market Makers, Opportunistic Traders, Fundamental

Buyers, Fundamental Sellers and Small Traders. As we argue earlier, the focus of our study

will be more on HFTs and Market Makers due to the large daily volume and their potential
28

impact to the financial markets. In Kirilenko et al. (2011)’s paper, there are only about from

16 to 20 HFTs on the S&P500 Emini market. Although this is a small population, their impact

to the market has drawn increased attention from policy makers, regulators and academia. That

is why we focus our attention on this small population. Among the roughly 10,000 trading

accounts for the S&P500 Emini market, we narrow down to about 120 accounts based their

high daily trading volume. In the first experiment, we select the top 10 trading accounts by their

volume and end-of-the-day positions. In this we guarantee our subjects are HTFs. In the second

experiment, we randomly select 10 out of the 120 accounts. This selection criterion ensures that

our subjects are of either HTF or market making strategies. With these two experimentation, we

show the performance of our IRL based approach to identify the high impact population of the

Algorithmic trading strategies.

A. Trader Behavior Identification

Yang et al. (2012) examine different trading behaviors using a linear IRL (LNIRL) algorithm

with the simulated E-Mini S&P 500 market data. That MDP model contains three variables: the

volume imbalance at the bid/ask prices, the volume imbalance at the 3rd best bid/ask prices,

and the position level. Although this MDP model is relatively simple, it is evident from the

experimental results that the IRL reward space is effective in identifying trading strategies with

a relatively high accuracy rate.

This paper tries to address two important issues during the modeling process to solve a realistic

market strategy learning problem using real market data. The first issue is that in reality, we often

do not have a complete set of observations of a trader’s policies. As the market presents itself as

a random process in terms of both prices and volume, it is unlikely that we will be able to capture

all possible states during our observation window. In contrast, the study performed by Yang et

al. (2012) assumes complete observation of a trader’s decision policies for the simulated trading

strategies. In other words, the policies simulated by a distribution can be completely captured

when the simulation is run long enough. The convergence of these simulated policies and the

testing results are consistent with their assumptions. However, when we use real market data to

learn about trading strategies, it is necessary to address the incomplete observation problem. The

second issue is to the assumption of deterministic policy vs. non-deterministic policy. Yang et al.

(2012) make a deterministic policy assumption. Under the linear feature optimization framework,
29

non-deterministic policies can be represented by a single maximum deterministic policy. In this

study, we relax the deterministic policy assumption and allow non-deterministic policies under a

Gaussian process framework. As we argue earlier, Gaussian process learning allows us to infer

policies even when we have a very limited set of observations. At the same time, we incorporate

Gaussian preference learning into our inference process. This approach helps us to incorporate

less frequently observed policies into our reward learning process. Together, the proposed GPIRL

approach results in a model that relies less on observations and makes fewer assumptions on the

polices we are to learn.

B. Multi-class SVM Trader Classifier using GPIRL vs. LNIRL

This section uses the support vector machine (SVM) classification method to identify traders

based on reward functions that we recover from the observations of the trader’s behaviors. We

select a group of traders whose behaviors are consistently observed during the period we study.

The primary reason for choosing the SVM classification method is its flexibility that allows us

to explore feature separation in different high dimensional spaces using kernel functions. We

aim to compare the performance of the two behavior learning algorithms LNIRL and GPIRL,

and to show that GPIRL perform better in real world trading strategy identification.

We constructed 80 sample trajectories for each of the top 10 trading accounts. While there are

121 trading accounts consistently traded over the 4-week period, this study focuses on the top

10 trading accounts. We apply both the LNIRL ( Ng et al. (2000) and Yang et al. (2012)), and

GPIRL (Qiao and Beling (2011)) to these 800 samples. And then we apply the SVM algorithm

to the 10 traders using pair-wise classification. For each pair, we first train a SVM classifier (with

Gaussian kernel) with 60 randomly selected samples, and test the classification on the remaining

20 samples. We repeat the sampling 100 times and then take the average classification accuracy.

We list both LNIRL classification results in Table III, and GPIRL results in Table IV. On average,

LNIRL gives a classification accuracy of 0.6039, while GPIRL achieves a classification accuracy

of 0.9650. This result confirms our earlier assumption that GPIRL performs better when we have

incomplete observations, and incorporate non-deterministic policies through Gaussian preference

learning.
30

TABLE III

PAIR-WISE TRADER CLASSIFICATION USING SVM BINARY CLASSIFICATION USING LNIRL

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]

[1,] 0.0000 0.5437 0.5187 0.4812 0.6375 0.4812 0.5312 0.5750 0.7750 0.5937[2,] 0.5437 0.0000 0.5250 0.5125 0.7437 0.5562 0.4937 0.4250 0.7625 0.6812[3,] 0.5187 0.5250 0.0000 0.4687 0.6875 0.5250 0.5187 0.5250 0.7312 0.6250[4,] 0.4812 0.5125 0.4687 0.0000 0.6937 0.5000 0.4937 0.5062 0.6562 0.6625[5,] 0.6375 0.7437 0.6875 0.6937 0.0000 0.6625 0.7375 0.6875 0.7750 0.5437[6,] 0.4812 0.5562 0.5250 0.5000 0.6625 0.0000 0.5500 0.5500 0.6500 0.6375[7,] 0.5312 0.4937 0.5187 0.4937 0.7375 0.5500 0.0000 0.4937 0.8000 0.6125[8,] 0.5750 0.4250 0.5250 0.5062 0.6875 0.5500 0.4937 0.0000 0.6437 0.6562[9,] 0.7750 0.7625 0.7312 0.6562 0.7750 0.6500 0.8000 0.6437 0.0000 0.7437[10,] 0.5937 0.6812 0.6250 0.6625 0.5437 0.6375 0.6125 0.6562 0.7437 0.0000Notes: The columns and rows of this table represent anonymous traders.
31

TABLE IV

PAIR-WISE TRADER CLASSIFICATION USING SVM BINARY CLASSIFICATION USING GPIRL

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]

[1,] 0.0000 1.0000 0.9875 0.9750 0.9500 0.9750 0.9625 1.0000 0.9750 1.0000[2,] 1.0000 0.0000 0.9750 0.9375 0.9875 0.9750 0.9625 0.9625 0.9875 1.0000[3,] 0.9875 0.9750 0.0000 0.9750 0.9625 0.9875 1.0000 0.9750 0.9750 0.9875[4,] 0.9750 0.9375 0.9750 0.0000 0.9750 0.9500 0.9375 0.9875 0.9875 0.9750[5,] 0.9500 0.9875 0.9625 0.9750 0.0000 1.0000 1.0000 0.9625 0.9875 1.0000[6,] 0.9750 0.9750 0.9875 0.9500 1.0000 0.0000 0.9625 0.8750 0.9125 0.9750[7,] 0.9625 0.9625 1.0000 0.9375 1.0000 0.9625 0.0000 0.8625 0.9625 0.9875[8,] 1.0000 0.9625 0.9750 0.9875 0.9625 0.8750 0.8625 0.0000 0.8000 1.0000[9,] 0.9750 0.9875 0.9750 0.9875 0.9875 0.9125 0.9625 0.8000 0.0000 0.9625[10,] 1.0000 1.0000 0.9875 0.9750 1.0000 0.9750 0.9875 1.0000 0.9625 0.0000Notes: The columns and rows of this table represent anonymous traders.
32

C. Trading Strategy Clustering and Comparison with the Summary Statistic-Based Approach

Next, we will show that our IRL based behavior identification approach is far superior to

the statistic-based approach. We will use the top 10 trading accounts as examples to demonstrate improvement of behavior-based trading strategy identification achieved using the Gaussian

Preference IRL model.

In the previous section, we discovered that using reward functions we can reliably identify a

particular trading strategy over a period of time with a relatively high accuracy. In this section,

we want to study the similarity of reward characterization among the different trading strategies.

This problem can be characterized as an unstructured learning problem - clustering. We have the

characterization of rewards over the state space and action space, and we aim to group trading

strategies based on their similarity over the Cartesian product of the state and action spaces.

We also attempt to establish connections between these trading strategy classification definitions

established by Kirilenko et al. (Kirilenko et al. (2011)) and our behavior-based trading strategy

clustering.

The first problem we have to address is the dimensionality of the feature space. We essentially

have a reward structure over a large set of feature sets. This feature set is a product of the state

space and the action space in our computational model. Fortunately, under the LNIRL algorithm,

we reduce the feature space to only the state space because in this linear feature expectation

optimization problem we only consider reward at a particular state. Under the deterministic policy

assumption, we assume that the value function converges at a particular state. In other words,

the reward function is not a function of actions. In this case, we have 243 features that must

be considered during the clustering. However, under the GPIRL framework we do not assume

deterministic policy, and we treat reward as a function of both states and actions. Therefore we

have 243x30 features for the latter approach. We also observe that the reward matrix is relatively

sparse where there are zero values at many states. To consider computational tractability and

efficiency, we first examine the data structure through Principal Component Analysis.

In the LNIRL case, the first two Principal Components (PCs) explain 79.78% of the data

variation, and from the upper left plot in Figure (6) (a) we see that the first 200 PCs provide

nearly 100% explanatory power. In the GPIRL case, the first two PCs only explain 38.98% of

the data variation. Looking at the upper left plot in Figure (6) (b), we see that more PCs are
33

needed to have better represent the data. To balance the accuracy and computational efficiency,

we choose the first 200 PCs for the LNIRL and the first 400 PCs for the GPIRL case. This

reduction leads to significant gain in computational efficiency and sacrifices less than 2% data

variation (lower left figure in both Figure (6) (a) and (b)). From the upper right plots in both the

LNIRL and the GPIRL spaces, we see that the first two PCs give a good representation along

the first PC and that in the LNIRL case, the feature vector representation is evenly distributed

between the first two PCs. The LNIRL space includes distinctly separated observations. On the

other hand, the GPIRL space contains concentrations of observations, but unclear boundaries. In

both cases, we would expect the PC dimension reduction technique to achieve relatively good

representation of the data variation.

Now we apply unsupervised learning method to group the trading behavior observed on a

selected group of trading accounts over the observation period. We select 10 trading accounts

with the highest average daily trading volume over a period of 4 weeks (20 days) in our first

experiment. We define an observation instance as a continuous period covering two hours over

which we take all the activities of a particular trader including placing new orders, modifying and

canceling existing orders, and placing market orders. For each trader, we collect four observation

instances on each trading date: two observation instances during the morning trading and two

observation instances during the afternoon trading. The two observation periods in the morning

and in the afternoon have an hour overlap time, but the observations in the morning and the

afternoon do not overlap. This observation distribution is selected based on the general theory

of intraday U-shaped patterns in volume - namely, that trading is heavy at the beginning and

the end of the trading day and relatively light in the middle of of the day (Ekman et al. (2006),

Admati et al. (1988), Lee et al. (2001), and Chordia et al. (2001)). We also examined traders’

actions throughout the entire trading day. We found that the two-hour observation time is a good

cut-off, and with the overlapping instances in both the morning and the afternoon we expect to

capture the U-shaped pattern of the market.

We then perform hierarchical clustering and generate a heat map and dendrogram of the

observations in both the LNIRL reward space and the GPIRL reward space. The simplest form

of matrix clustering clusters the rows and columns of a dataset using Euclidean distance metric

and average linkage. For both Figure (7) (a) and (b), the left dendrogram shows the clustering

of the observations (rows), and the top dendrogram shows the clustering of the PCs (columns).
34

(a)

(b)

Fig. 6. Principal Component Representation of the Reward Data: (a). Data representation under the first two PCs in the

LNIRL reward space; The upper left figure shows cumulative percentage of the data variance explained by the PCs. The lower

left figure plots the loadings of all the observations onto the first two PCs; The upper right figure shows the projection of the

observation and feature vector onto the first two PCs. The lower right shows the projection of the observation onto the first

two PCs with boundary point markers. (b). Data representation under the first two Principal Components in the GPIRL reward

space. The upper left figure shows the cumulative percentage of the data variance explained by the PCs. The lower left figure

plots the loadings of all the observations on to the first two PCs. The upper right figure shows the projection of the observation

and feature vectors onto the first two PCs. The lower right figure shows the projection of the observation onto the first two PCs

with boundary point markers.
35

It is evident that there is a clear division of the observations (rows) in both cases. Upon closer

examination, the left dendrogram contains two clusters: the top cluster and the bottom cluster

with a black dividing strip in the middle of the second small cluster. We then zoomed into the

small cluster and look for the sources of these observations2. In the LNIRL reward space, we

find that the small cluster consists mostly of observations from trader 1 (observations numbered

from 1 to 80) and trader 2 (observations numbered from 81 to 160). Observations from trader

9 (observations numbered from 641 to 720) form the black division between these two groups.

In the GPIRL reward space, we find that the small cluster consists of three traders: trader 1

(observations numbered from 1 to 80), trader 2 (observations numbered from 81 to 160), and

trader 5 (observations numbered from 321 to 400) with the observations from the rest of the

traders lying on the other side of the divide. Moreover, we find that the observations from trader

9 (observations numbered from 641 to 720) form the black division between these two groups.

These observations show that the majority of the top 10 traders form one group with 2 or 3

traders behaving a little differently. Furthermore, we observe that the clustering has less than

perfect purity. In other words, individual observations from the top cluster occasionally lie in

the small cluster at the bottom indicating that behavior changes over time. The interpretation

of this observation is that the HFTs may behave like Opportunistic Traders for a short period

of time. We also occasionally observe Opportunistic Traders behaving like HFTs. In this case,

observations cross the divide into the top cluster.

Next we propose a continuous measure of clustering using the hierarchical clustering method.

We use the summary statistic-based trader classification method proposed by Kirilenko et al.

(Kirilenko et al. (2011)) to create reference labels. For this market data, we do not have true labels

on those trading accounts. We aim to improve the labeling methods documented by Kirilenko

et al. The motivation for creating a continuous measure of clustering is to address the potential

changes in trading behavior over time. As we mentioned earlier, we applied the summary statisticbased classification rule on the 200 observations over the 4-week period and found we can

only consistently label the traders as a single type 40% of the time. We now define a weighted

scoring system to evaluate both the rule-based classifier and the behavior-based classifier. Among

2Note: In both Figure (6) (a) and (b), we group observations from the same trader together in our data matrix. We have 10

traders and each has 80 observations. From the lower left graph in both (a) and (b), observations are ordered sequentially by

trader IDs. For example, observations 1 through 80 come from trader 1, and observations 81 through 160 come from trader 2.

This continues along the X-axis up to observations 721 through 800 from trader 10.
36

the 6 types of traders defined in the data section, we only concerned with labeling HFTs,

Intermediaries, and Opportunistic Traders. The other three types of traders, e.g., Fundamental

Buyers, Fundamental Sellers, and Small Traders, can be reliably identified by their daily volume

and their end of day positions. Here, we assign score 2 if a trader is classified as a HFT; we

assign score 1 if a trader is classified as an Opportunistic Trader; and we assign 0 if a trader is

classified as an Intermediary. Labels for clustering are assigned using the majority voting rule

based on the summary statistic classification rule. We then combine the scores using a weight

defined as the frequency with which a particular score is assigned to a particular trader. Here, we

want to compare the summary statistic-based trader type classification with the behavior-based

trader type classification in an effort to find connections between these two methods.

The visual representations in Figure (8) (a) show that trader 1 and trader 5 have a wide range of

end of day positions, but their daily trading volumes remains at relatively the same levels. These

traders will likely be classified sometimes as HFTs and sometimes as Opportunistic Traders.

While trader 2 exhibits a smaller range of end of day positions than trader 1 and trader 5, the

general pattern is very similar to that of traders 1 and 5, and we should classify trader 2 as an

Opportunistic Trader. Based on this manual examination, traders 1, 2 and 5 should be classified

as Opportunistic Traders and the rest should be classified as HFTs. Now, we compare the results

of the summary statistic-based classification rule with those of the behavior-based classification.

Figure (9) (a) shows that two groups of traders exist in the LNIRL reward space. Eight out

of ten are identified as HFTs and only trader 1 and trader 2 are classified as Opportunistic

Traders. This result is consistent with our observation from the dendrogram in Figure (7) (a).

When we compare this result with the GPIRL reward space, we can locate all three traders

(1, 2, and 5) that we identified through the manual process. This result is also consistent with

our observation from the heat map in Figure (7) (b). The statistic-based classification method

misclassified trader 2 because the cut-off in the statistic-based approach is based on a simple

ratio between the trading volume and the end position. We can see that trader 2 has a relatively

small spread of end position. However, the behavior-based approach can identify this pattern

and is able to cluster this trader with other traders with similar patterns.

We run another experiment using 10 randomly selected traders out of the traders with the top

30 trading volumes. We know this selection will only result in three types of traders, i.e., HFTs,

Intermediaries and Opportunistic Traders. We feed these 800 observations to both LNIRL and
37

GPIRL algorithms to obtain reward representations of their trading behaviors. Based on visual

examination (see Figure (8) (b)), we see that trader 1, trader 2 and trader 7 are Opportunistic

Traders and the rest are HFTs. We apply the same techniques as before and we use the same cutoff scores (1.85 in the LNIRL reward space, and 1.75 in the GPIRL reward space). As a result,

we can accurately identify the two classes of traders using the same cut-off score we used for the

top 10 case (see Figure (10)). The classification in the LNIRL reward space gives the same result

as that in the GPIRL reward space, while the statistic-based classification method misclassified

trader 3 as an Opportunistic Trader. Based on the daily end position, daily total trading volume

and inventory variance, trader 3 should be classified as a HFT. Again, this misclassification is

due to the aggregate cut-off ratio. However, the behavior-based approach can identify this pattern

and is able to cluster trader 3 with other traders with similar behavioral patterns. Overall, we

argue that the GPIRL reward space score-based classification rule provides an advantage over

the summary statistic-based approach in that it is based on similarity in behavior and it can be

clearly interpreted. Because the GPIRL rule a better reflects traders’ choices of actions under

different market conditions than the summary statistics, it is well suited for the discovery of new

behavioral patterns of the market participants. We also conclude that the GPIRL reward space is

more informative and is a superior measure of trading behavior in terms of the LNIRL reward

space.

VI. CONCLUSION

We assume incomplete observation of algorithmic trading strategies and model traders’ reward

functions as a Gaussian process. We also incorporate traders’ action preferences under different

market conditions through preference graph learning. The aim of this study is to quantify

trader behavior-based on IRL reward learning under a Bayesian inference framework. We apply

both a linear approach (a linear combination of known features) (Abbeel et al. (2004)) and

GPIRL (Qiao and Beling (2011)) to a real market dataset (The E-Mini S&P 500 Futures),

and we conclude that GPIRL is superior to the LNIRL methods, with a 6% greater rate of

identification accuracy. Furthermore, we establish a connection between the summary statisticbased classification (Kirilenko et al. (2011)) and our behavior-based classification. We propose

a score-based method to classify trader types, and because of the transferable property of the

reward structure the cut-off score for classifying a group of traders can be applied to different
S. Y. YANG ET AL. 38

market conditions.

The implication of this research is that reward/utility-based trading behavior identification can

be applied to real market data to accurately identify specific trading strategies. As documented

by Abbeel et al. (Abbeel et al. (2004)) and confirmed by many other researchers, the reward

function is the most succinct, robust, and transferable definition of a control task. Therefore,

the behavior learned under the reward space has much broader applicability than observed

policies. Furthermore, these learned reward functions will allow us to replicate a particular trading

behavior in a different environment to understand their impact on the market price movement

and market quality in general.

We also want to note some future research on improving the identification accuracy and discuss

applications of this behavioral characterization:

- During our preference learning inference phase, we only considered a simple two layer

preference graph. However, traders’ preferences can be further distinguished with multilayer graphs or other preference learning techniques.

- Our study focused on the sampled algorithmic traders on a market. Future studies can

extend these results to a large scale experiment to include market participants (specifically

Opportunistic traders), and study their behavioral similarity through clustering. We can then

associate the group behavior with market quality measures.

- Under the GPIRL framework, we are able to recover a detailed reward structure. These

reward functions can be used to generate new policies under a simulated market condition

to understand the full behavior of certain trading strategies. This framework provides a

particularly interesting way for market regulators to see how the various trading strategies

will interact during stressed market conditions.

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(a)

(b)

Fig. 7. Hierarchical Clustering of Data Matrix: (a). Heat map of 800 observations of the LNIRL Rewards in the first 200

PCs. (b). Heat map of 800 observations in GPIRL Rewards in the first 400 PCs.
44

(a)

(b)

Fig. 8. Top 10 Traders: Trader’s Daily Trading Volume vs. Daily End Position during a 20 Day Period. (a) Randomly

Selected 10 Traders: Traders 1, 2 and 5 have varying end positions. (b) Traders 1, 2, and 7 have varying end positions.
45

(a) (b)

Fig. 9. Trader Type Classification Compared with the Summary Statistic Based Rule Classification for the Top 10

Traders. (a) Hierarchical clustering in the LNIRL reward space. (b) Hierarchical clustering in the GPIRL reward space.

(a) (b)

Fig. 10. Trader Type Classification Compared with the Summary Statistic-Based Rule Classification for the 10 Randomly

Selected Traders: (a) Hierarchical clustering in LNIRL reward space. (b) Hierarchical clustering in GPIRL reward space.