Four patterns of mispricing in binary markets

Florian Héraud BEcon, MEcon

Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

School of Economics and Faculty of Business and Law Queensland University of Technology

2021 ii Acknowledgements

First of all, I want to express my gratitude to Lionel Page for all his help and guidance over the last three years. I would like to thank him in particular for introducing me to the universe of binary options which eventually inter- ested me enough to dedicate my whole thesis to the topic. I am also very grateful to Christoph Siemroth who was a very rigorous and cooperative coau- thor. Besides, Annastiina Silvennoinen and Changxia Ke provided me very helpful advice. Besides, I acknowledge QUT for awarding me the QUTPRA scholarship.

I also would like to thank my colleagues and friends at QUT, in particular Alice, Ambroise, Ammarr, Anthony, Imke, Joe, Laura, Sébastien, Shupeng, Sylvain, Tanish, Thomas and the many others I have forgotten to mention. Besides, I am very grateful to Imran Qazi who proofread my thesis. Finally, I would like to give a special thanks to my family, as well as Crestine, Steven and Sylvain for their continuous support.

iii Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet requirements for an award at this or any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made.

QUT Verified Signature

09/03/2021

iv Abstract

Abstract: This thesis investigates four patterns of mispricing in markets. It is composed of four studies that theoretically and empirically investigate the extent and the causes for these biases.

The first study studies the presence of the left-digit bias in binary option markets. We analyse if the traders will pay more attention to the left-most digits of a price number and less attention to the right-most digits. We model this left-digit bias in a simple financial market with the presence of inatten- tion. Discontinuities in the performance of contracts at different traded prices suggest the presence of a left-digit bias in a real time financial market.

The second study investigates the existence of the favourite-longshot bias in binary option markets. We analyse how the relationship between the price and the actual probability of underlying events varies. High-likelihood events are underpriced, and low-likelihood events are overpriced. We attempt to explain the bias by creating a two-period model with a discount factor in which the favourite-longshot bias is a consequence of the time difference between the transaction and the of contracts. Our data show that the favourite- longshot bias extent depends positively on the time length between transaction and expiration.

The third study focuses on the variation of security return rates depend- ing on the-day-of-the-week in binary option markets. We observe lower return rates on Mondays by 2.317%. When controlling by the hourly standard de- viation of return rates, the differences of return rates between Monday and other days are not significant anymore. Our study favours the of the hourly return rate as an explanation of the Monday effect.

Finally, the fourth study highlights differences in average return rate be- tween sellers or buyers of contracts in binary option markets. We observe

v higher return rates for sellers than for buyers. The preferred reasons for the bias are the presence of an endowment effect and some judgemental heuristics caused by the structure of the platform of exchange.

Keywords: ; behavioural finance; behavioural economics; left-digit bias; favourite-longshot bias; Monday effect; endowment effect; judge- mental heuristics; anchoring effect.

vi CONTENTS CONTENTS

Contents

1 Introduction 1

1.1 Historical context ...... 1

1.2 Prediction markets ...... 2

1.3 Binary options ...... 5

1.4 Market efficiency ...... 6

2 Data description 10

2.1 Source of the dataset ...... 10

2.2 First descriptive statistics ...... 12

3 Left-digit bias 14

3.1 Introduction ...... 15

3.2Model...... 18

3.2.1 Setup ...... 18

3.2.2 Analysis ...... 22

3.3 Discontinuity estimates ...... 27

3.3.1 Raw data ...... 27

3.3.2 Merged data ...... 29

3.3.3 Robustness check: discontinuities at other thresholds . . 35

3.3.4 Density observation ...... 36

3.4 Discussion ...... 39

4 favourite-longshot bias 41

4.1 Introduction ...... 42

vii CONTENTS CONTENTS

4.2Model...... 48

4.3 Statistical results ...... 54

4.3.1 Study on the whole sample ...... 54

4.3.2 Time length ...... 57

4.4 Discussion ...... 62

5 Monday effect 64

5.1 Introduction ...... 65

5.2 Empirical Analysis ...... 70

5.2.1 A Monday effect ...... 70

5.2.2 An hourly analysis of the Monday effect ...... 75

5.3 Influence of the currency pair ...... 80

5.4 Discussion ...... 83

6 Return differences 85

6.1 Introduction ...... 86

6.2 Description of the market ...... 88

6.3 Empirical Analysis ...... 90

6.3.1 The effects of the favourite-longshot bias ...... 91

6.3.2 Remaining unexplained effect ...... 92

6.4 Discussion ...... 95

7 Conclusion 97

7.1 Summary ...... 97

7.2 Limits and extensions ...... 98

viii CONTENTS CONTENTS

7.3 Potential impact ...... 99

8 Bibliography 101

9 Annexes 121

9.1 Description of the fees on the platform ...... 121

9.2 Discontinuity estimates at threshold 5 with a smaller interval . . 123

9.3 Illustration of the discontinuity estimates for a small bandwidth 124

ix LIST OF FIGURES LIST OF FIGURES

List of Figures

1 An Arrow-Debreu security illustration with the underlying event being Donald Trump getting reelected as the president of the United States of America on 3rd November 2020 ...... 2

2 Perceived prices b(p) as a function of actual prices p with a

left-digit bias, using inattention parameters θ2 = θ3 = θ4 =0.4.22

3 Discontinuity regression for each multiple of 10 with bandwidth 2 28

4 Average outcome or return for merged prices between 5 and 15 . 30

5 Graph of regression discontinuity estimates given the bandwidth 34

6 Regression discontinuity results for bandwidth 3 in intervals 5 to 15, 15 to 25 etc... all merged in an interval (5,15) ...... 34

7 Frequency of transactions by price ...... 37

8 Local polynomial density estimator of price in intervals 5 to 15, 15 to 25 etc... all merged in an interval (5,15) ...... 38

9 Representation of an "efficient" price for both buyer and seller side ...... 50

10 Representation of an "average" price between the seller and the buyer indifference price with a discount factor of 1.2 ...... 51

11 Representation of an "average" price between the seller and the buyer indifference price for different discount factors ...... 52

12 Local polynomial (degree 3) regression of outcome by price (NB: A 95% confidence interval is included in the graph but appears overlapping the fitted value line) ...... 55

13 Distribution of transactions of contracts by time difference be- tween the expiration and the trade in hours ...... 58

x LIST OF FIGURES LIST OF FIGURES

14 Focus on the distribution of transactions of contracts by time difference between the trade and the settlement in hours .... 58

15 Local polynomial (degree 3) regression of the outcome on the price given various time differences between the transaction and the expiration time ...... 60

16 Difference of returns given the day-of-the-week ...... 71

17 Difference of average transaction prices given the day-of-the-week 72

18 Standard deviation of returns given the day-of-the-week ..... 73

19 Number of trades given the day-of-the-week ...... 74

20 Comparison of return rates on Monday VS other days by exe- cution hour ...... 76

21 Comparison of price on Monday VS other days by execution hour 77

22 Comparison of standard deviation of returns on Monday VS other days by execution hour ...... 78

23 Comparison of volume on Monday VS other days by execution hour ...... 79

24 Monday returns of contracts for each currency pair ...... 82

25 Display of the offered contracts ...... 88

26 Illustration of a ticket order ...... 89

27 Average return rates for buyers and sellers ...... 90

28 Differences of volume by price whether buyer or seller side of contract ...... 92

29 Differences of outcome by price whether buyer or seller side of contract ...... 93

xi LIST OF TABLES LIST OF TABLES

30 Differences of outcome by price whether buyer or seller side of contract with focus on price < $50 and price > $50 ...... 94

31 Outcome discontinuity illustration at threshold 10 with a band- width of 1 ...... 124

List of Tables

1 Distribution of contracts by pair of currencies ...... 12

2 Key figures of contracts by pair of currencies ...... 13

3 Regression discontinuity in intervals 15 to 25, 25 to 35,... all merged in an interval (5, 15) ...... 33

4 Regression discontinuity of returns at threshold 1, 2, 3, 4, 5, 6, 7,8,9 ...... 36

5 Regression of the return rate and Monday effect ...... 75

6 Regression of the return rate and Monday effect with hourly variables ...... 80

7 Regression discontinuity of returns at threshold 5 in interval (2, 8)...... 123

xii Introduction

1 Introduction

1.1 Historical context

The idea of extracting beliefs from price is very old and is one of the foundation of economics. We can find traces of connection between price and beliefs as early as in "Van Rekeningh in Spelen van Gelucken" (1658) ("On Reasoning in Games of Chance") [79] of Christiaan Huygens. The treaty is one of the first in probability theory to focus specifically on the expected value. The author relies on the examples of games of fortune to prove the relationship between the price and the expected outcome of a game. Few years later, another figure from the Dutch Golden Age, Johan de Wittt in "Waardije van Lyf-renten naer Proportie van Los-renten" (1671) ("Value of life annuities in proportion to redeemable rents") [40] went even further and extracted surviving probabilities by comparing bonds with life annuities. It is not surprising that the theories come from the 17th century Netherlands. Since back then the country was very wealthy from international trade and saw the birth of corporate finance.

Many centuries and theories later, Arrow and Debreu (1954) [9] developed the concept of state-price security (also called Arrow-Debreu security) during the process of proving the existence of a unique general equilibrium. Such a financial product offers one unit of a numeraire (a currency or a commodity) if a particular state occurs in the future and zero in all the other states. An example of an Arrow-Debreu security is one which will offer $1 if the incumbent candidate Donald Trump would be reelected president of the United States of America during the presidential election held on 3rd November 2020 and 0 otherwise. The price of such securities becomes an indicator of the beliefs of agents.

1 1.2 Prediction markets Introduction

Figure 1: An Arrow-Debreu security illustration with the underlying event being Donald Trump getting reelected as the president of the United States of America on 3rd November 2020

1.2 Prediction markets

The prediction markets can be considered as extensions of the Arrow-Debreu security. They are markets where agents exchange contracts in which the pay- off will depend on the outcome of future events. The source of inspiration can be found in Galton (1907) [59] which describes how in a weight-judging competition, the median of people estimates for the weight of an ox was sur- prisingly very close (1%) from the truth and the mean of people estimates was even closer. Later, Hayek (1945) highlights the importance of the sum of individual knowledge of people:"It is with respect to this that practically every individual has some advantage over all others in that he possesses unique in- formation of which beneficial use might be made, but of which use can be made only if the decisions depending on it are left to him or are made with his ac- tive cooperation". The greater accuracy of the "crowd’s" prediction compared to the expert’s one has been largely documented. The aggregation of all the trades in a prediction market is then supposed to give better predictions than the experts of the underlying event. For example, Using data from the Iowa

2 1.2 Prediction markets Introduction

Electronic Markets Berg et al. (2008) [15] showed that the prediction market outperformed the average of polling organisation results to predict political outcomes on a majority of elections.

Wolfers and Leigh (2002) [182] proved the prediction market was even more accurate when studying Australian elections at an individual district scale. These very promising results motivated various fields to take profit from this "crowd wisdom". Several original ways of encouraging people to participate emerged in the last two decades. In addition of prediction market, the gamifi- cation of useful tasks is another way to make individuals engage in a collective activity. It consists of inciting people to fulfil exercises under the form of a game. One of the most famous examples is "Fold.it"1. Created by researchers, it uses the time and the logic of individuals to help them predict protein struc- tures by solving puzzles (the reader interested in the use of games to make agents participate to tasks can refer to Cooper et al. (2010) [34]). We can also quote the example of the CAPTCHA (Completely Automated Public Turing test to tell Computers and Humans Apart). It makes the participation in simple useful exercises, like image or text recognition, necessary for accessing some internet services requiring human identification (e.g. recaptcha in Von Ahn et al. (2008) [177]).

This dissertation will be centred on prediction markets, which incentivize agents with payments to know their beliefs on events. Wolfers and Zitzewitz (2004) [183] separate the contracts in three types:

• "Winner-takes-all", which pays a fixed amount if and only if an event happens at a certain date. The price of the exchanged contract should reveal the probability of the event to happen.

• "Index", which pays an amount proportional to the score of an event (for

1https://fold.it/portal/

3 1.2 Prediction markets Introduction

example $1 by percentage of vote in favour of a certain candidate during an election). The price of the contract should disclose the mean value of the score in this case.

• "Spread", which pays a fixed amount if and only if the score is superior to score∗ (for example a certain candidate having a percentage of votes > 30% at an election). People will bid according to their belief on the value of the score. The price of the contract should then reflect the median value of the score.

The prediction markets are various and numerous. The most famous among economists might be the Iowa Electronic market 2 run by the University of Iowa which started in 1988. The common underlyings are US presidential election results or the decisions of Federal Reserve. More recently, another very popular prediction market is run by the Victoria University of Wellington: PredictIt 3. Private companies and governments also developed their own platforms of trading on the happening of political, intelligence, financial, sport or even entertainment events. However, prediction markets can also raise some moral- ity question. For example, in 2001, the Defence Advanced Research Project Agency (DARPA) wanted to create prediction markets useful for intelligence called Future Markets Applied to Prediction (FutureMAP). Yet, this initia- tive was quickly cancelled in summer 2003 under the accusation of offering “terrorism betting parlors” (Yeh (2006) [187]). The first private companies to introduce prediction markets also suffered from a loss of popularity after the initial buzz. For example, TradeSports and Newsfutures created in the 2000’s are now closed and are replaced by more mature platforms. Cantor Exchange and Nadex specialising in binary options are examples of such ma-

2https://iemweb.biz.uiowa.edu/ 3https://www.predictit.org/

4 1.3 Binary options Introduction ture platforms. These two companies offer binary option markets that can be assimilated to prediction markets with "winner-takes-all" type of contract.

1.3 Binary options

The binary options are a specific form of the Arrow Debreu Security. A binary option is a standard traded security whose payoff is either a fixed positive payment (usually $100) or $0 depending on the value of a collateral at the expiration of the contract. If the value is above a predetermined amount, the outcome of the binary option holder is positive, otherwise null. This is a "winner-takes-all" type of contracts in which the trade price is meant to reflect the probability of the underlying event to happen. For instance, a binary option called "AUD/USD > 0.80 (7PM)" will stem in either a $100 payoff if the exchange rate of one Australian dollar in USD is above 0.8 at 7 PM today, either $0.

In 2008, the Securities and Exchange Commission approved the public exchange of binary options in USA. This decision quickly led the Chicago Board Options Exchange to offer binary options on Standard & Poor’s 500 Index and Volatility Index (VIX). Rapidly, platforms specialising on binary options are created such as 24Option, and AnyOption. However, the popularity of this type of platforms suffered as some of them were fraudulent and unregulated. As a response to this, the regulatory authority started to alert investors on the existence of scams connected to binary options. This pushed well-regarded binary option platforms such as Nadex towards more transparency. Today Nadex publicly publishes all the trades to increase the trust of investors, making it a rich dataset for researchers

5 1.4 Market efficiency Introduction

1.4 Market efficiency

Traditionally in economics, the financial markets have been held as paragons of the rational and efficient markets. The large number of sophisticated agents and the high amount of money implied should cancel out any isolated irrational error of individuals. There are two main reasons for this. First, each individual agent decision has a negligible weight in the whole market. Second, if rational market participants are present they will act as arbitragers which will drive out irrational agents.

Fama (1965) [44] and Samuelson (1965) [143] launched the debate about the possibility of forecasting the returns in the financial markets. They pro- vided some clues, theoretical for Samuelson and empirical for Fama, about how prices in stock markets should follow a random walk. Fama (1970) [45] further developed this idea by defining for the first time the concept of market effi- ciency. This is the property of a market in which at any time all security prices fully reflect all available information. Besides, Fama generalises the concepts in 3 sub-fields: the weak form efficiency only includes historical prices in the set of information, the semi-strong form where pr