ASYMMETRY. Vasiliki A. Makropoulou and Raphael N. Markellos 1. Athens University of Economics and Business

Transcription

1 THE COMPETITIVE MARGIN OF BOOKMAKERS UNDER INFORMATION ASYMMETRY Vasiliki A. Makropoulou and Raphael N. Markellos 1 Athens University of Economics and Business ABSTRACT. In fixed-odds betting bookmakers run the risk that public information changes. We present a long-run competitive equilibrium model in which bookmakers set their margin by balancing the expected revenues from noise bettors against the expected losses to informed bettors. The model suggests that expected returns to bettors increase as a monotonic function of winning probabilities. Therefore, the favorite-longshot bias that has been widely reported in empirical studies can be fully explained by our model solely on the basis of information asymmetry. Using an extensive data-set from two bookmakers on football betting, we estimate an average degree of informed betting. Keywords: Fixed-odds betting, favorite-longshot bias, asymmetric information, contingent pricing. JEL Classification: D82 (Asymmetric and Private Information), D84 (Expectations; Speculations), G13 (Contingent Pricing; Futures Pricing). 1 The authors belong to the Financial Engineering Research Centre (FRC), Department of Management Science and Technology. Corresponding author is Markellos, 47A Evelpidon & 33 Lefkados, Athens, Greece. tel , fax , rmarkel@aueb.gr 1

2 INTRODUCTION Since the first half of last century the economics of gambling, and particularly those of betting, have attracted increased research attention. This is due to two main reasons. First, gambling is nowadays a significant global market reporting turnover of $433 billion in 2003 with betting accounting for the 37% of the market. Second, the fact that gambling markets have many parallels with the financial markets and yet are simpler provides great opportunities for economic analysis and specifically for testing market efficiency. If betting markets are efficient, then the expected return should be equal at all probabilities. However, a well-documented anomaly in the betting markets literature is the so-called favorite-longshot bias, according to which returns on favorites are higher than returns on longshots. The specific setting of this paper is the fixed-odds betting market on UK football. The last decade witnessed the emergence of Britain as a global leader in the rapidly growing betting industry, due to both the technological advances that have taken place and the British liberal gaming legislation. Particularly, customer location has become increasingly less important since it is possible nowadays to place cross-border bets without incurring significant costs. Additionally, the abolishment of the betting tax in October 2001 has brought almost all British-owned betting businesses back to their home country. As a result innovation and growth have spurred, turning the local British betting industry into one of the most profitable businesses. The most common form of betting in Britain is via fixed-odds coupons. Bookmakers set prices and as a compensation for underwriting the risk require a margin in the range of 10-13% of the total bets placed. The particularity of this market is that once the odds are posted, usually a few days before the match, they remain fixed. However, the traditional bookmaking system is seriously threatened by the emergence of betting exchanges. In this setting, gamblers wanting to bet for a particular outcome are matched to those that want to bet against it. As a result players assume all the risk whereas the betting exchange companies receive from the winner a commission that is between 2% and 5%, as a reward for their services. The fact that the margins charged by the betting exchanges are lower than those required by the traditional bookmakers makes the former choice more appealing to bettors than the latter. 2

3 Under the scope of the aforementioned developments, the accurate pricing of bets by the bookmakers has become crucial in order to preserve competitiveness and market shares. The focus of this paper is therefore the determination of the competitive margin of bookmakers in the betting market on UK football under information uncertainty. Given that the odds remain fixed throughout the betting period, bookmakers run the risk that the public information set may change and that some odds become more favorable to bettors than those that would have been set conditional upon the new information. Informed agents can exploit this information and are analogous to insiders in other gambling markets in which the odds vary continuously during the betting period. We develop a model of perfectly competitive equilibrium in which bookmakers set their margin by balancing the expected revenues from noise bettors against the expected losses to informed bettors. By modeling the loss to informed bettors as writing a call option to them, we determine a fair margin that compensates bookmakers for the uncertainty of information flow. We find that even under the assumptions of zero transaction costs and zero long-run expected bookmaker profit, a positive margin is still required implying that the margin is a purely informational phenomenon. The computation of the competitive margin is made possible by assuming a stochastic process for the probability of a specific outcome occurring and performing Monte Carlo simulations. We show that expected returns to bettors increase monotonically with winning probabilities due to the higher risk that bookmakers face at lower probabilities, which in turn induces them to require a higher margin. Therefore, the favorite-longshot bias is naturally recovered and explained as an optimal pricing response by bookmakers to information uncertainty. The rest of this paper is organized as follows. In Section I a brief review of the literature is presented. Section II discusses the assumptions underlying the framework of the analysis followed, while, a description of the model developed for the determination of the competitive margin is found in Section III. Section IV transforms the competitive equilibrium model into an optionpricing framework and presents a methodology in order to calculate the competitive margin. Section V demonstrates how the model predicts the existence of the favorite-longshot bias and provides a rational foundation for this. Section VI extends the preceding analysis to account for 3

4 time-varying odds. Section VII presents the results of an empirical application using data from football betting, while, the final section concludes the paper. I. LITERATURE REVIEW The approach followed in this paper for modeling the competitive margin of bookmakers can be traced back to the literature dealing with the informational source and optimal level of the bid-ask spread under asymmetric information (e.g. Copeland and Galai, 1983; Glosten and Milgrom, 1985). Particularly, Copeland and Galai (1983) determined the bid-ask spread that maximized the difference between the expected revenues from liquidity-motivated traders, and the expected losses to information-motivated traders in financial markets. In their framework, informed traders possess private information unknown to the dealer. When an actual informed trade takes place the private information is revealed and as a response the dealer might revise his spread. Shin (1991, 1992, 1993) recognizing the informational source of bookmakers margin, addressed the problem of determining the optimal margin of a monopolistic bookmaker under asymmetric information in the UK racetrack betting market. In a more recent study, Ottaviani and Sorensen (2005a), following an analysis closely related to that of Shin (1991, 1992) examine the information effects on equilibrium prices in a competitive bookmaking market of racetrack betting. To the best of our knowledge, this is the first paper that explicitly models in continuous-time the competitive margin of bookmakers in the betting market on UK football under information uncertainty and heterogeneously informed bettors. Whereas in racetrack betting, bookmakers can vary their odds during the betting period, in fixed-odds betting the odds do not change. Furthermore, the racetrack betting market convenes for about half an hour whereas in the market on UK football there is a betting period of about one week. Therefore, the information effects on the margin of bookmakers in football betting are even more important considering that public information changes constitute an additional risk and that the longer time period increases information uncertainty. As mentioned before, a well-documented anomaly in betting markets is the favorite-longshot bias. A significant number of papers have provided evidence of the bias and many have attempted to explain it. Evidence of this bias was first observed by Griffith (1949) who studied the pari- 4

5 mutuel system in racetrack betting. Snyder (1978) surveyed six studies including 50,000 races in North America, each exhibiting the favorite-longshot bias, and calculated the aggregate returns at various levels of odds. Dowie (1976) found evidence of the bias in the British racetrack betting market where odds are set by bookmakers, making this finding not unique to pari-mutuel betting. Actually, the bias is more prominent in Dowie s data (Sauer, 1998). In a more recent study, Bruce and Johnson (2001) also observed that the bias tends to be more pronounced in bookmaking than in pari-mutuel markets. Different theories have been developed to explain the favorite-longshot bias. Griffith gave a psychological explanation by arguing that individuals may have the tendency to underestimate the chances of favorites and overestimate those of longshots. Weitzman (1965), Ali (1977) and Quandt (1986) viewed the favorite-longshot bias observed in the pari-mutuel system as evidence of local risk preference in the bettors utility functions, while, Shin (1992, 1993) focused on the supply side and explained the bias observed in the bookmaking market of racetrack betting in terms of information asymmetry. Particularly, he argued that the presence of insiders who hold private information creates a favorite-longshot bias through an optimal pricing response by the monopolistic bookmaker. Focusing on the same market, Ottaviani and Sorensen (2005a) showed that information asymmetry creates a similar bias even when bookmakers are competitive. They concluded that the bias arises because relative to favorites, longshots attract a relatively higher proportion of insiders. They also extended their analysis to pari-mutuel markets (2005a, 2005b) and constructed a theoretical model of pari-mutuel betting that provides an informational explanation of the favorite longshot bias observed in this market. Cain, Law and Peel (2000) showed that the fixed-odds betting market on UK football exhibits the same favorite longshot bias, as that found in racetrack betting. In another study (Cain, Law and Peel, 2003), the authors examined a variety of sports betting markets where odds are set by bookmakers and after estimating the average degree of insider trading, they concluded that it is higher in markets where the odds are fixed. This was explained as a consequence of the existence of two possible sources of inefficiency. Firstly, bookmakers are exposed to the danger that some bettors hold superior information, what is called insider trading and secondly, they are exposed to the additional danger that the public information set may change after the odds have been declared. 5

6 Therefore, bookmakers should protect themselves against this possibility, ust as they would do in the presence of insiders. Kuypers (2000), focusing on fixed-odds football betting in UK, shows that an expected profit maximizing bookmaker could set market inefficient odds to take account of bettors biases. Under such a profit maximizing strategy, a favorite longshot bias could arise. This paper adds to the existing literature of betting markets by explicitly modeling public information flow in a continuous-time setting and deriving the optimal margin of bookmakers. The model developed here indicates that even under the assumption of zero expected profit, a favorite longshot bias could still arise. Therefore, the bias can be explained solely on the basis of information uncertainty. II. A FRAMEWORK FOR ANALYZING THE MARGIN OF BOOKMAKERS A. Bookmakers price setting The purpose of this subsection of the paper is to analyze how bookmakers set their odds. At first, it would be reasonable to assume that the odds should reflect the probability of a certain outcome occurring. However, the bookmaker is not trying to predict the outcome of a match. Neither does he have to correctly assess the probability of each possible outcome. In reality, he is trying to quote odds that will be inversely distributed to bets. If he has to predict something, that is the expectations of bettors. Therefore, the bookmaker assigns his initial odds according to his expectations about the distribution of bets. From there after, he reads and interprets the information he is getting from bettors, and adusts his odds accordingly. Therefore, ideally the bookmaker would like to continuously adust his odds according to the wagers in order to balance his book. However, in fixed-odds betting, the bookmaker cannot revise his odds according to the flow of bets. His odds reflect an initial subective probability assigned to each outcome. This probability, rather than being his estimation of the probability that a certain outcome is realized, is his estimation of the proportionate amount of money that will finally be bet on this outcome. However, at the end of the betting period, the final distribution of bets might deviate from the initial probability he had assigned to each outcome. There are three possible reasons for such deviations: prediction errors on the part of bookmakers, changes in public information after the odds are set, 6

7 and the presence of insiders who possess superior information. No matter what is the actual source of this uncertainty, bookmakers run the risk that some posted odds will be more favorable to bettors than those that would have been set conditional upon the information set at the time the bet is placed. In this paper, it is assumed that bookmakers can predict with accuracy the initial expectations of punters, i.e., there are no prediction errors. 2 Therefore, any deviations will be due to information that arrives after the declaration of the odds. Consequently, the margin should reflect the uncertainty of new information, both public and private, which accrues after the odds are posted and a brokerage fee. The approach developed here implies that the true probabilities of match outcomes evolve according to the information flow throughout the betting period until the event takes place. In the simplest case, bookmakers offer odds against three possible outcomes: home team win (H), away team win (A) and drawn match (D). The quoted prices against outcome, where =1 is for a home win, =2 for an away win and =3 for a drawn match, of a match can be expressed in terms of the return θ on a one-unit stake if is realized. If a bet is successful then, ignoring taxes, the bettor receives (1+θ ) for every one unit staked. From the perspective of a bookmaker, the prices (φ ) are modeled as depending on the subective probability for outcome (p ) 3 and a margin (λ ), in the following manner: 1 φ = = p + λ 1+ θ (1) As aforementioned, this margin, λ, consists of two parts; the first part is the bookmaker s brokerage fee whereas the second part accounts for the impact of new information on the odds. If the bookmaker did not wish to earn any profit for providing his brokerage service and if no information were released after the odds were posted, then λ would be zero. B. The informational source of bookmakers margin 2 Levitt (2004) provides evidence that the bookmaker is at least as good as any of the bettors at predicting game outcomes. 3 As mentioned previously, in reality, p is not the bookmaker s subective probability of outcome occurring but his estimation of the final proportion of money bet on this outcome. 7

8 The rest of this section analyzes the framework that will be used in order to model the information effects on the margin as well as the underlying assumptions. In the present model, the bookmaker is assumed to face two different types of punters, namely noise bettors and informed bettors. Just like noise traders (Black, 1986), noise bettors do not bet on the basis of information but rather randomly. On the other hand, informed bettors observe the public information flow and make their decisions based on the news that they receive by the market. 4 The bookmaker is assumed to also be an informed agent. It is necessary to emphasize the differences between insiders and informed bettors as two distinct types of punters: insiders bet on the basis of private information they possess, whereas informed bettors do not hold any private information and place their bets according to the public information (e.g., forecasts by analysts, news about players inuries, weather changes, etc). However, given that in this specific betting market, bookmakers cannot adust their odds to account for public information changes, informed bettors are considered analogous to insiders. 5 As mentioned earlier in this paper, the expectations of informed bettors are being shaped according to the release of new information. Therefore, in fixed-odds betting, informed bettors have an incentive to bet at the last minute, when all information is available. Moreover, they bet only if their expected return is positive. On the other hand, since noise bettors are uninformed, they ignore changes in the information set. Taking this into account and assuming that there are no prediction errors on the part of bookmakers we can infer that the latter can accurately predict the expectations of noise bettors and hence their final distribution of bets. Therefore, the bookmaker will have an expected loss (profit) from informed bettors (noise bettors). The optimal margin of bookmakers is determined as a tradeoff between expected losses to informed bettors and expected gains from noise bettors. A set of additional assumptions are necessary to proceed: a) There are no transaction costs. 4 Compared to noise bettors, informed bettors are assumed to be individuals particularly skillful in processing public information. Thus, noise bettors may also observe public information flow but do not assimilate it and their decisions are not based on it. 5 In reality, there are three types of punters: noise bettors, informed bettors and insiders. However, this would imply that the information flow and the true probabilities have a second stochastic component related to private information. For simplicity, in the present model it is assumed that there is no private information and hence no insider trading. 8

9 b) The true probability of outcome occurring, P, follows a stochastic process. This could be either a continuous process or a discontinuous ump or a mixture of the two. c) E t= 0 P() t = P(0) for any t>0. Hereafter the time subscript is dropped from the expectation and it is always assumed t=0. d) The stochastic process contains only public information. e) All agents are assumed to be risk-neutral. The bookmaker s obective is to set a margin that maximizes his profit. If he is a monopolist, he will maximize his expected profit by maximizing the difference between expected revenues from noise bettors and expected losses to informed bettors. If there is free entry, the long-run competitive equilibrium will be established where the expected revenues from noise bettors equal the expected losses to informed bettors. To illustrate how competition will lead to such equilibrium, suppose there is a single bookmaker in the market and that there is free entry. If the bookmaker sets a margin which is higher than the competitive one, his expected profit is positive. This positive profit encourages entry of a second bookmaker who will rationally undercut the first by setting a lower margin. An equilibrium is established when all bookmakers in the market earn zero long-run profits. The information arrival process and the reaction of agents to it can be described as follows. The bookmaker declares his initial odds according to the information set that exists today. Informed bettors have an incentive to wait for the revelation of future information and bet only if the future states of nature favor them. We assume that noise bettors do not observe any information changes and that the bookmaker gains from them. As mentioned before, the bookmaker is assumed to be an informed agent and thus observes the information flow, although he cannot revise his book. The basic intuition behind our model is that given the fact that the bookmaker cannot revise his book, he has to be compensated for the uncertainty of information and hence requires a margin. Furthermore, we expect this margin to be larger the greater the uncertainty and the longer the time period between the declaration of the odds and the occurrence of the match. For example, posting the odds two weeks before the match, rather than one hour, is more risky for the bookmaker, so the latter should require a higher margin. 9

10 III. A MODEL OF COMPETITIVE EQUILIBRIUM A. No informed bettors Assume that only noise bettors exist in the market, W is the total amount of money bet by them on the three possible outcomes and w is the amount bet on outcome where =1, 2, 3. Assume also that the bookmaker gives odds of (1+θ ) for each one of the outcomes. Then, ignoring the time-value of money, the bookmaker s expected profit at time zero is: [ ] [ ] [ ] E( Π ) = W E P( T) w (1 + θ ) E P( T) w (1 + θ ) E P( T) w (1 + θ ) (2) or equivalently using assumption (c): E( Π ) = W P(0) w (1 + θ ) P(0) w (1 + θ ) P(0) w (1 + θ ) (3) where P (0) and P (T) are initial ant terminal values (time T) for the true probability that outcome is realized, respectively. Given that P1 P2 P = for each t [ 0, T] expected profit, it is sufficient that for each :, then for the bookmaker to have a positive W w w (1 + θ ) φ (4) W Therefore, if only noise bettors exist in the market and as assumed earlier the bookmaker can accurately predict their expectations, then for the latter to have zero expected profit it is sufficient that prices are set equal to the bookmaker s expectation about the final proportion of money bet on each outcome, i.e. φ = p, where p = wn, Wn. As a result, the competitive margin of the bookmaker in the absence of informed bettors is zero. B. Informed bettors If informed bettors also exist in the market, then the final distribution of bets will depend upon both the expectations of noise and informed bettors. As aforementioned, the bookmaker can predict with 10

11 accuracy the expectations of noise bettors but not those of informed bettors since they change over time according to the information flow. It is assumed that at the end of the betting period, the expectations of informed bettors fully reflect all publicly available information and, hence, are an unbiased estimator of the true probability of outcome being realized. Consider an informed bettor who always waits until the end of the betting period when all information is available and bets only if the true probability at time T is greater than the quoted prices. 6 Therefore, the bettor has the option to wait and place his bet, only if he expects a positive return. As a result, the bookmaker is expected to lose from informed bettors. If we assume that the time value of money is negligible over this small period, the informed bettor s expected profit on a one unit bet is: Therefore, the expected profit of the bookmaker is: { ( )} E( Π ) = E max 1 + P ( T)(1 + θ ),0 (5) i 3 3, θ { (,, θ )} (6) E( Π ) = W E P ( T) w (1 + ) E max w + w P ( T)(1 + ),0 n n i i =1 = 1 For the bookmaker to have zero expected profit, from equation (6) we obtain the following condition: 3 3, (0)(1 θ ) +, { max( 1 ( )(1 θ ),0)} (7) W = w P + w E + P T + n n i =1 = 1 Rearranging the terms in equation (7) we obtain: 3 wn, w i, 1 = P(0) (1 + θ ) + E{ max( 1 + P( T)(1 + θ ),0) } (8) =1 Wn P(0) Wn For this equation to hold it is sufficient that for each : 6 The subective probability of the informed bettor at time T is the same with that of the bookmaker since they both reflect all available information until that time. Further, given that it fully reflects all publicly available information, it is also equal to the true probability of outcome being realized. 11

12 w w 1 = (1 + ) + { max( 1 + P ( T)(1 + θ ),0)} (9) W P W n, i, θ E n (0) n The second part of the right-hand side of this equation is greater or equal to zero. Therefore, the first part should be lower than unity implying that: W w n, n w, n (1 + θ ) φ (10) Wn Therefore, if informed bettors also exist in the market, then for the bookmaker to have zero expected profit prices should be set greater than the expected proportionate amount of money placed by noise bettors on each outcome, i.e. φ p. Thus, the competitive margin of the bookmaker is greater than zero. Rearranging the terms in equation (9), we get bookmakers competitive price: w n, wi, φ P( T) φ = 1+ E max 1 +,0 Wn w, n P(0) φ (11) or 1 φ = p 1+ q E max P( T),0 P (0) { ( φ )} (12) w, i where q =. w, n Solving for the margin, λ, we obtain: p λ = q E{ max ( P( T) φ,0) } (13) P (0) In the above equation, the ratio p P(0) captures biases in bettors expectations. If there were no such biases, the subective probability assigned by bettors to outcome should coincide with the 12

13 true probability of outcome occurring. Moreover, the expectation captures information uncertainty and q the existence of informed bettors who can potentially benefit from it. If there was no uncertainty or if all bettors in the market were noise bettors, then the margin would be zero. This is a very important finding considering that although the informational source of the margin in fixedodds betting has been well recognized (Pope and Peel, 1989), it had not been modeled. IV. THE MARGIN OF BOOKMAKERS IN AN OPTION-PRICING FRAMEWORK The basic idea is that the commitment made by bookmakers to sell at fixed prices, the quoted odds, can be analyzed as a call option. The bookmaker gives a prospective bettor a call option, i.e., the right to buy at the asking price φ>p 7 0. Note that in order for the bookmaker to be compensated for the uncertainty of information this option is issued out-of-the money and is similar to an American call option on a stock that pays no dividends. Informed bettors wait for updated information and place their bets only if the true probability at maturity, P(T), is greater than the initial odds, φ 8. If we further assume that informed bettors are risk-neutral, then today s option price can be determined by discounting the expected value of the terminal option price by the riskless rate of interest. Therefore, neglecting the time-value of money, the value of the call option is: { max ( ( ),0)} C = E P T φ (14) Equation (14) simply says that the value of the option at maturity will be either P ( T) φ or zero, whichever is greater. If the true probability at time T is greater than the exercise price, the option will expire in the money. This simply means that the informed bettor will exercise it by placing his bet. Otherwise, the option expires unexercised. In an option-pricing framework, equation (12) can now be transformed into the equation: 7 P 0 is the initial true probability of a certain outcome occurring. 8 This is consistent with the option pricing theory for American options on a stock that pays no dividends, according to which it is never optimal to exercise the option before the expiration date. Note that an American option is an option that can be exercised at any time up to the expiration date in contrast to a European option that can be exercised only on the expiration date. 13

14 C φ = p 1 + q P (0) (15) A. Computing the value of the option and bookmakers margin In order to derive the value of the option we need to know the stochastic process followed by the true probability. As mentioned before, this could be a ump process or a continuous one or even a mixture of the two. We will assume that the true probability follows a continuous process. Given that the probability can only take values in the range [0, 1], assuming a Geometric Brownian Motion would be inappropriate, since this would imply that true probabilities can take values in the range [0, ). Consider instead that the true probability evolves according to the following stochastic equation: dp () t = ( 1 P ( t) ) σ dz (16) P () t where σ, is the instantaneous standard deviation of the change in the random variable P (t) and dz, is a standard Wiener process. It can be easily shown that P (t) takes values in the range [0, 1]. Furthermore, given that this process is driftless, the expected value of P (t) at any t>0 is equal to the initial value P (0) and hence any deviations from this value are white noise. Knowing the stochastic process followed by P (t) and performing Monte Carlo simulations we can calculate the value of the call option. The competitive margin φ can be found from equation (15) by solving the optimization problem. Our computations of the competitive margin are based on the assumption that the subective probability of noise bettors is equal to the true probability, i.e p = P (0). B. Properties and implications It is a well known fact in the option-pricing literature that the value of the option, C, increases with the uncertainty over the true probability and with the time to maturity. It is interesting to note that, in the above model, the specific risk of the true probability, i.e., the uncertainty related to it, is an 14

15 important factor that affects the margin of bookmakers. In the case where there is no uncertainty or T goes to zero, the value of the option goes to zero and hence the margin is driven to zero as well. In other words, if there is no uncertainty about future information or if there are no informed agents in the market (w,i = 0), i.e., bookmakers could know ex ante the exact distribution of wagers, then, at any equilibrium, the prices should coincide with the distribution of bets, i.e. φ = p. Furthermore, the margin of bookmakers depends upon the life of the bet, i.e., the period from the time the odds are posted until the match takes place. This is a very classic issue in option pricing: the value of the option increases with the time to maturity. However, the life of the bet is rarely discussed in fixed-odds sports betting. It would be natural though, to assume that as the life of the bet increases, the margin also increases, given that more information might accrue after the posting of the odds when the time interval is significant. In this case, informed bettors can take advantage of the variance of new information and as a result bookmakers should require a higher margin. Figures 1, 2 and 3 plot the competitive margin as a function of the volatility, time to maturity and ratio of informed to noise betting. Competitive Margin, φ % 20% 40% 60% 80% 100% Volatility, σ Figure 1. Competitive Margin vs. Volatility: The line plots the competitive margin at various levels of volatility and for initial probability 0.5. It is assumed that the life of the option T = 5 and q =

16 0.55 Competitive Margin, φ Time to Maturity, Τ (days) Figure 2. Competitive Margin vs. Time to Maturity: The line plots the competitive margin vs. the time to maturity of the option for initial probability 0.5. It is assumed that the volatility σ = 30% and q = 5. Competitive Margin, φ Ratio of Ιnformed to Νoise Βetting, q Figure 3. Competitive Margin vs. Ratio of Informed to Noise Betting: The line plots the competitive margin vs. the ratio of informed to noise betting for initial probability 0.5. It is assumed that the volatility σ = 30% and T = 5. 16

17 As mentioned before, the above model assumes dealer risk neutrality. Clearly, this is a simplification. However, even with risk neutral bookmakers and zero transaction costs, the margin will be positive, as long as there is information uncertainty and informed bettors who can potentially benefit from the release of future information. Therefore, bookmaker risk aversion is not necessary in order to explain the existence of a positive margin. This is a very important result considering that most research in this field assumes that bookmakers are risk-averse and hence require a positive margin. V. THE FAVORITE-LONGSHOT BIAS The aim of this section is to show that our model predicts that when bookmakers set their competitive margin, expected returns will exhibit the favorite-longshot bias, i.e. expected returns on favorites will be higher than those on longshots. If there is no bias in betting markets, then the efficient markets hypothesis would argue that the expected return at all probabilities is the same. The expected return of a noise bettor to a unit bet is: E P ( T) P (0) ER ( ) = 1+ = 1+ (17) φ φ Formally, we shall say that equilibrium prices exhibit the favorite-longshot bias when, ϕ ϕ > P(0) P (0) if and only if P(0) < P (0). In other words, the odds understate the winning i i chances of favorites relatively less than the winning chances of longshots. i Suppose first that only noise bettors exist in the market. As shown earlier, for the bookmaker to have zero expected profit, it is sufficient that he sets prices equal to the subective probability, i.e. φ = p. Therefore, the expected return is: P (0) ER ( ) = 1+ (18) p 17

18 If the expectations of bettors are equal to the obective winning probabilities, then the expected return is zero for all probabilities. Suppose that p P (0). Then for expected returns to exhibit the favorite-longshot bias it is necessary that de( R) dp (0) or equivalently d P(0) p dp(0) > 0. This means that as P (0) increases, the ratio of subective to obective probability should decrease. In other words, if bettors tend to overestimate the winning chances of longshots relative to those of favorites and hence bookmakers quote relatively shorter odds on longshots, then returns will exhibit the favorite-longshot bias. This is what we call behavioral biases. We now assume that informed bettors also exist in the market. As shown before, prices will deviate from subective probabilities. In order to demonstrate that expected returns will exhibit the favorite-longshot bias, we take the first derivative of the ratio of the price, φ, to the initial obective probability, P (0) with respect to this probability. Equation (15) can be written: φ p C = 1 + q P(0) P(0) P(0) (19) Differentiating this equation with respect to P (0), we obtain: φ p d d P (0) P (0) qc d( q) C q dc 1 p dp dp P dp P P dp P P = qc 2 (0) (0) (0) (0) (0) (0) (0) (0) (0) p In the case where bettors do not exhibit behavioral biases then d dp (0) = 0. However if P (0) bettors tend to overestimate the wining chances of longshots and underestimate those of favorites as it is commonly argued in the literature this derivative is negative. Therefore, the first term of the right hand side of equation (20) explains the part of the favorite-longshot bias that can be attributed to behavioral biases. Behavioral explanations fall into two sub-categories: bettors risk preferences (20) 18

19 (e.g. Weitzman, 1965; Ali, 1977; Quandt, 1986; Golec and Tamarkin, 1998; Cain and Peel; 2004) and psychological explanations (e.g. Griffith, 1949). The second term can be explained by information asymmetries. The first term in the bracket models the intensity of the adverse selection problem with respect to the true probability. Specifically, if q falls with P (0) the bias becomes more prominent. Finally, as shown in figure 4, the sum of the last two terms in the bracket is always negative Quant Initial Probability Figure 4. The line plots the value of dc C quant = dp (0) P (0) vs. the initial probability when σ = 30%, T = 5 and q = 0.5. Information asymmetries and behavioral biases have an additive effect to the favoritelongshot bias. Therefore the model developed in this paper predicts that returns to bettors will exhibit the favorite-longshot bias as a consequence of two factors. First, bettors tend to overbet longshots and as a result bookmakers adust their margin accordingly. Second, bookmakers require a compensation for the uncertainty of information and the potential existence of informed bettors who exploit it. Figure 5 depicts bettors returns at various initial probabilities and for two levels of volatility, 20% and 40% when the betting period is 5 days and the ratio of informed to noise betting is equal to one at all initial probability levels. Note that as uncertainty increases, the expected returns to 19

20 bettors decrease due to the fact that bookmakers have to be compensated for the increased uncertainty. 0% -2% Expected Return -4% -6% -8% -10% Initial Probability Figure 5. Returns vs. Probabilities: The lines plot the returns at various probabilities for two levels of volatility: 20% and 40%. It is assumed that the life of the option T = 5, and that q = 1. VI. AN EMPIRICAL APPLICATION The data set analyzed includes the results of 8,363 football matches played during the period from 2002 to 2004, together with the associated odds against each outcome (home win, away win or draw) quoted by a leading online bookmaker in UK (hereafter called A). The data set includes also odds by a fixed-odds bookmaker (hereafter called B) for the same set of football matches. The maor difference between online bookmakers and fixed-odds bookmakers is that online bookmakers can vary their odds at any time before a match takes place, whereas the odds offered by fixed-odds bookmakers remain fixed throughout the betting period. The odds from the online bookmaker are the closing odds. Descriptive statistics are presented in Table 1. As expected, bookmaker B operates at a greater margin, due to the additional risk that he faces. Bookmaker A Bookmaker B Sample Size 8,363 8,363 Average Margin 11.93% 17.19% Standard Deviation 0.82% 1.50% 20

21 Table 1. Margin Summary Statistics First, the efficiency of the bookmaker odds is analyzed. For the full sample of matches there is some evidence of the favorite longshot bias, as shown by the average returns to bets at different price levels in Table 2: bets on longshots generated substantially lower returns than bets on favorites. Bookmaker A Bookmaker B Range of prices, φ Ν Returns Ν Returns Δ(R) 0 φ % % -4.99% 0.2<φ < % % -5.53% 0.4<φ % % -3.78% 0.6<φ % % -3.55% 0.8<φ % % -3.45% 25,089 25, % Table 2. Returns to a unit bet on match outcomes Our goal is to obtain an estimate for the average degree of informed betting by assuming that the higher margin of bookmaker B is due to information uncertainty. Since bookmaker A does not face the risk of public information changes, he sets prices φ Α = p A. For bookmaker B equation (12) holds: pb φb = pb + qc ( P(0), φb, σ, T ) (21) P(0) Therefore, pb φb φa = ( pb pa) + qc( P(0), φb, σ, T) (22) P(0) If we further assume that there are no behavioral biases, i.e. P(0) = p B we obtain a measure for the ratio of informed to noise betting: 21

22 q B A = (23) C φ φ ( φ, φ, σ, T) A B We have found that the average φ Α = and the average φ B = In order to calculate the value of the option we need to know the volatility. Assuming that bookmaker B quotes his odds five days before the match and given that the odds of bookmaker A are the closing odds, we can obtain a measure for the daily volatility of the true probability of a certain outcome occurring, given that this probability is reflected in the odds. We get σ = 6.32% for the probability of a Home win. The value of the option for the above parameters and T = 5 is C = Therefore, from equation (25) we find q = This means that 73.55% of all bettors are informed in the sense that they observe public information and place their bets according to it. VII. SUMMARY AND CONCLUSIONS This paper has formulated a theoretical model of price formation in betting markets where the odds are set by bookmakers. More specifically, we have examined the information effects on the margin of bookmakers that operate in the fixed-odds betting market on UK football and have shown that the long-run competitive margin is determined as a trade-off between expected revenues from noise bettors and expected losses to informed bettors. The bookmaker is assumed to offer prospective bettors an out-of-the money call option during a fixed time interval. Throughout the analysis it is assumed that the bookmaker can accurately predict the expectations of noise bettors, given that these reflect the information set that exists at the time the odds are posted, and thus always gains from them. On the other hand, informed bettors wait until the end of the betting period when all information is available and place their bet only if they expect a positive return by exercising the inthe-money call option. Consequently, the bookmaker always loses from informed bettors. This paper demonstrates that even in the case of perfect competition with zero expected profits and zero transaction costs, there is a positive margin related to the uncertainty of the information flow. Furthermore, we find that the magnitude of the margin is an increasing function of both the variance of new information and the ratio of informed to noise betting. We extended our analysis to 22

23 account for time-varying odds as in racetrack betting and developed a similar model considering that only private information changes constitute a risk for the bookmaker. Our model has implications that are consistent with the existing literature on empirical evidence. First, we show that bookmakers require a proportionately higher margin in low probabilities. This results in the prediction of the so called favorite-longshot bias found in a variety of sports betting markets. Given that we provide an aggregate explanation based on both behavioral biases and the adverse selection problem faced by the bookmaker, our model also ustifies the fact that the bias is more intense in bookmaking than in pari-mutuel betting markets (Bruce and Johnson, 2001). 9 Most important, modeling public information risk in fixed-odds betting provides an explanation for the higher degree of insider trading observed in these markets (Cain, Law and Peel, 2003). Using data on football betting, we find the daily volatility of the true probability of a certain outcome occurring to be 6.32% and the implied degree of informed betting 78.92%. The implications of the model derived in this paper are very important for the bookmaking business in UK, especially in view of the growing interest of consumers and competitors on betting exchanges. It can be easily inferred that betting exchanges are advantageous against traditional bookies. Even perfectly competitive bookies with zero expected profits and zero transaction costs will not be able to beat the exchanges given that the former will always require a positive margin for the uncertainty of future information. Bookmakers operating in the betting market on UK football should probably consider changing their odds over time in order to balance their book. In this case, the only risk they run is that some bettors hold superior information. REFERENCES 9 For an information-based explanation of the bias in pari-mutuel systems see Ottaviani and Sorensen (2005). 23

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