Optimal Bookmaking, Which Takes Into Account the Situations Described Above

arXiv:1907.01056v3 [q-fin.MF] 5 Mar 2021 [email protected] elrto finterest: of Declaration maximization Keywords: trs T02910,USA. 06269-1009, CT Storrs, ∗ † ‡ colo ahmtc n ttsis nvriyo Sydney of University Statistics, and Mathematics of School eateto ple ahmtc,Uiest fWashingt of University Mathematics, Applied of Department orsodn uhr 4 asedRa 10,Department U1009, Road Mansﬁeld 341 author. Corresponding e nvrositrsigmodels. interesting bo various the in to expe lem characterizations his or maximizes solutions bet that explicit of process obtain intensity We price or optimal rate an the outcomes seeks aﬀect on maker bookmaker bets the of by prices set prices the control dynamically can maker eitoueagnrlfaeokfrcniuu-iebetti continuous-time for framework general a introduce We tcatcpormig oso rcs;sot betting; sports process; Poisson programming; Stochastic ate Lorig Matthew . e-mail uoenJunlo prtoa Research Operational of Journal European none. : [email protected] pia Bookmaking Optimal cetdfrpbiainin publication for Accepted hsVrin ac 2021 March Version: This is eso:Jl 2019 July Version: First ∗ Abstract huZhou Zhou 1 hn:+1-860-486-3921. Phone: . n etl,W,USA. WA, Seattle, on, kae’ pia okaigprob- bookmaking optimal okmaker’s † fMteais nvriyo Connecticut, of University Mathematics, of lcdb abes h book- The gamblers. by placed s td(tlt f emnlwealth. terminal of) (utility cted frno vns ntr,the turn, In events. random of gmres nwihabook- a which in markets, ng yny Australia. Sydney, , i Zou Bin tcatccnrl utility control; stochastic e-mail ‡ : [email protected] e-mail . : 1 Introduction

Sports betting is a large and fast-growing industry. According to a recent report by Zion (2018), the global sports betting market was valued at around 104.31 billion USD in 2017 and is estimated to reach approximately 155.49 billion by 2024. As noted by Street and Smith (2018), growth within the United States is expected to be particularly strong due to a May 2018 ruling by the Supreme Court, which deemed the Professional and Amateur Sports Protection Act (PASPA) unconstitutional and thereby paved the way for states to rule individually on the legality of sports gambling. Less than a year after the PASPA ruling, eleven states had passed bills legalizing gambling on sports compared to just one (Nevada) prior to the PASPA ruling. At the time of the writing of this paper, several additional states have drafted bills to legalize sports betting within their borders.1 In this paper, we analyze sports betting markets from the perspective of a bookmaker. A bookmaker sets prices for bets placed on different outcomes of sporting events (or random events in general), collects revenue from these bets, and pays out winning bets. Often the outcomes are binary (team A wins or team B wins). But, some events may have more than two outcomes (e.g., in an association football match, a team may win, lose or tie). Moreover, the outcomes need not be mutually exclusive (e.g., separate bets could be placed on team A winning and player X scoring). Ideally, a bookmaker strives to set prices in such a way that, no matter what the outcome of a particular event, he collects sufficient betting revenue to pay out all winning bets while also retaining some revenue as profit. However, because a bookmaker only controls the prices of bets – not the number of bets placed on different outcomes – he may lose money on a sporting event if a particular outcome occurs. Consider, for example, a sporting event with only two mutually exclusive outcomes: A and B. Suppose the bookmaker has received many more bets on outcome A than on outcome B. If outcome A occurs, then the bookmaker would lose money unless the revenue he has collected from bets placed on outcome B is sufficient to cover the payout of bets placed on outcome A. Thus, if a bookmaker finds himself in a situation in which he has collected many more bets on outcome A than on outcome B, he may raise the price of a bet placed on outcome A and/or lower the price of a bet placed on outcome B. This strategy would lower the demand for bets placed on outcome A and increase the demand for bets placed on outcome B. As a result, the bookmakers would reduce the risk that he incurs a large loss, but he would do so at the cost of sacrificing expected profits. Traditionally, bookmakers would only take bets prior to the start of a sporting event. In such cases, the probabilities of particular outcomes would remain fairly static as the bookmaker received bets. More recently, however, bookmakers have begun to take bets on sporting events as the events occur. In these cases, the probabilities of particular outcomes evolve stochastically in time as the bookmaker takes bets. This further complicates the task of a bookmaker who, in addition to

1Please see update on the American Gaming Association https://www.americangaming.org/research/state-gaming-map/.

2 considering the number of bets he has collected on particular outcomes, must also consider the dynamics of the sporting event in play. For example, scoring a point near the end of a basketball game when the score is tied would have a much larger effect on the outcome of the game than scoring a point in the first quarter. And, the effect of scoring a goal in the first half of an association football match would be greater than the effect of scoring a goal early in a game of basketball. In this paper, we provide a general framework for studying optimal bookmaking, which takes into account the situations described above. In our framework, the probabilities of outcomes are allowed to either be static or to evolve stochastically in time, and bets on any particular outcomes may arrive either at a deterministic rate or at a stochastic intensity. This rate or intensity is a decreasing function of the price the bookmaker sets and an increasing function of the probability that the outcome occurs. In this setting a bookmaker may seek to maximize either expected wealth or expected utility from wealth. We provide several examples of in-game dynamics. And we analyze specific optimization problems in which the bookmaker’s optimal strategy can be obtained explicitly. Existing literature on optimal bookmaking in the context of stochastic control appears to be rather scarce. In one of the few existing papers on the topic, Hodges et al. (2013) consider the bookmaker of a horse race. In their paper, the probability that a given horse wins the race is fixed and the number of bets placed on a given horse is a normally distributed random variable with a mean that is proportional to the probability that the horse will win and inversely proportional to the price set by the bookmaker. In this one-period setting, the bookmaker seeks to set prices in order to maximize his utility from terminal wealth. Unlike our paper, Hodges et al. (2013) do not allow the bookmaker to dynamically adjust prices as bets come in nor do they find an analytic solution for the bookmaker’s optimal control. As far as we are aware, our paper is the first to provide a general framework for studying optimal bookmaking in a dynamic setting. Similar to our work, Divos et al. (2018) consider dynamic betting during an association football match. However, rather than approach bookmaking as an optimal control problem, they use a no-arbitrage replication argument to determine the value of a bet whose payoff is a function of the goals scored by each of the two teams. As “hedging assets” they consider bets that pay the final goal tally of the two teams. By contrast, in our work, the bookmaker has no underlying assets to use as hedging instruments. Arbitrage opportunities in soccer betting markets are also studied by Grant et al. (2018). As mentioned above, we study a stochastic control problem for a bookmaker who seeks to set prices of bets optimally in order to maximize his expected profit or utility. The sports betting system in our framework is fundamentally different from a parimutuel betting system, in which all winning bets share the funds in the pool. Bayraktar and Munk (2017) study the impacts of large bettors on the house and small bettors in parimutuel wagering events. Note that Bayraktar and Munk (2017) model parimutuel wagering as a static game and assume the event has two mutually exclusive

3 results. We refer to Thaler and Ziemba (1988), Hausch and Ziemba (2011), and the references therein for further discussions on parimutuel betting. We comment that our research of optimal bookmaking, in terms of motivation, problems, and methodology, is fundamentally different from those statistical modeling works in sports betting. On the statistical modeling side, most existing papers attempt to model and then forecast the performance of a player (or a team) and the outcomes of random (sporting) events, and analyze the impact of various factors on individual or team performance in sports. Some works further apply the fitted model to construct a betting strategy to exploit opportunities, mostly from the perspective of a bettor. For instance, in an early work, Dixon and Coles (1997) use a regression model of Poisson distributions to fit the football (soccer) data in England and obtain a betting strategy that delivers a positive return. Klaassen and Magnus (2003) propose a statistical method to forecast the winner of a tennis game. Bozóki et al. (2016) apply pairwise comparison matrices to rank top tennis players. A recent paper by Song and Shi (2020) applies a gamma process to model the dynamic in-game scoring of a National Basketball Association (NBA) game. In other related areas, Müller et al. (2017) apply multilevel regression analysis to estimate the market value of professional soccer players in top European leagues. We refer to the survey article Wright (2014) and the references therein for a comprehensive review of the application of operations research and analytics in the laws and rules of various sports. In certain respects, the optimal bookmaking problem we consider is similar to the optimal market making problem considered by Avellaneda and Stoikov (2008); Guéant et al. (2012); Adrian et al. (2020) and the optimal execution problem analyzed in Gatheral and Schied (2013); Bayraktar and Ludkovski (2014); Cartea and Jaimungal (2015). In these papers, a market maker offers limit orders to buy and sell a risky asset whose reference price is a stochastic process. The intensity at which limit orders are filled is a decreasing function of how far below (above) the reference price the limit order to buy (sell) is. In this setting, a market maker seeks to maximize either his expected wealth or his expected utility from wealth, which includes both cash generated from filled limit orders and the value of any holdings in the risky asset. In some cases, the market maker also seeks to minimize his holdings in the risky asset. In a sense, a market maker offering limit orders to buy and sell a risky asset is akin to a bookmaker taking bets on mutually exclusive outcomes during a sporting event. And, a market maker seeking to minimize holdings in a risky asset is similar to a bookmaker seeking to have equal money bet on mutually exclusive outcomes. A recent paper Capponi et al. (2019) unifies the above mentioned two strands of literature. They consider a large uninformed seller who wants to optimally execute an order, accounting for the response of a group of competitive market makers, under a continuous-time Stackelberg game framework. The large seller determines his sell intensity in order to maximize his expected net proceeds, while market markers decide their optimal buy/sell amount to maximize their expected gross wealth minus a quadratic cost of run- ning inventory. On the modeling side, Capponi et al. (2019) model the clearing bid and ask prices

4 as a function of the market makers’ controls (amount offered), which then affect the end-users’ buy/sell intensity and the large seller’s sell intensity. Similarly, we model the betting intensity as a function of the bookmaker’s control (price). Both the market makers in Capponi et al. (2019) and the bookmaker in our paper want to maximize their expected (utility of) wealth, but we do not consider the competition among different bookmakers. We remark that the optimal bookmaking problem considered in this paper is also related to stochastic control problems with jump-diffusion dynamics from a methodological point of view, as both attempt to control the “jump events” in an optimal way. Zou and Cadenillas (2014) study an optimal investment and risk control problem for an insurer whose risk process is modeled by a jump-diffusion process. Capponi and Figueroa-López (2014) study optimal investment in a defaultable market with regime switching, where an additional defaultable bond with recovery is available for investment. Bo and Capponi (2016) consider a power utility maximizing agent who invests in several credit default swaps (CDSs) and a money market, where the default intensity is piece-wise constant between consecutive jumps to defaults. Bo et al. (2019) further extend the setup in Bo and Capponi (2016) by applying a diffusion process to model the default intensity between consecutive default events. Apart from the obvious different context of problem formulation, there exists a significant difference between this paper and the above-mentioned papers: the jump intensity in this paper is endogenous and depends on the control (prices of bets), but is exogenous and cannot be controlled in the above-mentioned papers. The rest of this paper proceeds as follows. In Section 2, we present a general framework for continuous-time betting markets, state the bookmaker’s optimization problem and define his value function. In Section 3, we provide a characterization of the bookmaker’s value function as the solution of a partial (integro)-differential equation (PDE). In Section 4, we study the bookmaker’s optimization problem in a semi-static setting. In this section, the probabilities of outcomes remain constant in time, but bets arrive at rates that depend on the prices set by the bookmaker. In Section 5, we study the wealth maximization problem for a risk-neutral bookmaker. In Section 6, we focus on the bookmaker’s optimization problem when his preferences are characterized by a utility function of exponential form. Lastly, in Section 7, we offer some concluding remarks. Several technical proofs and results are collected in an online appendix.

2 A General Framework for Continuous Time Betting Markets

Let us fix a probability space (Ω, F, P) and a filtration F = (Ft)0 t T completed by P-null sets, ≤ ≤ where T < is a finite time horizon. We shall suppose that all the stochastic processes and ∞ random variables introduced in this paper are well-defined and adapted under the given filtered probability space, which may be further characterized for specific examples or models in the sequel. We will think of P as the real world or physical probability measure. Consider a finite number of

5 N F F subsets (Ai)i Nn (where n := 1, 2, ,n ) of Ω, each of which is T -measurable (i.e., Ai T ∈ { · · · } ∈ for all i). We can think of Ai as a particular set of outcomes of a sporting event, which ﬁnishes at P time T . To avoid trivial cases, we suppose (Ai) (0, 1) for all i. Note that the sets (Ai)i Nn need ∈ ∈ not be a partition of Ω. In particular, there may be outcomes ω such that ω / n A . Moreover, ∈∪i=1 i sets may overlap (i.e., we may have A A = where i = j). i ∩ j 6 ∅ 6 i i F We will denote by P = (Pt )0 t T the t-conditional probability of Ai. We have ≤ ≤ i E 1 Pt = t Ai , where E := E( F ) denotes conditional expectation. Note that, by the tower property of condi- t · · | t tional expectation, P i is a (P, F)-martingale. We will denote by P = (P 1, P 2, , P n) the vector · · · of conditional probabilities. In general, the conditional probability P i is a stochastic process. How- i ever, we will also consider scenarios where it is reasonable to assume that Pt is a ﬁxed constant pi for all t < T . Let us take a look at some examples of sporting events and show how we can describe them probabilistically.

Example 2.1. Suppose the number of goals scored by player X in an association football match is a Poisson process with intensity µ. Consider an in-game bet on a set of outcomes of the form Ai = “player X will ﬁnish the match with exactly i goals.” We can model the conditional probability that

Ai occurs as follows. Fix a probability space (Ω, F, P) and let F = (Ft)0 t T be the augmented ≤ ≤ µ µ ﬁltration generated by the Poisson process N = (Nt )0 t T with intensity µ. Then we have µ µ ≤ ≤ i i P 1 µ i 1 µ(T t) i n 1 µ Pt = n=0 (NT Nt = i n) N =n = n=0 (i n)! e− − [µ(T t)] − N =n , from which − − { t } − − { t } we canP easily construct the conditional probabilitiesP of outcomes of the form “player X will ﬁnish the match with between i and j goals (inclusive).” For example, if Aj = “player X will score at j 1 µ 1 µ µ(T t) j least one goal” then we have Pt = N 1 + N =0 (1 e− − ). The dynamics of P can be { t ≥ } { t } − j 1 µ(T t) µ j µT easily deduced as dPt = P j <1 e− − (dNt µdt), with P0 = 1 e− . Observe that the { t− } − − above P j is a martingale, as it must be.

Example 2.2. Consider an National Basketball Association (NBA) game. Although points in NBA games are integer-valued, as the number of points in an NBA game is on the order of 100, it is reasonable to approximate the number of points scored as a R-valued process. To this end, let us consider a probability space (Ω, F, P) equipped with a ﬁltration F = (Ft)0 t T for a Brownian ≤ ≤ motion W = (Wt)0 t T . We can model the point diﬀerential ∆ = (∆t)0 t T between team A and ≤ ≤ ≤ ≤ team B as a Brownian motion with drift ∆t = µt + σWt where the sign and size of µ captures how much team A is favored by. Now, consider a bet on a set of outcomes of the form Ai = “team A will win by i points or more.” Then we have

i i ∆t µ(T t) ∆t + µ(T t) i P = P(∆T i ∆t) = 1 Φ − − − =Φ − − , (2.1) t ≥ | − σ√T t σ√T t − −

6 where Φ is the cumulative distribution function (c.d.f.) of a standard normal random variable. We can easily obtain the dynamics of P i using Itô’s Lemma by dP i = Φ (Φ 1(P i))/√T t dW . t t ′ − t − t Observe that P i is a martingale, as it must be.

Having seen a few examples of how we can describe sporting events probabilistically, let us now focus on the payoﬀ structure of bets. Throughout this paper, we will assume that a bet placed on a set of outcomes A pays one unit of currency at time T if and only if ω A (i.e., if A occurs). i ∈ i i Thus, we have

1 i “payoﬀ of a bet placed on Ai”= Ai = PT . (2.2)

Remark 2.3. In the US, if a bookmaker quotes odds of +120 on outcome A, then a $100 bet on outcome A would pay $120 + $100 = $220 if A occurs. If a bookmaker quotes odds of -110 on outcome B, then a $110 bet on outcome B would pay $100 + $110 = $210 if B occurs. Using our setup, the price of a bet quoted at +120 is 100/220 0.4545 and the price of a bet quoted at ≈ 110 is 110/210 0.5238. In general, a quote of +x (x > 100) is equivalent to setting the price − 100 ≈ to x+100 < 0.5 (underdogs in sports betting) and a quote of x (x> 100) is equivalent to setting x − the price to x+100 > 0.5 (favorites in sports betting). The payoﬀ in (2.2) is simply a convenient normalization, which can be applied to any bet that pays a ﬁxed (i.e., non-random) amount if a particular outcome or set of outcomes occurs.

In our framework, the bookmaker cannot control the number of bets that the public places on a set of outcomes Ai directly. However, the bookmaker can set the price of a bet placed on Ai and this will aﬀect the rate or intensity at which bets on Ai are placed. Such a setup (i.e., controlling the price to inﬂuence demand) is similar to the setup used in operations research literature to study the problem of selling seasonal and style goods in various industries; see, e.g., Pashigian (1988) and i i Gallego and Van Ryzin (1994). We will denote by u = (ut)0 t

A F A n (t, T ) := u = (us)s [t,T ) : u is progressively measure w.r.t. and us := [0, 1] , (2.3) { ∈ ∈ } with t [0, T ). Note that we do not include the control u in the deﬁnition of A(t, T ), as the case ∈ T at time T is trivial. Without loss of generality, we set ui = 1 for all i N . T Ai ∈ n u u Let us denote by X = (Xt )0 t T the total revenue generated by the bookmaker and by ≤ ≤ u,i u,i Q = (Qt )0 t T the total number of bets placed on a set of outcomes Ai. Note that we have ≤ ≤ indicated with a superscript the dependencies of Xu and Qu on bookmaker’s pricing policy u. The u u u n i u,i u u,i relationship between X , Q and u is dXt = i=1 ut dQt . Observe that X and Q for all P 7 i N are non-decreasing processes. Typically, we will have Xu X = 0 and Qu,i Qi = 0 for ∈ n 0 ≡ 0 0 ≡ 0 all i N . However, we do not require this (to account for the fact that the bookmaker may have ∈ n taken the bets before time 0). u,i In this paper, we consider two models for Q . In one model, bets on a set of outcomes Ai arrive at a rate per unit time, which is a function λ : A A R¯ of the vectors of conditional probabilities i × → + P of outcomes and prices u set by the bookmaker. Here, A = [0, 1]n and R¯ = R + . Under + + ∪ { ∞} this model, we have

t u,i i Continuous arrivals : Qt = λi(Ps, us)ds + Q0. (2.4) Z0

In another model, bets on a set of outcomes Ai arrive as a state-dependent Poisson process u,i u,i A A R¯ N = (Nt )0 t T whose instantaneous intensity is a function λi : + of the vec- ≤ ≤ × → tors of conditional probabilities P of outcomes and prices u set by the bookmaker. Under this model, we have

t u,i u,i i E u,i Poisson arrivals : Qt = dNt + Q0, tdNt = λi(Pt, ut)dt. (2.5) Z0

Throughout the paper, we will refer to the function λi as the rate function when bets arrive continuously and the intensity function when bets arrive as a state-dependent Poisson process. To summarize, we model the bet arrivals Qu,i as a controlled deterministic process in (2.4) or a controlled Poisson process in (2.5). Such a modeling is similar to that of demand process in the production-pricing literature; see Li (1988) and Gallego and Van Ryzin (1994) for details, and the survey article Elmaghraby and Keskinocak (2003) and the references therein for general discussions.

Remark 2.4. For sporting events with a large betting interest (e.g., the Super Bowl, the UEFA Champions League ﬁnal, etc.), the continuous arrivals model given by (2.4) is suﬃcient to capture the dynamics of bet arrivals. However, for sporting events with limited betting interest (e.g., curling in the Winter Olympics, the Westminster Dog Show, etc.), the dynamics of bet arrivals are better captured by the Poisson arrivals model in (2.5).

Although our framework is suﬃciently general to allow for the rate/intensity function λi to depend on the entire vector of conditional probabilities P and vector of prices u, the case in which i i λi depends only on the conditional probability P and price u will be of special interest. In cases for which λ (P , u ) λ (P i, ui), where i N and t [0, T ], we expect λ to be an increasing i t t ≡ i t t ∈ n ∈ i i i function of the conditional probability P of outcome Ai and a decreasing function of the price u set by the bookmaker. Examples of rate/intensity functions λ : [0, 1] [0, 1] R¯ that satisfy i × → + those analytical properties include

p 1 u λ (p , u ) := i − i , (2.6) i i i 1 p u − i i 8 log ui λi(pi, ui) := . (2.7) log pi

i The functions (2.6) and (2.7) have reasonable qualitative behavior because (i) as the price ut of a bet on outcome Ai goes to zero, the intensity of bets goes to inﬁnity

lim λi(pi, ui)= , (2.8) ui 0 → ∞

i (ii) as the price ut of a bet on outcome Ai goes to one, the intensity of bets goes to zero

lim λi(pi, ui) = 0, (2.9) ui 1 →

i i and (iii) all fair bets ut = Pt have the same intensity λi(pi,pi)= λi(qi,qi). Another rate/intensity function λi we shall consider in this paper is

β(ui pi) λi(pi, ui) := κe− − , κ,β> 0. (2.10)

As we shall see, the form of λi in (2.10) facilitates analytic computation of optimal pricing strategies. The exponential form (2.10) of the intensity function is also used in Gallego and Van Ryzin

(1994)[Section 2.3] to obtain optimal pricing strategies in closed-form. Note, however, that λi in (2.10) does not satisfy (2.8) or (2.9). We mention that we can also multiple the intensity functions in (2.6) and (2.7) by a positive scaling factor κ as in (2.10). u Let us denote by YT the total wealth of the bookmaker just after paying out all winning bets, assuming he follows pricing policy u. Then we have

n n n T u u i u,i u i i i i u,i YT = XT PT QT = X0 PT Q0 + ut PT dQt , (2.11) − − 0 − Xi=1 Xi=1 Xi=1 Z u,i where Qt is given by either (2.4) or (2.5). We will denote by J the bookmaker’s objective function. We shall assume J is of the form

E u J(t,x,p,q; u)= t,x,p,qU(YT ), where U : R R is either the identity function or a utility function. Here, we have introduced the → notation E = E( Xu = x, P = p,Qu = q). t,x,p,q · · | t t t The bookmaker seeks an optimal control or pricing policy u∗ to the problem:

V (t,x,p,q):= sup J(t,x,p,q; u), (2.12) u A(t,T ) ∈ where the admissible set A(t, T ) is deﬁned by (2.3). We shall refer to the function V as the book-

9 maker’s value function. When we take U to be the identity function, the bookmaker is risk neutral and his objective is to maximize the expected terminal wealth; see, e.g., Gallego and Van Ryzin (1994) for the same criterion. When we take U to be an increasing and concave utility function, the bookmaker is risk-averse and his objective is to maximize the expected utility of terminal wealth.

Remark 2.5. As pointed out by an anonymous referee, the bookmakers and bettors often have different predictions about the probability of an event. In our framework, the (conditional) probabilities P are the bookmaker’s best predictions on betting outcomes. Note that we indirectly take into account the fact that bettors may have different views on the probabilities of outcomes and may have different objective functions. These are all captured in a reduced form through λi.

3 PDE Characterization of the Value Function

M Let us denote by the inﬁnitesimal generator of the process P , and by ∂t, ∂x and ∂qi the partial derivative operators with respect to the corresponding arguments. For any u A, deﬁne operator ∈ Lu by either of the following

n Lu M := λi(p, u)(ui∂x + ∂qi )+ , (3.1) i=1 Xn Lu := λ (p, u)(θx θqi 1) + M, (3.2) i ui 1 − Xi=1

qi qi where θz is a shift operator of size z in the variable qi, that is, θz f(q) := f(q1,...,qi + z,...,qn). u Suppose the bookmaker were to fix the prices of bets at a constant ut = u. Then L as defined in (3.1) is the generator of (Xu,P,Qu) assuming the dynamics of Qu are described by the continuous arrivals model (2.4), and Lu as defined in (3.2) is the generator of (Xu,P,Qu) assuming the dynamics of Qu are described by the Poisson arrivals model (2.5). As is standard in stochastic control theory, we provide a verification theorem to Problem (2.12) in a general setting. We refer the reader to Fleming and Soner (2006)[Section III.8] and Yong and Zhou (1999)[Section 4.3] for proofs.

Theorem 3.1. Let v : [0, T ] R A Rn R be a real-valued function which is at least once × + × × + → diﬀerentiable with respect to all arguments and satisﬁes

∂ v> 0 and ∂ v< 0, i N . x qi ∀ ∈ n

Suppose the function v satisﬁes the Hamilton-Jacobi-Bellman (HJB) equation

n Luˆ E 1 u u ∂tv + sup v = 0, v(T,x,p,q)= U x qi Ai XT = x, PT = p,QT = q , (3.3) uˆ A − − − − ∈ h i=1 i X

10 where Luˆ is given by either (3.1) or (3.2). Then v(t,x,p,q) = V (t,x,p,q) is the value function to

Problem (2.12) and the optimal price process u∗ = (us∗)s [t,T ] is given by ∈ Luˆ us∗ = argmax v(s, Xs∗, Ps,Qs∗). (3.4) uˆ A ∈ Remark 3.2. The PDE characterization in (3.3) to the value function and the optimal price process given by (3.4) are obtained without any assumptions on function U, the arrival rate/intensity function λi, or the conditional probabilities P . With additional assumptions, we may simplify (3.3) and (3.4) to more tractable forms. For instance, when Qu is deﬁned by the continuous arrivals model (2.4), we can simplify (3.4) and obtain for all i N that ∈ n

n n j, λ (P , u∗)+ u ∗ ∂ λ (P , u∗) ∂ V + ∂ λ (P , u∗) ∂ V = 0. (3.5)  i s s s · ui j s s  x ui j s s · qj jX=1 jX=1   Equation (3.5) reduces the problem of ﬁnding the optimal price process u∗ in feedback form to solving a system of n equations. We shall apply the results of Theorem 3.1 to obtain the value function and/or the optimal price process in closed forms in the subsequent sections.

4 Analysis of the Semi-static Setting

In this section, we solve the main problem (2.12) in a semi-static setting. The standing assumptions of this section are as follows.

Assumption 4.1. The arrivals process Qu,i is given by the continuous arrivals model (2.4). The vector of conditional probabilities is a vector of constants, namely, P p (0, 1)n for all t [0, T ). t ≡ ∈ ∈ i The utility function U is continuous and strictly increasing. The rate function λi = λi(u ) is a continuous and decreasing function of ui for all i N . ∈ n i For notational simplicity, we write the rate function as λi(ut) in the rest of this section because the conditional probabilities P p are ﬁxed constants. Under Assumption 4.1, we have from (2.11) t ≡ u that the bookmaker’s terminal wealth YT is given by

n n T n T u i 1 i i i 1 YT = Xt Qt Ai + λi(us) us ds λi(us)ds Ai . (4.1) − · t − t · Xi=1 Xi=1 Z Xi=1 Z 1 As λi is decreasing by Assumption 4.1, its inverse λi− exists. Deﬁning the function fi by

1 f (x) := x λ− (x), x> 0, i N , (4.2) i · i ∈ n

11 u we are able to rewrite YT in (4.1) as

n n T n T u i 1 i i 1 YT = Xt Qt Ai + fi(λi(us)) ds λi(us)ds Ai . − · t − t · Xi=1 Xi=1 Z Xi=1 Z ˆ ˆ u Denote by fi the concave envelope of fi in (4.2) and deﬁne YT by

n n T n T ˆ u i 1 ˆ i i 1 YT := Xt Qt Ai + fi(λi(us)) ds λi(us)ds Ai . (4.3) − · t − t · Xi=1 Xi=1 Z Xi=1 Z It is obvious that Yˆ u Y u for any pricing policy u. We are now ready to present the main results T ≥ T of this section.

Theorem 4.2. Let Assumption 4.1 hold, we have

n n n E 1 ˆ 1 ˆ V (t,x,p,q) = sup t,x,p,qU x qi Ai + (T t) fi(λi(ˆui)) (T t) λi(ˆui) Ai := V , uˆ A − − − − ! ∈ Xi=1 Xi=1 Xi=1 (4.4) n n n E 1 ˆ 1 = sup t,x,p,qU x qi Ai + (T t) fi(Λi) (T t) Λi Ai , Λ Dλ − − − − ! ∈ Xi=1 Xi=1 Xi=1 where A = [0, 1]n and D = (D , D , , D ) with D being the range of λ for all i N . λ λ1 λ2 · · · λn λi i ∈ n Remark 4.3. The results in Theorem 4.2 help us reduce the original problem, which is a dynamic optimization problem over an n-dimensional stochastic process, into a static optimization problem over an n-dimensional constant vector.

The proof of Theorem 4.2 relies on the following two lemmas.

Lemma 4.4. Let Assumption 4.1 hold, we have

E ˆ uˆ E ˆ u sup t,x,p,qU YT = sup t,x,p,qU YT , uˆ A u A(t,T ) ∈ ∈ ˆ u A where YT is deﬁned by (4.3) and set (t, T ) is deﬁned by (2.3).

Proof. As the “ ” is obvious, we proceed to show the converse inequality is also true. Let u be ≤ any pricing policy adopted by the bookmaker. As λi is continuous, there exists a constant ui∗ such 1 T i ˆ that λi(ui∗)= T t t λi(us)ds. By the concavity of fi, we obtain − R T T 1 ˆ i ˆ 1 i ˆ fi(λi(us)) ds fi λi(us)ds = fi(λi(ui∗)). T t t ≤ T t t − Z − Z !

12 As a result, we have

n n n E ˆ u E i 1 ˆ 1 t,x,p,qU(YT ) t,x,p,qU Xt Qt Ai + (T t) fi(λi(ui∗)) (T t) λi(ui∗) Ai ≤ − · − − − ! Xi=1 Xi=1 Xi=1 E uˆ sup t,x,p,qU(YT ). ≤ uˆ A ∈ Then the result follows from the arbitrariness of u.

Lemma 4.5. Let Assumption 4.1 hold, we have

E ˆ uˆ V (t,x,p,q) sup t,x,p,qU YT . ≥ uˆ A ∈ ε ε ε ˆ Proof. For any ε> 0, let u = (u1,... ,un) be an ε/2-optimizer for V in (4.4). Let δ > 0 be small ε ε enough so that E U(Yˆ u δ) > E U(Yˆ u ) ε/2 > Vˆ ε. Next choose v , w , ρ [0, 1] t,x,p,q T − t,x,p,q T − − i i i ∈ for each i N so that the following conditions are met: ρ λ (v ) + (1 ρ ) λ (w )= λ (uε) and ∈ n i · i i − i · i i i i ˆ ε δ ρi fi(λi(vi)) + (1 ρi) fi(λi(wi)) > fi(λi(ui )) n(T t) . We construct a pricing strategyu ˜ by · − · − − i 1 1 u˜s = vi t s YT δ, which in turn implies V (t,x,p,q) t,x,p,qU YT ε − ≥ ≥ E U Yˆ u δ > Vˆ ε. Hence, the desired result follows. t,x,p,q T − − Proof of Theorem 4.2. The result follows from Lemmas 4.4 and 4.5 and the fact that f fˆ. i ≤ i Corollary 4.6. If uˆ = (ˆu , uˆ , , uˆ ) is an optimizer for Vˆ and ∗ 1∗ 2∗ · · · n∗

f (λ (û∗)) = fˆ(λ (û∗)), i N , (4.6) i i i i i i ∈ n then uˆ∗ is an optimizer for V . That is, a static strategy u = (us)s [t,T ], where us uˆ∗, is an optimal ∈ ≡ strategy to Problem (2.12). In general, the pricing policy u˜ defined in (4.5) is an ε-optimizer for V .

Remark 4.7. If fi is concave, (4.6) is satisﬁed. For example, this is the case if λi is given by (2.6).

However, if λi is given by (2.7), fi in general is not a concave function. In fact, we have

2 2 fi′′(x) < 0 if 0 0 if x> . −log pi −log pi

As mentioned in Remark 4.3, Theorem 4.2 allows us to reduce the target optimization problem into a much simpler static optimization one, but the general characterization (either explicit or numerical solutions) of the optimal constant vectoru ˆ∗ to Vˆ is still not available. Below we provide an example where the optimizeru ˆ∗ is given by a system of equations. In the example, we choose

13 the rate function to be given by (2.6), and, by Corollary 4.6 and Remark 4.7, the static strategy u uˆ is an optimal strategy to the bookmaker’s problem. ≡ ∗

Corollary 4.8. Let Assumption 4.1 hold. Assume further that (i) the sets (Ai)i Nn form a partition ∈ of Ω, (ii) the rate function λ is given by (2.6), and (iii) U is given by U(y)= e γy, where γ > 0 i − − Then the optimizer uˆ∗ of Vˆ , deﬁned in (4.4), solves the following equation

1 p 1 g(t,p ,q ;u ˆ∗)= p g(t,p ,q ;u ˆ∗), i N , (4.7) i (ˆu )2 i i i j j j j n · i∗ − ! · j=i · ∀ ∈ X6 where function g is deﬁned by

p 1 g(t,p ,q ; u ) := exp γq + γ(T t) i 1 , i i i i − 1 p u − − i i for all t [0, T ), p (0, 1), q R , and u (0, 1). Moreover, if we replace assumption (i) with ∈ i ∈ i ∈ + i ∈ (i’) the sets (Ai)i Nn are independent, then uˆ∗ solves the following equation ∈ 1 p 1 p 1 exp γ(T t) i 1 = 1 p , i N . (4.8) i · (ˆu )2 − · − 1 p uˆ − − i ∀ ∈ n i∗ ! − i i∗ !! Proof. Equations (4.7) and (4.8) are the ﬁrst-order conditions of the corresponding optimization problems under the given assumptions.

4.1 Case Studies

In this subsection, we apply the results from Corollary 4.8 to study two cases, which correspond to the two diﬀerent conditions (i) and (i’). 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 1: A plot ofu ˆ∗ given by (4.8) as a function of p with γ = 2 and T t =1 −

First, we study the case of independent sets (Ai)i Nn in Corollary 4.8. According to (4.8), for ∈ ﬁxed γ and T t, the optimal priceu ˆ of a bet on set A is fully determined by the set probability − ∗ i p . In Figure 1, we plotu ˆ as a function of p, given γ = 2 and T t = 1. As expected, the i ∗ − optimal priceu ˆ∗ is an increasing function of the outcome probability. In addition,u ˆ∗ is a concave

14 function of p, indicating that the bookmaker’s optimal price is more sensitive to the changes of small probabilities. The economic explanation is that, the increment of bets on an event is more signiﬁcant when its probability increases from 0.1 to 0.2 than that from 0.8 to 0.9. In other words, the changes at small probabilities cause more imbalance in the bookmaker’s book than the same amount of changes but at large probabilities. Therefore, to counter such an eﬀect, the bookmaker increases the price at a faster scale at small probabilities.

0.8 0.8

0.6 0.6 * 1 * 2

0.4 0.4

Optimal Price u 0.2 Optimal Price u 0.2

0 0 0.5 0.5 0.4 0.5 0.4 0.5 0.3 0.4 0.3 0.4 0.2 0.3 0.2 0.3 0.2 0.2 0.1 0.1 0.1 Probability p Probability p 0.1 2 0 0 Probability p 2 0 Probability p 1 0 1 ∗ ∗ Figure 2: Surface plot ofu ˆ1 andu ˆ2 with γ = 2, T t = 5 and q = (0, 0, 0) −

Next we analyze the results of Corollary 4.8 when the sets (Ai)i Nn form a partition of Ω. By ∈ Theorem 4.2, the optimizeru ˆ∗ solves a static problem given in (4.4) and is then an n-dimensional scalar vector independent of time t. It seems from (4.7) thatu ˆ∗ may depend on t through function g. However, noting the deﬁnition of λi in (2.6), we have

∗ ∗ uˆ∗,i uˆ ,i pi 1 uˆ ,i uˆ∗,i Q Q = 1 (s t) and g(t,p ,Q ;u ˆ∗) g(s,p ,Q ;u ˆ∗), s − t 1 p uˆ − − i t i ≡ i s i − i i∗ ! which in turn showsu ˆ∗ is indeed independent of time t. To solve (4.7), we consider an example with three mutually exclusive outcomes A , A and A . We consider p ,p (0, 0.5) and solve 1 2 3 1 2 ∈ (4.7) numerically given γ = 2, T t = 5 and q = (0, 0, 0). The surface plots of the optimal prices − uˆ1∗ andu ˆ2∗ are given in Figure 2. For fixed p2 (resp. p1), the optimal priceu ˆ1∗ (resp.u ˆ2∗) is a strictly increasing and concave function of the outcome probability p1 (resp. p2). From (4.7), with all ∗ other parameters fixed, we obtain ∂uî > 0. To visualize this result, we fix the model parameters as ∂qi

15 Optimal Prices Optimal Terminal Wealth in the Worst Scenario 1 0.4

0.9 0.38

0.8 u* 0.36 1 u* 2 0.7 0.34 u* 3

0.6 0.32

0.5 0.3

0.4 0.28

0.3 0.26

0.2 0.24

0.1 0.22 0 0.5 1 1.5 2 0 0.5 1 1.5 2

∗ ∗ Figure 3: Left panel plots the optimal prices ui (=u ˆi ), i = 1, 2, 3. Right panel plots the bookmaker’s terminal wealth after paying out the winning bets under the optimal price strategy u∗ in the worst scenario. In both panels, the x-axis is the current number of bets q1 accumulated on outcome A1, and the remaining model parameters are set as in (4.9) with the initial wealth x = 0. follows

1 1 1 p = , p = , p = , q = q = 0, γ = 2, T t = 1, (4.9) 1 2 2 3 3 6 2 3 − but allow the current accumulated bets on outcome A1 to vary from 0 to 2. Recall that qi is the current accumulated bets on outcome Ai by the initial time. We then plot the optimal pricesu î∗, where i = 1, 2, 3, in the left panel of Figure 3. It is readily seen that the optimal priceu ˆ1∗ on outcome A1 is an increasing function of its accumulated number of bets q1. Such an increasing relation is consistent with our intuition that, if the bookmaker has received too many (few) bets on event A1, the bookmaker should increase (decrease) the price on A1 to balance the books. In fact, Figure 3 also reveals that the bookmaker should simultaneously decrease the prices on mutually exclusive outcomes A2 and A3, which further helps maintain a balanced book. The monotonic relation ofu î∗ w.r.t. probability pi in Figure 3 also confirms our findings in Figures 1 and 2. In the right panel of Figure 3, we plot the bookmaker’s terminal wealth (profit) after paying out the ∗ ∗ ∗ winning bets in the worst scenario (i.e., min Y uˆ = Xuˆ max Quˆ ,i ) against the inventory T T − i=1,2,3 { T } q1 on outcome A1, with the initial wealth setting to 0. Here, since outcome A1 has the largest probability among the three outcomes, it also has the largest total number of bets by the terminal time T . In consequence, the worst scenario to the bookmaker is when outcome A1 happens at time ∗ ∗ ∗ T , i.e., min Y uˆ = Xuˆ Quˆ ,1. As shown in the right panel of Figure 3, the bookmaker will pocket T T − T positive profits from the bettors in regardless of the final result of the gamble.

16 5 Wealth Maximization

In this section, we consider Problem (2.12) when U(y)= y (i.e., the objective of the bookmaker is to maximize his expected terminal wealth). Henceforth, we shall refer to this problem as the wealth maximization problem. We solve the wealth maximization problem using three diﬀerent methods, and obtain the solutions to Problem (2.12) in Theorem 5.1 and Corollaries 5.3, 5.5 and 5.6.

5.1 Method I

The following theorem transforms the bookmaker’s dynamic optimization problem into a static optimization problem.

Theorem 5.1. Assume U(y)= y, and the bet arrival process Qu,i is given by either (2.4) or (2.5) for all i N . Then we have ∈ n T n E i V (t,x,p,q)= x p q + t,x,p,q sup λi(Ps, uˆ) (ˆui Ps)ds, (5.1) − · t uˆ A − Z ∈ Xi=1 n where p q = piqi. · i=1 P Proof. As Qu,i is given by either (2.4) or (2.5), we have

E u,i E 1 i tdQt = λi(Pt, ut)dt, t Ai = Pt , and, as a result,

n T E i u,i u,i1 V (t,x,p,q)= x p q + sup t,x,p,q us dQs QT Ai − · u A(t,T ) t − ! ∈ Xi=1 Z n T E i i = x p q + sup t,x,p,q λi(Ps, us) (us Ps)ds − · u A(t,T ) t − ! ∈ Xi=1 Z T n E i = x p q + t,x,p,q sup λi(Ps, uˆ) (ˆui Ps )ds, − · t uˆ A − Z ∈ Xi=1 where the last equality can be shown by a measurable selection argument (see Wagner (1977)).

Remark 5.2. Due to the assumption of U(y) = y, the value function obtained in (5.1) depends linearly on both the current wealth x and the current books q. To be precise, we have

∂V (t,x,p,q) ∂V (t,x,p,q) = 1 > 0 and = pi < 0, i Nn, ∂x ∂qi − ∀ ∈ which verify the derivatives conditions in Theorem 3.1.

17 ε ε ε,1 ε,n For any ε> 0, u = (us(p))t s

In light of Theorem 5.1, we are able to reduce the complexity of the wealth maximization problem signiﬁcantly. As mentioned above, Theorem 5.1 allows us to transform a dynamic optimization problem (over u A(t, T )) into a static optimization problem (overu ˆ A), and shows that the ∈ ∈ value function are the same for these two problems. Under the general setup, the characterization in (5.1) is not enough for us to obtain the optimal price process u∗ explicitly. However, when the rate or intensity function λi is given by (2.6) or (2.7), we are able to ﬁnd u∗ in closed forms; see the corollary below.

Corollary 5.3. Assume U(y)= y, and the arrival process Qu,i is given by either (2.4) or (2.5) for all i N . ∈ n

(i) If the rate or intensity function λi is given by (2.6), then the optimal price process to the 1, 2, n, i, wealth maximization problem is u∗ = (ut ∗, ut ∗, , ut ∗)t [0,T ), where ut ∗ is given by · · · ∈

i, i u ∗ = P , i N . (5.2) t t ∀ ∈ n q If we assume further that P p (0, 1)n for all t [0, T ), then the optimal price process u t ≡ ∈ ∈ ∗ is

i, u ∗ √p , i N , (5.3) t ≡ i ∀ ∈ n

and the value function V is given by

n pi 2 V (t,x,p,q)= x p q + (T t) (1 √pi) . (5.4) − · − 1 pi − Xi=1 −

(ii) If the rate or intensity function λi is given by (2.7), then the optimal price process to the wealth 1, 2, n, i, maximization problem is u∗ = (ut ∗, ut ∗, , ut ∗)t [0,T ), where ut ∗ is the unique solution on ∈ 1 · · · (e− , 1) to the equation

r(1 + log r)= P i, i N . (5.5) t ∀ ∈ n

i, 1 uî i ∗ Proof. According to Theorem 5.1, we have ut = argmaxuî [0,1] −uˆ uî Pt , when λi is given ∈ i − i, i by (2.6); and ut ∗ = argmaxuî [0,1] log(ûi)(ûi Pt ), when λi is given by (2.7). The rest follows ∈ −

18 naturally.

If λi is given by (2.6) and Pt p, we obtain the value function in (5.4), from which we easily ∂V ∂V ≡ see = 1 and = pi as in Remark 5.2. In addition, we have ∂x ∂qi − n ∂V pi 2 ∂V pi + √pi 1 N = (1 √pi) < 0, = qi −2 , i n. ∂t − 1 pi − ∂pi − − 1+ √pi ∀ ∈ Xi=1 − ∂V Due to ∂t < 0, we conclude that, all things being equal, it is always a better decision to start pi+√pi 1 accepting bets earlier. Straightforward calculus shows that ϕ(pi) := −2 is an increasing (1+√pi) function of p and takes values in ( 1, 1/4). When p > ((√5 1)/2)2 38% or when q > 1, we i − i − ≈ i have ∂V < 0. ∂pi 1 i, i Let U : (0, 1) (e , 1) be the solution function to (5.5), namely, u ∗ = U(P ). We easily → − t t verify that

∂U(z) 1 1 ∂2U(z) 1 0 < = , 1 and = − < 0. U 2 U U 3 ∂z 2 + log (z) ∈ 2 ∂z 2 + log (z) i, Thus, the function U is increasing and concave. As such, the optimal price u ∗ on set of outcomes i Ai increases when the conditional probability P increases and the rate of increase is larger when P i is small.

5.2 Method II (Dynamic Programming Method)

Now, we use the PDE characterization of the bookmaker’s value function (Theorem 3.1) to solve the wealth maximization problem under the continuous arrivals model (2.4). We begin the analysis by establishing some analytical properties for the value function V .

Proposition 5.4. Let U(y) = y for all y R. If the value function V (t,x,p,q) is diﬀerentiable ∈ with respect to t, x and q, we have

∂ V (t,x,p,q) < 0, ∂ V (t,x,p,q) = 1, and ∂ V (t,x,p,q)= p , i N . t x qi − i ∀ ∈ n

Proof. Recalling the deﬁnition of Y u in (2.11), for any price process u A(t, T ), we obtain T ∈ n T n n T E u E i u,i i i u,i t,x,p,qYT = t,x,p,q x + usdQs PT qi PT dQs " t − − t # Xi=1 Z Xi=1 Xi=1 Z n n T E i i i i = x piqi + t,x,p,q (us Ps )λ(Ps, us)ds. − t − Xi=1 Xi=1 Z

19 It is clear that, for the optimal price process u , we have ui, P i > 0 for all s [t, T ] and i N . ∗ s ∗ − s ∈ ∈ n The desired results are then obvious.

We now present the explicit solutions to the wealth maximization problem.

Corollary 5.5. Assume U(y) = y and the bets arrive according to the continuous arrivals model

(2.4). (i) If the rate function λi is given by (2.6), then the optimal price process u∗ is given by

(5.2). (ii) If the rate function λi is given by (2.7), then the optimal price process u∗ is the solution to (5.5)

Proof. (i) Under the continuous arrivals model (2.4) with rate function (2.6), we derive from (3.4) 2 that ∂ V ui, − ∂ V = 0, i N . Together with the results from Proposition 5.4, we obtain, x − s ∗ · qi ∀ ∈ n for all s [t, T ), that ui, = ∂ V/∂ V P i (0, 1), i N . (ii) Under the continuous ∈ s ∗ − qi x ≡ s ∈ ∀ ∈ n q q i, i, arrivals model (2.4) with rate function (2.7), we derive from (3.4) that (1+log ut ∗) ∂xV +∂qi V/ut ∗ = 0, i N . The desired results are then obtained. ∀ ∈ n

5.3 Method III

Lastly, we use the results from Theorem 4.2 to solve the wealth maximization problem.

Corollary 5.6. Let Assumption 4.1 hold and suppose U(y) = y. If an interior optimizer Λ∗ in Theorem 4.2 exists, then we have

fˆ′(Λ∗)= p , i N . i i i ∀ ∈ n

In particular, if λi is given by (2.6), we obtain

pi 1 N Λi∗ = 1 and uˆi∗ = √pi, i n 1 pi √pi − ∀ ∈ − ! and if λi is given by (2.7), then

∗ Λi p (1+Λ∗ log p )= p and uˆ∗(1 + logu ˆ∗)= p , i N . i i i i i i i ∀ ∈ n

Proof. The ﬁrst general result is immediate thanks to Theorem 4.2 and the assumption of U(y)= y.

As pointed out in Remark 4.7, if λi is given by (2.6), we have fi = fˆi. If λi is given by (2.7), we 1 calculate fˆ′(x)= f ′(x) > 0 for 0

5.4 Comparison of the Three Methods

In terms of the model generality, Theorem 5.1 obtained via Method I is the most general one, since (i) the conditional probabilities P can be modeled by any stochastic process taking values

20 in (0,1) and (ii) the arrival process is given by either the continuous arrival model (2.4) or the Poisson arrivals model (2.5). Corollary 5.5 of Method II holds when the arrival process is given by the continuous arrival model (2.4). Corollary 5.6 of Method III is further restricted to P being constants. In terms of application scope, Method I is the most restricted one, as it only applies to the wealth maximization problem. Both Methods II and III are developed to solve the main problem for a general function U, and hence can be applied to solve concave utility maximization problem (see for instance Corollary 4.8). Both Theorem 5.1 of Method I and Theorem 4.2 of Method III provide an explicit characterization to the value function, but do not provide an explicit solution to the optimal price process. Method II (dynamic programming method) provides characterizations to both the value function and the optimal price process. However, solving the HJB equation (3.3) in Method II is often a challenging task. From a computational point of view, the static optimization problem in Method I (see (5.1)) or in Method III (see (4.4)) is easy to solve, while ﬁnding the numerical solutions to the HJB equation (3.3) under the feedback strategy (3.4) may be diﬃcult computationally.

5.5 Numerical Analysis of a Coin Example

In this subsection, we consider a simple coin tossing example, which has two mutually exclusive 1 outcomes A1 = Heads and A2 = Tails . Suppose Pt pˆ (0, 1) for all t [0, T ), and then { } { } ≡ ∈ ∈ ∗ P 2 1 pˆ. We carry out the analysis based on the two models of the arrival process Qu ,i – the t ≡ − continuous arrivals model (2.4) and the Poisson arrivals model (2.5). The rate or intensity function

λi is given by (2.6). In addition, we shall assume that the bookmaker has taken zero bets at time t = 0 and has zero initial wealth:

P = (ˆp, 1 pˆ), t

Under these conditions, the bookmaker’s optimal policy u∗ is given by (5.3). In what follows, we analyze the probability that the bookmaker makes proﬁts when he follows the optimal price process, P denoted by (YT∗ > 0).

Case 1: Continuous arrivals model (2.4) with λi given by (2.6). In this case, the bookmaker’s terminal wealth YT∗ is given by

Y ∗(Heads) = ψ (ˆp) T and Y ∗(Tails) = ψ (1 pˆ) T, T 1 · T 1 − ·

21 where function ψ1 is deﬁned by

pˆ 1 1 pˆ ψ (ˆp) := 2 pˆ + − (1 1 pˆ), pˆ (0, 1). 1 √ 1 pˆ − − pˆ pˆ − − ∈ − p p 1 It can be shown that ψ1 is a decreasing function over (0, 1) with limpˆ 0 ψ1(ˆp)= and limpˆ 1 ψ1(ˆp)= → 2 → 0. Hence, regardless of the probability that the coin toss results in a head, the bookmaker is guar- anteed to make a proﬁt by following the optimal price policy given by (5.3). Ifp ˆ = 0.5 (the coin is fair), the bookmaker’s proﬁt is a constant given by

Y ∗ = ψ (0.5) T 0.171573 T. T 1 · ≈ ·

Case 2. Poisson arrivals model (2.5) with λi given by (2.6). In this case, the bookmaker’s terminal wealth YT∗ is given by

u∗,1 u∗,2 Y ∗(Heads) = ( pˆ 1) Q + 1 pˆ Q , T − · T − · T u∗,1 u∗,2 Y ∗(Tails) = ppˆ Q + ( 1 ppˆ 1) Q , T · T − − · T p p u∗,i where the total number of bets placed on Heads and Tails QT (i = 1, 2) are independent Poisson √pi(1 √pi) random variables with expectations given by λ∗ T = − T , where p1 =p ˆ and p2 = 1 pˆ i · 1 pi · − P − Let us compute (YT∗ > 0), the probability that the bookmaker obtains a proﬁt. We have

∗ ∗ ∗ ∗ P P P u ,1 √1 pˆ u ,2 P P u ,2 √pˆ u ,1 (YT∗ > 0) = (Heads) QT < − QT + (Tails) QT < QT 1 √pˆ · ! 1 √1 pˆ · ! − − − M M j 1(j) i i 2(i) j ∞ λ∗T (λ2∗T ) λ∗T (λ1∗T ) ∞ λ∗T (λ1∗T ) λ∗T (λ2∗T ) =p ˆ e− 2 e− 1 + (1 pˆ) e− 1 e− 2 , j! i! − i! j! jX=1 Xi=0 Xi=1 jX=0

√1 pˆ where M (k) is the largest integer less than − k and M (k) is the largest integer less than 1 1 √pˆ · 2 √pˆ − P P 1 √1 pˆ k for all k = 0, 1, . The above expression of (YT∗ > 0) allows us to compute (YT∗ > 0) − − · · · · numerically in an eﬃcient way. For instance, givenp ˆ = 0.5, we compute

P (YT∗ > 0) = 33.67% (T = 1); 54.43% (T = 2); 76.82% (T = 5); 86.49% (T = 10). (5.6)

5.6 Numerical Analysis of a Diﬀusion Example

In this section, we consider a diffusion example, in which the conditional probabilities are given by diffusion processes and the betting arrivals are given by the Poisson model (2.5). The arrival intensity λi is assumed to take the form in (2.6). Our main focus is to investigate the bookmaker’s terminal payoff YT∗, after paying out the winning bets, when he follows the optimal price strategy u∗ given by (5.2).

22 In particular, we follow the setup in Example 2.2 and consider a spreads betting on an NBA game, which oﬀers three outcomes to bet on: A = home team wins by 3 points or more , A = 1 { } 2 home team wins by less than 3 points , and A = away team wins . The reason behind such a { } 3 { } choice of outcomes is that, home advantage is well documented and empirically tested in the sports literature, and the latest home advantage is 2.33 (points) per game during the 2019-2020 NBA 2 season. Let ∆ = (∆t)0 t T denote the scores of the home team minus the scores of the away ≤ ≤ team, and recall from Example 2.2 that ∆t = µt + σWt. We have

P 1 = E 1 = P(∆ 3 ∆ ), P 2 = E 1 = P(∆ 0 ∆ ) P 1, P 3 = 1 P(∆ 0 ∆ ), t t A1 T ≥ | t t t A2 T ≥ | t − t t − T ≥ | t where P(∆ n ∆ ) is given by (2.1). We set the model parameters by T ≥ | t

µ = 2.33, σ = 10, x = 0, q = (0, 0, 0), T = 1, κ = 10000, (5.7) where κ is the scaling factor of the intensity function in (2.6). Since we normalize T = 1, the factor of κ = 10000 predicts that the average number of bets is about 4142 if the probability of the outcome is 0.5 and the price is set to √0.5. We comment that our objective is to obtain qualitative, not quantitative, ﬁnding, since real market data are business secretes and are not available to us. The ﬁndings are robust and hold under any positive scaling factor κ.

1 4

0.9 P1 2 t 2 0.8 P t 0 P3 0.7 t -2

t 0.6 -4 0.5 -6 Probability 0.4 Points Difference -8 0.3

-10 0.2

0.1 -12

0 -14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 4: A simulated path of the points diﬀerence ∆ and the conditional probabilities P i, i =1, 2, 3, over time t [0, 1]. The y-axis on the left is probability, while the y-axis on the right is points diﬀerence ∆. The parameters∈ are set in (5.7).

We first draw one simulated sample path of the points difference ∆ and compute the conditional probabilities P i for i = 1, 2, 3 over time t [0, 1] in Figure 4. In this particular path, the points t ∈ difference (black solid line) eventually goes to negative, meaning the away team wins the game 1 and outcome A3 is the winning bet. Accordingly, we observe that the conditional probabilities Pt 2See statistics here https://www.usatoday.com/sports/nba/sagarin/.

23 3 3 (red dashed line) and Pt (blue dash-dot line) converge to 0, and Pt (pink dotted line) converges ,i i to 1. Recall ut∗ = Pt , the dynamic movements of the conditional probabilities imply that the bookmaker indeed shouldq adjust the prices dynamically to account for the changes of ∆t. Along the particular path in Figure 4, we compute

,1 ,2 ,3 XT∗ = 5471, QT∗ = 2356, QT∗ = 2032, QT∗ = 4323,

,3 which shows the final profit of the bookmaker is Y = X Q∗ = 1148 > 0. T∗ T∗ − T Next, we use the Monte Carlo method and repeat the same simulation process N times (we take N = 10, 000). We report the descriptive statistics of YT∗ in Table 1. The key finding is that, regardless which outcome turns out to be the winning one, the bookmaker is able to make a positive profit when following the optimal dynamic price policy u∗. The median of YT∗ is 3134, greater than the mean 2433, indicating the distribution is left-skewed. Comparing this example with the coin example considered in Section 5.5, we see the importance of dynamically adjusting the prices for the bookmaker. In the coin example, the conditional probabilities remain unchanged and the bookmaker’s optimal price strategy is a constant policy. In such a case, the probability of profitability is strictly less than 1; see (5.6). However, in the example considered here, the dynamic movements of the points difference lead to dynamic changes of the conditional probabilities. The bookmaker then adjusts its offering on odds in a dynamic manner, and is able to achieve strict profitability by following the optimal strategy.

mean s.d. min Q25% median Q75% max 2433 1048 163 1400 3134 3322 4035

∗ Table 1: Descriptive statistics of the bookmaker’s proﬁt YT . Here, s.d. is standard deviation, and Q25% and Q75% refer to 25% and 75% quantiles, respectively. The parameters are set in (5.7).

6 Exponential Utility Maximization

In this section, we study Problem (2.12) when U(y) = e γy, with γ > 0, henceforth called the − − exponential utility maximization problem. We solve this problem assuming Qu is described by the Poisson arrivals model (2.5).3 We summarize the key results in Theorem 6.2.

6.1 Main Results

Let us begin by stating some assumptions, which are assumed to hold throughout the analysis that follows. 3Notice Corollary 4.8 solves the exponential utility maximization problem under the continuous arrivals model (2.4), among other model assumptions.

24 Assumption 6.1. The utility function U is given by U(y) = e γy, where γ > 0. The vector of − − conditional probabilities is a vector of constants, namely, P p (0, 1)n for all t [0, T ). The t ≡ ∈ ∈ u,i bet arrivals process Q is given by the Poisson arrivals model (2.5). The intensity function λi is β(ui pi) given by (2.10), i.e., λi(pi, ui)= κe− − , where κ,β > 0.

As in the previous section, because the conditional probability of set Ai is assumed to be a i ﬁxed constant pi, in order to simplify the notation, we write the intensity function as λi(u ) for all i N . Under Assumption 6.1, with X = x, P = p and Q = q, the bookmaker’s terminal wealth ∈ n t t t u YT is given by

n T n T u i u,i u,i 1 YT = x + utdNt qi + dNt Ai . t − t ! ! Xi=1 Z Xi=1 Z The standard method of solving a stochastic control problem is to develop a verification theorem first, solve the associated HJB PDE with appropriate boundary conditions, and verify all the conditions in the verification theorem are met. We have followed exactly this standard method previously; see Theorem 3.1 and Corollary 5.5. It is clear that the general verification theorem obtained in Theorem 3.1 also applies to the problem we are considering in this section, with operator Lu given by (3.2) and M = 0. Please refer to Øksendal and Sulem (2005)[Chapter 3] for standard stochastic control theory with jumps. The HJB associated with the exponential utility maximization problem is

n ∂tV (t,x,p,q)+ sup (λi(ui)[V (t,x + ui,p,q + ei) V (t,x,p,q)]) = 0, (6.1) ui − Xi=1 and the boundary condition is

γx V (T,x,p,q)= e− a(q), (6.2) − · where e Nn is the vector whose i-th component is 1 and other components are 0, and i ∈ n E 1 a(q) := exp γ qi Ai . ! Xi=1

To better present the solutions in Theorem 6.2, we deﬁne constants c, bi and hi by

β γ β βpi γ β c := , bi := κe , and hi := cbi, i Nn, (6.3) γ · β + γ · β + γ ∈

25 and functions d(q) and α (q), k = 0, 1, 2, , by k · · · n c 1 ji d(q) := [a(q)]− , α (q) := d(q), α (q) := h d(q + j), 0 k k! i · j Nn: j =k i=1 ! ∈ X| | Y where j = (j ,... ,j ) Nn and j := j + . . . + j . For instance, when k = 1, we have α (q) := 1 n ∈ | | 1 n 1 n h d(q + e ). i=1 i · i P Theorem 6.2. Let Assumption 6.1 hold. The value function V of Problem (2.12) is given by

γx 1/c V (t,x,p,q)= e− [G(t,q)]− , (6.4) − where function G is deﬁned by

∞ G(T,q)= d(q), G(t,q)= α (q) (T t)k, t [0, T ). (6.5) k · − ∈ kX=0

The optimal price process u∗ = (us∗)s [t,T ] to Problem (2.12) is given by ∈

i, 1 β H(s,Qs∗) u ∗ = log · , i N , (6.6) s −γ (β + γ) H(s,Q + e ) ∈ n · s∗ i 1/c where H(t,q) := [G(t,q)]− .

Proof. As the utility function is of exponential form, we make the following Ansatz

γx V (t,x,p,q)= e− H(t,q). − ·

Inserting the Ansatz into the HJB equation (6.1) and the boundary condition (6.2), we obtain

n γui ∂tH(t,q)+ inf λi(ui)[e− H(t,q + ei) H(t,q)] = 0, and H(T,q)= a(q). (6.7) ui i=1 − X Solving the inﬁmum problems in (6.7) gives

1 β H(t,q) u∗ = log · > 0, (6.8) i −γ (β + γ) H(t,q + e ) · i which proves the optimal price process in (6.6) once H is found. Next, substituting ui∗ in (6.8) back into (6.7), we obtain

n [H(t,q)]c+1 ∂tH(t,q) bi c = 0, (6.9) − · [H(t,q + ei)] Xi=1

26 c where constants bi and c are deﬁned by (6.3). Now let G(t,q) := [H(t,q)]− . We establish the equation for G as follows

n ∂ G(t,q)+ h G(t,q + e ) = 0, and G(T,q)= d(q). (6.10) t i · i Xi=1 where hi = cbi as in (6.3). The results from Lemma 6.3 verify that G given by (6.5) solves (6.10), and therefore the value function to Problem (2.12) is given by (6.4).

Lemma 6.3. We have

k 1. The series x ∞ α (q)x is convergent with radius + , and hence, G(t,q) given by (6.5) 7→ k=0 k ∞ is well deﬁned andP is in fact analytic.

2. G(t,q) given by (6.5) is a solution to (6.10).

We place the proof to Lemma 6.3 in online Appendix, along with a technical corollary that deals with the scenarios when the bookmaker sets an upper bound on the number of bets Q.

6.2 Sensitivity Analysis

In this subsection, we focus on the sensitivity analysis of the value function and the optimal strategy on the model parameters. By Theorem 6.2, we obtain that

∂V γx 1/c 1 ∞ k 1 = e− [G(t,q)]− − k αk(q) (T t) − < 0, ∂t − − ! kX=1 ∂V γx 1/c = γe− [G(t,q)]− > 0, ∂x and

n ∂d(q) c 1 E 1 1 ∂αk(q) ∂V = c a(q) − − γ Aj qi Ai < 0 < 0, ∂qj − " i=1 # ⇒ ∂qj ⇒ ∂qj X which are consistent with the ﬁndings in the previous section; see Remark 5.2, the comments on

Corollary 5.3, and Proposition 5.4. In deriving Theorem 6.2, we assume the intensity function λi β(ui pi) is given by (2.10), i.e., λi(pi, ui) = κe− − , where β is the key parameter while κ is merely a scaling factor. Indeed, κ affects the constants bi and hi proportionally, and we easily obtain ∂V/∂κ > 0 as expected. However, the impact of β on the value function is rather complex, and no analytical result is available, noting β affects both c and G. Our conjecture is that the bookmaker benefits more from a smaller β, as the smaller the β, the more the expected number of bets.

27 0.9 1 0.9 u1,* 0.99 0.85 0.85 u2,* 1,* u 0.98 0.8 0.8 u2,* 0.97 H(t,q) 0.75 0.75 0.96 0.7 0.7 0.95 0.65

0.94 Function H Optimal Price 0.65 Optimal Price 0.6 0.93 0.6 0.55 0.92 0.55 0.91 0.5

0.5 0.9 0.45 5 10 15 20 0 1 2 3 4 5 6 7 8 9 10 Time T-t Figure 5: Left panel plots the impact of β on the optimal price strategies u1,∗ (red solid line) and u2,∗ (blue dashed line), and the function H(t, q) (black dotted line). The left y-axis is for optimal prices and the right y-axis is for the function H. Right panel plots the impact of the remaining time T t on the optimal price strategies u1,∗ (red solid line) and u2,∗ (blue dashed line). All the parameters other− than β (left) or T t (right) are set as in (6.11). −

To continue the investigations, we consider a coin example as in Section 5.5, and denote P(A1)= P(Heads) =p ˆ (0, 1) and P(A )= P(Tails) =p ˆ. We set the base parameters as follows: ∈ 2

pˆ = 0.6, q = q = 0, T t = 1, β = 10, κ = 1, γ = 2. (6.11) 1 2 −

In the sensitivity analysis, we will allow the parameter of the study to vary around its base value while fix the values of all other parameters as in (6.11). We first study how β affects the bookmaker’s price strategies, which in turn helps answer the impact of β on the value function. We comment that the bookmaker controls the prices (intensities) to attract more bets and to balance the book, which are the key to understand the following results. We draw the plot of the optimal price strategies (u1, , u2, ) and the function H(t,q) over β [5, 20] in the left panel of Figure 5. As β increases, the ∗ ∗ ∈ expected number of bets on both outcomes A1 and A2 decreases, and, to attract more bettors, the bookmaker lowers the prices on both A and A , as observed in Figure 5. Recall V = e γx H(t,q), 1 2 − − the fact that H is an increasing function of β implies that ∂V/∂β < 0, which means a decrease in the amount of bets is adverse to the bookmaker and hence verifies our previous conjecture. We next study the impact of the remaining time T t on the optimal prices and obtain the results in − the right panel of Figure 5. When the bookmaker has enough time (i.e., large T t) to balance his − book, he charges bettors a lower margin (commonly referred to juice, vig, cut, take, among others), but when the current time t approaches the finishing time T , the bookmaker charges a very high margin trying to stay in profit. Recall from (6.11) that p1 =p ˆ = 0.6 and p2 = 0.4, Figure 5 shows that at the far end T t = 10, the bookmaker charges a margin of about 10% but aggressively − increases to 30% or more when t gets close to T . We also observe that the scale of increase is more significant on outcome A2 than on outcome A1, which is because outcomes of small probabilities

28 impose higher risk to the bookmaker. In this section, the bookmaker is risk averse equipped with an exponential utility function U(y) = e γy, where γ > 0 is the risk aversion parameter. In the next study, we investigate how − − the bookmaker’s risk aversion γ aﬀects the optimal strategy u∗ and the value function V . Since both u∗ and V also depend on γ through the function H or G, an analytical sensitivity result is not available. We then resort to numerical analysis and plot both u∗ and V as a function of γ in Figure 6. With all other model parameters ﬁxed, we observe that the bookmaker’s value function increases as γ increases, i.e., a bookmaker with higher risk aversion achieves higher expected utility when following the optimal strategy. As the risk aversion parameter γ increases, the bookmaker becomes more risk averse and sets the optimal prices on both outcomes at a higher level.

1 0 u1,* -0.1 u2,* 0.9 V -0.2

-0.3 0.8 -0.4

-0.5 0.7 Optimal Price Value Function -0.6

-0.7 0.6

-0.8

0.5 -0.9 0 1 2 3 4 5

Figure 6: The graph plots the impact of the risk aversion parameter γ on the optimal price strategies u1,∗ (red solid line) and u2,∗ (blue dashed line), and the value function V (black dotted line). The left y-axis is for the optimal price u∗ and the right y-axis is for the value function V . All the parameters other than γ are set as in (6.11) and x = 1.

We also conduct sensitivity analysis on the probabilityp ˆ (the probability of “Heads”) and the inventory q1 (the number of bets on “Heads”), with results plotted in Figure 7. The main ﬁndings 1, are summarized as follows: (i) Whenp ˆ = p1 increases, the bookmaker increases the price u ∗ 2, on outcome A1 and simultaneously decreases the price u ∗ on outcome A2. Notice that when p1 = p2 = 0.5, the optimal prices on two outcomes are the same. (ii) Keeping the inventory q2 1, on outcome A2 constant (we set q2 = 5), the optimal price u ∗ on outcome A1 is an increasing 2, function of its inventory q1 while the optimal price u ∗ on outcome A2 is a decreasing function of

A1’s inventory q1, which is consistent with the result in the left panel of Figure 3.

7 Conclusion

In this paper, we introduce a general framework for continuous-time betting markets. A bookmaker takes bets on random (sporting) events and pays oﬀ the winning bets at the terminal time T . The

29 1 1 u1,* u1,* 0.9 0.9 u2,* u2,*

0.8 0.8

0.7 0.7 0.6 0.6 0.5 0.5

Optimal Price Optimal Price 0.4 0.4 0.3

0.3 0.2

0.2 0.1

0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 Probability Inventory Figure 7: The graph plots the impact of the probabilityp ˆ (the probability of “Heads”) in the left panel and the impact of the inventory q1 (the number of bets on “Heads”) in the right panel, on the optimal price strategies u1,∗ (red solid line) and u2,∗ (blue dashed line). All the parameters are set as in (6.11), exceptp ˆ as a variable in the left panel and q1 as a variable and q2 = 5 in the right panel. conditional probability of a set of outcomes is exogenous and may be stochastic, and the bets placed on a set of outcomes may arrive either at a continuous rate or as a state-dependent Poisson process. The bookmaker controls (updates) the prices of bets dynamically. In turn, the prices set by the bookmaker affect the rates or intensities of bet arrivals. The bookmaker seeks to set prices in order to maximize his expected terminal wealth or expected utility of terminal wealth. We are able to obtain either explicit solutions or characterizations of such optimal bookmaking problems in a number of distinct settings. To solve the optimal bookmaking problem (2.12), we consider two settings for the bookmaker’s utility function U: the risk neutral setting and the exponential utility (risk averse) setting. We end this paper by commenting on the challenges when the bookmaker is risk averse with a power utility 4 1 α function. Let us illustrate the main difficulty under the power utility setting of U(y)= α y , where α< 1 and α = 0, via a simple example when Assumption 4.1 holds (i.e., under the continuous arrival 6 model (2.4) and the constant conditional probabilities). Suppose (Ai)i=1,2, ,n form a partition ··· of the sample space with P(A ) = p (0, 1), and the rate function λ is given by (2.6). Let i i ∈ us further restrict to time-independent constant strategies only u = (u , u , , u ) [0, 1]n. 1 2 · · · n ∈ For any strategy u, the bookmaker’s final wealth Y u is given by Y u = x n q 1 + (T T T − i=1 i Ai − n pi(1 ui) n pi(1 ui) u t) − − 1 . Maximizing E [U(Y )] leads to the following system of i=1 1 pi i=1 ui(1 pi) Ai t,x,p,q T P − − − n non-linearP equations:P 2 2 u α 1 u α 1 (1 u )/u Y − p = Y − p , − i i · i i j j j=i X6 where Y u is the bookmaker’s profit when A occurs at T (i.e., Y u = Y u(ω), where ω A ). Notice i i i T ∈ i u u u that Yi depends on the state variables and controls of all outcomes, i.e., Yi = Yi (t,x,p,qi, u). 4We are indebted to an anonymous referee for motivating us discuss these challenges faced in the power utility setting.

30 Hence, the above non-linear system is fully coupled, while in comparison, the ﬁrst-order condition in the exponential utility case leads to a decoupled system and ui∗ can be explicitly obtained; see (6.7) and (6.8). The exact same issue appears when we analyze the HJB equation in more complicated cases, which prevents us from guessing an ansatz in a separable form. On the numerical side, the corresponding HJB equation is at least (5+1)-dimensional (t,x,p1,p2,q1,q2). Solving an HJB equation in such high-dimension with any sort of accuracy is extremely diﬃcult due to the curse of dimensionality. In addition, even if the HJB equation could be solved numerically, it is really the optimal control rather than the value function that is sought after. And deriving the optimal control from a value function that has been solved numerically is even more challenging than solving the HJB equation numerically because the control is written in terms of derivatives of the value function.

Acknowledgments

The authors would like to thank two anonymous referees, Editor Steﬀen Rebennack, Agostino Capponi, Xuedong He, and Steven Kou for valuable comments and suggestions. We also thank seminar participants at Rutgers University and the University of Michigan, and the organizers (Kim Weston, Kasper Larsen, Erhan Bayraktar, and Asaf Cohen) for their hospitality. Bin Zou acknowledges a start-up grant from the University of Connecticut.

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33 Online Appendix for “Optimal Bookmaking” by Matthew Lorig, Zhou Zhou and Bin Zou

A Technical Proofs and Results in Section 6

Proof to Lemma 6.3. Denote by b := max b ,... ,b . For k large enough, we have { 1 n} 1 1 k + n 1 1 kn 1 kn bk n α (q) bk = − bk − bk < bk − bn, | k |≤ k! k! n 1 ∼ k! (n 1)! k! ∼ (k n)! j Nn: j =k ! ∈ X| | − − − which proves the ﬁrst result. To show the second result, we obtain

∞ k 1 ∞ k ∂ G(t,q)= k α (q) (T t) − = (k + 1) α (q) (T t) , t − · k − − k+1 · − kX=1 kX=0 n n ∞ k hiG(t,q + ei)= hi αk(q + ei) (T t) . · ! − Xi=1 kX=0 Xi=1 Since we have

n n 1 h α (q + e )= h hj1 ...hjn d(q + e + j) i · k i i k! 1 n · i i=1 i=1 j Nn: j =k X X ∈ X| | 1 n = h hj1 ...hjn d(q + e + j) k! i · 1 n · i i=1 j Nn: j =k X ∈ X| | ′ ′ 1 j1 j = h ...h n d(q + j′) k! 1 n · j′ Nn: j′ =k+1 ∈ X| | = (k + 1)αk+1(q), the second result follows.

In Section 6, we do not impose any upper bound on the number of bets the bookmaker takes. However, if the bookmaker sets an upper for each betting event, we need to modify the results in Theorem 6.2 as described in the corollary below.

Corollary. Let Assumption 6.1 hold. Assume the total number of bets placed on set of outcomes A is at most m , where i N . The value function to Problem (2.12) is given by i i ∈ n 1/c γx − V (t,x,p,q)= e− Gˇ(t,q) , − h i

i where Gˇ is deﬁned by

m q | |−| | Gˇ(T,q)= d(q), Gˇ(t,q)= αˇ (q) (T t)k, t [0, T ), k · − ∈ kX=0 with functionsα ˇk(q) given by

1 n αˇ (q) := d(q), αˇ (q) := hji d(q + j), 0 k k! i · j I(k,q) i=1 ! ∈X Y for all k = 1, 2, , n m and · · · i=1 i P I(k,q) := j Nn : j = k, q + j m := (m ,m , ,m ) . { ∈ | | ≤ 1 2 · · · n }

The optimal price process u∗ = (us∗)s [t,T ] to Problem (2.12) is given by ∈ ˇ i, 1 β H(s,Qs∗) us ∗ = log · , −γ " (β + γ) Hˇ (s,Q + e ) # · s∗ i for all i I(q), where Hˇ (t,q) := [Gˇ(t,q)] 1/c and ∈ −

I(q) := i : q

Proof. With the extra upper bound assumption, the bets on Ai will arrive according to a Poisson process at intensity λi(ui) if the total number is less than mi; and 0 if otherwise. Notice that all the equations (6.1), (6.7), (6.9), and (6.10) still hold, except that the summation over index i in these equations will be restricted to the set I(q), deﬁned by (A.1). All the results follow naturally by Theorem 6.2.