Optimal Bookmaking

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

Optimal Bookmaking

Bin Zou University of Connecticut

Financial/Actuarial Mathematics Seminar University of Michigan, Ann Arbor November 13, 2019 INTRODUCTION Framework Semi-Static Wealth Exp Utility References

COAUTHORS

Zhou Zhou University of Sydney (Ph.D. of Erhan from UM) Matt Lorig University of Washington INTRODUCTION Framework Semi-Static Wealth Exp Utility References

HIGHLIGHTS

I Propose a general framework for continuous-time betting markets

I Formula an optimal control problem for a bookmaker Control (strategy): price or odds of bet

I Two objectives: maximize proﬁt or maximize utility of terminal wealth

I Different mathematical techniques

I Obtain explicit (or characterizations to) solutions in interesting market models INTRODUCTION Framework Semi-Static Wealth Exp Utility References

OUTLINE

INTRODUCTION

Framework

Analysis of the Semi-static Setting

Wealth Maximization

Exponential Utility Maximization INTRODUCTION Framework Semi-Static Wealth Exp Utility References

BETTINGMARKETISENORMOUS!

I Zion Market Research: $104.31 (bn) sports betting in 2017, with annual growth rate 9% - 10% I American Gaming Association (AGA): only $5 (bn) legally and at least $150 (bn) illegally in US I UK Gambling Commission: £14.5 (bn) from 10/17 to 9/18 UK population: 66 M versus US population: 327.2 M UK data ⇒ US market size $87.39 (or £71.88) (bn) [Comparison] $38.1 (bn) on dairy products in US (2017) I Asia-Paciﬁc: most signiﬁcant percentage of market shares population: +4 bn I Sports betting is the biggest cake (+40%, still growing) NCAA, NFL, MLB, NBA, Soccer, Golf ... I Other betting markets: casinos $41.7 (bn) in 2018, up 3.5% lotteries $73.5 (bn) in 2017 INTRODUCTION Framework Semi-Static Wealth Exp Utility References

REGULATIONS ON THE WAY

I Sports betting was banned for a long time in US. In fact, “pool” betting among friends and co-workers was actually illegal by 2018 in almost all states (except NV).

I American Gaming Association: 97% of the $10 (bn) betting on 2018 NCAA men’s basketball tournament was illegal (3% with NV bookies).

I Supreme Court 2018 May ruling: the Professional and Amateur Sports Protection Act (PASPA) unconstitutional

I Prior to the PASPA ruling, sports betting was only legal in one state (NV).

I Now, the number is 13 and still counting, with a dozen more states in serious consideration (including MI). INTRODUCTION Framework Semi-Static Wealth Exp Utility References

SPORTS BETTINGIN U.S.

Live Legal Single Game Sports Betting (13 states)

Authorized Sports Betting, but Not Yet Operational (5 states + DC)

Active 2019 Sports Betting Legislation/Ballot (6 states)

Lt. Blue Dead Sports Betting Legislation in 2019 (18 states)

No Sports Betting Bills in 2019 (8 states)

American Gaming Association, Aug. 27, 2019 www.americangaming.org/resources/state-gaming-map/ INTRODUCTION Framework Semi-Static Wealth Exp Utility References

INTUITIONONSETUP

I One can bet on n outcomes of a game, Ai, i = 1, ··· , n, e.g., n = 2, A1 = {win} and A2 = {lose}. Note. (Ai) do NOT need to be a partition of Ω. I The bookmaker can set (control) the price (odds) of an i outcome, say u of Ai. The bookmaker cannot directly control the number of bets i i placed on Ai, denoted by Q ; but the price u apparently affects Qi (ui ↑ ⇒ Qi ↓).

I The bookmaker would like to see that, no matter which Ai occurs, the revenues he collected are sufﬁcient to pay off the winning bets (ideally with leftovers). I To balance the book, dynamically adjusting the price ui may be necessary.

Question: Is there an optimal price u∗ to the bookmaker? INTRODUCTION Framework Semi-Static Wealth Exp Utility References

ALESSON

Story from a Bookie Norway The prop bet: 175-to-1 odds that Luis Suarez would bite someone during the 2014 FIFA World Cup tournament.

Jonathan, among other lucky 167 people, placed 80 Kronas (£7) and won 14,000. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

COMPLEXITYOFDYNAMICMARKETS

I Traditionally, bookmakers would only take bets prior to the start of a sports game. Hence, the probabilities of outcomes remain fairly static. I We allow in-game betting, i.e., bookmakers take bets as the events occur. Now, the probabilities of outcomes evolve stochastically. I This complicates the task of a bookmaker who, in addition to considering the number of bets he has collected on particular outcomes, must also consider the dynamics of the sporting event in play. Example: The goal scored by Mario Gotze¨ in World Cup 2014 Final (7 minutes to the end) nearly put the odds to 1. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

LITERATURE (MORE RELATED)

I Hodges et al. (2013): horse race games with ﬁxed winning i probabilities (P(Ai) ≡ pi); # of bets Q ∼ N (Normal); one-period setup; look for u∗ to maximize utility

I Divos et al. (2018): dynamic betting during a football match; no-arbitrage arguments to price a bet whose payoff is a function of the two teams’ scores

I Bayraktar and Munk (2017): parimutuel betting with two mutually exclusive outcomes; continuum of minor players and ﬁnite major players; look for equilibrium (betting amount on each outcome)

New research topics to mathematical ﬁnance Not surprising that literature is scarce INTRODUCTION Framework Semi-Static Wealth Exp Utility References

LITERATURE (LESS RELATED)

I Optimal market making: Ho and Stoll (1981); Avellaneda and Stoikov (2008); Adrian et al. (2019) ...

I Optimal execution: Gatheral and Schied (2013); Bayraktar and Ludkovski (2014); Cartea and Jaimungal (2015) ...

Buyers ⇔ Market Makers ⇔ Sellers Buyers (bettors) ⇔ Bookmakers (hold the book) INTRODUCTION Framework Semi-Static Wealth Exp Utility References

OUTLINE

INTRODUCTION

Framework

Analysis of the Semi-static Setting

Wealth Maximization

Exponential Utility Maximization INTRODUCTION Framework Semi-Static Wealth Exp Utility References

PROBABILITY PREPARATION

I Fix a ﬁltered probability space (Ω, F, F = (Ft)0≤t≤T, P), where P as the real world or physical probability measure.

I Consider a ﬁnite number of outcomes Ai, i = 1, ··· , n, each of which ﬁnishes at T (Ai ∈ FT).

Note: Ai ∩ Aj 6= ∅ and ∪Ai 6= Ω are allowed.

i E 1 I Denote Pt = t[ Ai ], Ft-conditional probability of Ai. Note: Pi is a martingale. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

ANEXAMPLE

I Suppose the goals scored during a soccer game arrive as a Poisson process with intensity µ. µ Notation Nt : the number of goals scored by time t. I Consider outcome A1, where

A1 = {the game will ﬁnish with at least one goal}.

I We have

1 −µ(T−t) P = 1 µ + 1 µ (1 − e ). t {Nt ≥1} {Nt =0}

I The dynamics of P1 can be easily deduced

1 −µ(T−t) µ dP = 1 1 e (dN − µdt). t {Pt−<1} t INTRODUCTION Framework Semi-Static Wealth Exp Utility References

STRUCTUREOFBETS

We assume that a bet placed on outcome Ai pays one unit of currency at time T if and only if ω ∈ Ai (Ai occurs). Namely, 1 i “payoff of a bet placed on Ai” = Ai = PT.

i i u = (ut)0≤t

However, the bookmaker can set the price of a bet placed on Ai and this in turn will affect the rate or intensity at which bets on Ai are placed. Generally, higher prices will result in a lower rate or intensity of bet arrivals. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

BOOKMAKER’SREVENUE

u u Denote by X = (Xt )0≤t≤T the total revenue received by the bookmaker. u,i u,i Let Q = (Qt )0≤t≤T be the total number of bets placed on a set of outcomes Ai, given price u. Note that we have indicated with a superscript the dependency of Xu and Qu on bookmaker’s pricing policy u. The relationship between Xu, Qu and u is

n u X i u,i dXt = ut dQt . i=1

Observe that Xu and Qu,i are non-decreasing processes for all i. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

TWO ARRIVAL MODELS We consider two models for Qu,i: 1. Continuous arrivals Z t u,i i Qt = λi(Ps, us)ds + Q0. 0 2. Poisson arrivals Z t u,i u,i i E u,i Qt = dNs + Q0, tdNt = λi(Pt, ut)dt. 0

We will refer to the function λi as the rate or intensity function. In general, the function λ(p, u) could depend on vectors p and u

prob. p = (p1, p2,..., pn), price u = (u1, u2,..., un).

i We expect (1) pi ↑ ⇒ λi ↑ and (2) u ↑ ⇒ λi ↓. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

EXAMPLES OF RATE/INTENSITYFUNCTIONS ¯ Examples of rate/intensity functions λi :[0, 1] × [0, 1] → R+ are

pi 1 − ui log ui λi(pi, ui) := , λi(pi, ui) := . 1 − pi ui log pi These examples have reasonable qualitative behavior. i (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) = ∞. 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. ui→1 i i (iii) All fair bets ut = Pt have the same intensity

λi(pi, pi) = λi(qi, qi). INTRODUCTION Framework Semi-Static Wealth Exp Utility References

BOOKMAKER’S VALUE FUNCTION u Let YT denote the bookmaker’s terminal wealth after paying u Pn i i out all winning bets. (Add X0 − i=1 PTQ0, if non-zero)

n n Z T u u X i u,i X i i u,i YT = XT − PTQT = ut − PT dQt . i=1 i=1 0

Suppose the bookmaker’s objective function J is of the form

E u u u J(t, x, p, q; u) := [U(YT)|Xt = x, Pt = p, Qt = q],

where U is either the identity function or a utility function. We deﬁne the bookmaker’s value function V as

V(t, x, p, q) := sup J(t, x, p, q; u). u∈A(t,T)

n Admissible set A(t, T): non-anticipative and us ∈ [0, 1] . INTRODUCTION Framework Semi-Static Wealth Exp Utility References

INFINITESIMAL GENERATOR

Let P be the inﬁnitesimal generator of the process P. Deﬁne

n u X L := λi(p, u)(ui∂x − ∂qi ) + P, (1) i=1 n X q Lu := λ ( , )(θx θ i − ) + P, i p u ui 1 1 (2) i=1

q where θz is a shift operator of size z in the variable q. Lu as deﬁned in (1) is the generator of (Xu, P, Qu) assuming the dynamics of Qu are described by the continuous arrivals model. Lu as deﬁned in (2) is the generator of (Xu, P, Qu) assuming the dynamics of Qu are described by the Poisson arrivals. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

PDE CHARACTERIZATION Theorem 1 Letv be a real-valued function which is at least once differentiable with respect to all arguments and satisﬁes

∂xv > 0, and ∂qi v < 0, ∀ i.

Suppose the functionv satisﬁes the HJB equation ( A = [0, 1]n)

h n i uˆ E X i 0 = ∂tv + sup L v, v(T, x, p, q) = T− U x − qiPT . ˆ∈A u i=1

Thenv (t, x, p, q) = V(t, x, p, q) is the bookmaker’s value function and the optimal price process u∗ is given by

∗ uˆ ∗ ∗ us = arg max L v(s, Xs , Ps, Qs ). uˆ∈A INTRODUCTION Framework Semi-Static Wealth Exp Utility References

OUTLINE

INTRODUCTION

Framework

Analysis of the Semi-static Setting

Wealth Maximization

Exponential Utility Maximization INTRODUCTION Framework Semi-Static Wealth Exp Utility References

Assumptions (1) Qu is given by the continuous arrival model. n (2) Pt ≡ p ∈ (0, 1) for all t ∈ [0, T). Conditional probabilities are static during the entire game. (3) U is continuous and strictly increasing. We do not require U to be concave. i (4) λi = λi(u ) is continuous and decreasing. Recall that, in general, we have λi = λi(p, u). What we assume is that, the betting rate of outcome Ai only depends on its own price ui. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

MAIN RESULTS −1 ˆ Deﬁne fi(x) := x · λi (x) for all x > 0 and i. Denote by f the concave envelope of f . Theorem 2 Given the previous assumptions, we have

n n E X 1 Xˆ ˆ V(t, x, p, q) = sup t,x,p,qU x − qi Ai + (T − t) fi(λi(ui)) ˆ∈A u i=1 i=1

n ! X ˆ 1 ˆ − (T − t) λi(ui) Ai := V, i=1

where A = [0, 1]n. This is a STATIC optimization problem.

Remark: We can deﬁne Λi := λi(uˆi) and optimize over Λi. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

SKETCH

u 1. Express the bookmaker’s terminal wealth YT using û functions fi, and define the “concave envelope version” YT ˆ û u by replacing fi with fi.(YT ≥ YT) 2. Show that

E ˆu E ˆuˆ sup t[U(YT)] = sup t[U(YT)]. u∈A(t,T) uˆ∈A

Note: the r.h.s. is a constrained problem, namely, the price process is a constant vector. 3. Show that E ˆuˆ V(t, x, p, q) ≥ sup t[U(YT)]. uˆ∈A INTRODUCTION Framework Semi-Static Wealth Exp Utility References

EXTRAREMARKS

∗ ˆ ∗ I A sufficient condition: fi(λi(uî )) = fi(λi(uî )), ∗ ˆ where uˆ is the optimizer to V. If fi is concave itself, the above automatically holds (e.g., λ (x) = pi 1−x ). i 1−pi x I If we further assume (1) Sets (Ai) form a partition of Ω; (2) λ (x) = pi 1−x ; and (3) U(x) = −e−γx, γ > 0, then we have i 1−pi x

1 X p · − 1 · g(t, p , q ; uˆ∗) = p · g(t, p , q ; uˆ∗), i (uˆ∗)2 i i i j j j j i j6=i where g(t, p , q ; u ) := exp γq + γ(T − t) pi 1 − 1 . i i i i 1−pi ui ∂uˆ∗ ∂uˆ∗ We deduce i > 0 and observe by graph that i > 0 . ∂qi ∂t INTRODUCTION Framework Semi-Static Wealth Exp Utility References

OUTLINE

INTRODUCTION

Framework

Analysis of the Semi-static Setting

Wealth Maximization

Exponential Utility Maximization

The Standing Assumption of this section is U(x) = x. Inferring the bookmaker is risk neutral. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

METHOD I

Theorem 3 For both continuous and Poisson arrival models, we have

Z T n E X i V(t, x, p, q) = x − p · q + t sup λi(Ps, uˆ) · (uˆi − Ps)ds, ˆ∈A t u i=1

Pn where p · q = piqi. i=1

Proof. By measurable selection theorems. Transform a dynamic optimization problem over A(t, T) to a static one over A. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

Corollary 4

(i) If λ (p, u) = pi 1−ui , we obtain i 1−pi ui q i,∗ i ut = Pt ∀ i = 1, ··· , n.

n If further Pt ≡ p ∈ (0, 1) for all t ∈ [0, T), then

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

(ii) If λ (p, u) = log ui , we obtain ui,∗ as the unique solution on i log pi (e−1, 1) to the equation

i,∗ i,∗ i ut (1 + log ut ) = Pt ∀ i = 1, ··· , n. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

METHOD II(DPP) Assuming regularity, we compute

∂tV < 0, ∂xV = 1, and ∂qi V = −pi, ∀ i = 1, ··· , n.

We then use the HJB equation in the PDE characterization Theorem1 to obtain the same results as in Theorem3. For instance, under the continuous arrival model with λ (p, u) = pi 1−ui , we simplify the HJB into i 1−pi ui

−2 i,∗ ∂xV − us · ∂qi V = 0,

and obtain, for t ≤ s < T, that s −∂ V(s, X∗, P , Q∗) q i,∗ qi s s s i us = ∗ ∗ ≡ Ps ∈ (0, 1). ∂xV(s, Xs , Ps, Qs ) INTRODUCTION Framework Semi-Static Wealth Exp Utility References

METHOD III Recall Theorem2 in Section Semi-Static: V(t, x, p, q) = Vˆ. Corollary 5

Let the assumptions of Theorem2 hold, we obtain

ˆ0 ∗ fi (Λi ) = pi, (recall Λi = λi(ui)).

In particular, if λ (p, u) = pi 1−ui , we obtain i 1−pi ui ∗ pi 1 ∗ √ Λi = √ − 1 and uˆi = pi, 1 − pi pi

and if λ (p, u) = log ui , i log pi

Λ∗ i ∗ ˆ∗ ˆ∗ pi (1 + Λi log pi) = pi and ui (1 + log ui ) = pi. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

COMPARISON OF METHODS I-III

Prob. P Arrival Q Use Sol. u∗ Problem I – both wealth X static II – continuous general X dynamic III constant continuous general X static

I “Model Generality”: I > II > III I “Application Scope”: II = III > I Method I only applies to the wealth max problem. I “Solutions u∗”: II > I = III Methods I and III only characterize the value function. I “Computational complexity”: I ≈ III < II The HJB in Method II is difﬁcult to solve. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

PROFITABILITY ANALYSIS

P ∗ Purpose: study (YT > 0)

pi 1−ui Assumptions: (1) λ (p, u) = ; (2) Pt ≡ p = (pˆ, 1 − pˆ); i 1−pi ui i (3) two mutually exclusive sets A1 and A2; and (4) X0 = Q0 = 0. Case 1: continuous arrival model

∗ ∗ YT(Heads) = ψ1(pˆ) · T and YT(Tails) = ψ1(1 − pˆ) · T,

where function ψ1 is deﬁned by ! pˆ q 1 1 − pˆ q ψ1(pˆ) := 2 − pˆ − p + 1 − 1 − pˆ . 1 − pˆ pˆ pˆ INTRODUCTION Framework Semi-Static Wealth Exp Utility References

ψ1 is a decreasing function over (0, 1) with 1 lim ψ1(pˆ) = and lim ψ1(pˆ) = 0. ˆp→0 2 ˆp→1 Conclusion: the bookmaker always makes proﬁts by following the optimal price u∗. If pˆ = 0.5 (the coin is fair), the bookmaker’s proﬁt is a constant ∗ given by YT = ψ1(0.5) · T ≈ 0.171573 · T. 0.5 0.4 0.3 0.2 0.1

0.0 0.2 0.4 0.6 0.8 1.0

Figure 1: Graph of ψ1 over (0, 1) INTRODUCTION Framework Semi-Static Wealth Exp Utility References

Case 2: Poisson arrival model

q ∗ q ∗ ∗ ˆ u ,1 ˆ u ,2 YT(Heads) = ( p − 1) · QT + 1 − p · QT , q ∗ q ∗ ∗ ˆ u ,1 ˆ u ,2 YT(Tails) = p · QT + ( 1 − p − 1) · QT ,

u∗,i where QT (i = 1, 2) are independent Poisson r.v.’s with expectations given by √ √ ∗ pi(1 − pi) λi · T = · T, where p1 = pˆ, p2 = 1 − pˆ. 1 − pi P ∗ We can express (YT > 0) as the sum of two inﬁnite series. P ∗ If pˆ = 0.5, we get (YT > 0) = 33.6747% (if T = 1); 54.4348% (if T = 2); and 86.4919% (if T = 10). INTRODUCTION Framework Semi-Static Wealth Exp Utility References

OUTLINE

INTRODUCTION

Framework

Analysis of the Semi-static Setting

Wealth Maximization

Exponential Utility Maximization INTRODUCTION Framework Semi-Static Wealth Exp Utility References

Assumptions (1) The utility function is U(x) = −e−γx, where γ > 0. The bookmaker is risk averse. n (2) Pt ≡ p ∈ (0, 1) for all t ∈ [0, T). Conditional probabilities are static during the entire game. (3) The arrivals Qu,i is given by the Poisson model. (4) The intensity function is

−β(ui−pi) λi(pi, ui) = κ e , where κ, β > 0.

You may assign different parameters κi and βi for different outcomes Ai. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

HEURISTIC DERIVATION

The HJB associated with the exponential utility max problem is

n X ∂tV(t, x, p, q)+ sup λi(ui)[V(t, x+ui, p, q+ei)−V(t, x, p, q)] = 0, u i=1 i and the boundary conditions are

V(T, x, p, q) = −e−γx · a(q),

where ei = (0, ··· , 0, 1, 0, ··· , 0) (1 in ith) and

n ! E X 1 a(q) := exp γ qi Ai . i=1 INTRODUCTION Framework Semi-Static Wealth Exp Utility References

Theorem 6 The value function V is given by

V(t, x, p, q) = −e−γx [G(t, q)]−1/c ,

β where c := γ and function G is deﬁned by

∞ −c X k G(T, q) = [a(q)] , G(t, q) = αk(q) · (T − t) , t ∈ [0, T). k=0

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

∗ i,∗ 1 β · H(s, Qs ) us = − log ∗ , i = 1, ··· , n, γ (β + γ) · H(s, Qs + ei)

where H(t, q) := [G(t, q)]−1/c. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

REFERENCES I Adrian, T., Capponi, A., Vogt, E., and Zhang, H. (2019). Intraday market making with overnight inventory costs. http://dx.doi.org/10.2139/ssrn.2844881. Avellaneda, M. and Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3):217–224. Bayraktar, E. and Ludkovski, M. (2014). Liquidation in limit order books with controlled intensity. Mathematical Finance, 24(4):627–650. Bayraktar, E. and Munk, A. (2017). High-roller impact: A large generalized game model of parimutuel wagering. Market Microstructure and Liquidity, 3(01):1750006. Cartea, A.´ and Jaimungal, S. (2015). Optimal execution with limit and market orders. Quantitative Finance, 15(8):1279–1291. Divos, P., Rollin, S. D. B., Bihari, Z., and Aste, T. (2018). Risk-neutral pricing and hedging of in-play football bets. Applied Mathematical Finance, 25(4):315–335. Gatheral, J. and Schied, A. (2013). Dynamical models of market impact and algorithms for order execution. HANDBOOK ON SYSTEMIC RISK, Jean-Pierre Fouque, Joseph A. Langsam, eds, pages 579–599. INTRODUCTION Framework Semi-Static Wealth Exp Utility References

REFERENCES II

Ho, T. and Stoll, H. R. (1981). Optimal dealer pricing under transactions and return uncertainty. Journal of Financial Economics, 9(1):47–73. Hodges, S., Lin, H., and Liu, L. (2013). Fixed odds bookmaking with stochastic betting demands. European Financial Management, 19(2):399–417.

Preprint. Lorig, Matthew, Zhou, Zhou and Zou, Bin (2019). Optimal Bookmaking. Available at SSRN: https://ssrn.com/abstract=3415675.

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