optimal bookmaking - university of michigan

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

Optimal Bookmaking

Bin ZouUniversity of Connecticut

Financial/Actuarial Mathematics SeminarUniversity of Michigan, Ann Arbor

November 13, 2019


COAUTHORS

Matt LorigUniversity of Washington

Zhou ZhouUniversity of Sydney(Ph.D. of Erhan from UM)


HIGHLIGHTS

I Propose a general framework for continuous-time bettingmarkets

I Formula an optimal control problem for a bookmaker

Control (strategy): price or odds of bet

I Two objectives: maximize profit or maximize utility ofterminal wealth

I Different mathematical techniques

I Obtain explicit (or characterizations to) solutions ininteresting market models


OUTLINE

INTRODUCTION

Framework

Analysis of the Semi-static Setting

Wealth Maximization

Exponential Utility Maximization


BETTING MARKET IS ENORMOUS!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 USI UK Gambling Commission: £14.5 (bn) from 10/17 to 9/18

UK population: 66 M versus US population: 327.2 MUK 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 sharespopulation: +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


REGULATIONS ON THE WAY

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

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

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

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

I Now, the number is 13 and still counting, with a dozenmore states in serious consideration (including MI).


SPORTS BETTING IN U.S.

Active 2019 Sports Betting Legislation/Ballot (6 states)

Live Legal Single Game Sports Betting (13 states)

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

Dead Sports Betting Legislation in 2019 (18 states)

Lt. Blue

No Sports Betting Bills in 2019 (8 states)

American Gaming Association, Aug. 27, 2019www.americangaming.org/resources/state-gaming-map/

www.americangaming.org/resources/state-gaming-map/


INTUITION ON SETUP

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 anoutcome, say ui of Ai.The bookmaker cannot directly control the number of betsplaced on Ai, denoted by Qi; but the price ui apparentlyaffects Qi (ui ↑ ⇒ Qi ↓).

I The bookmaker would like to see that, no matter which Aioccurs, the revenues he collected are sufficient to pay offthe 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?


A LESSON

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

Jonathan, among other lucky 167 people, placed 80 Kronas (£7)and won 14,000.


COMPLEXITY OF DYNAMIC MARKETS

I Traditionally, bookmakers would only take bets prior tothe start of a sports game. Hence, the probabilities ofoutcomes remain fairly static.

I We allow in-game betting, i.e., bookmakers take bets as theevents occur. Now, the probabilities of outcomes evolvestochastically.

I This complicates the task of a bookmaker who, in additionto considering the number of bets he has collected onparticular outcomes, must also consider the dynamics ofthe sporting event in play.Example: The goal scored by Mario Gotze in World Cup2014 Final (7 minutes to the end) nearly put the odds to 1.


LITERATURE (MORE RELATED)

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

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

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

New research topics to mathematical financeNot surprising that literature is scarce


LITERATURE (LESS RELATED)

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

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

Buyers⇔Market Makers⇔ Sellers

Buyers (bettors)⇔ Bookmakers (hold the book)


OUTLINE

INTRODUCTION

Framework


Wealth Maximization



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, eachof which finishes at T (Ai ∈ FT).

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

I Denote Pit = Et[1Ai ], Ft-conditional probability of Ai.

Note: Pi is a martingale.


AN EXAMPLE

I Suppose the goals scored during a soccer game arrive as aPoisson process with intensity µ.Notation Nµ

t : 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

P1t = 1Nµ

t ≥1 + 1Nµt =0(1− e−µ(T−t)).

I The dynamics of P1 can be easily deduced

dP1t = 1P1

t−<1e−µ(T−t)(dNµ

t − µdt).


STRUCTURE OF BETS

We assume that a bet placed on outcome Ai pays one unit ofcurrency at time T if and only if ω ∈ Ai (Ai occurs). Namely,

“payoff of a bet placed on Ai” = 1Ai = PiT.

ui = (uit)0≤t<T denotes the price set by the bookmaker of a bet

placed on Ai. Let u = (u1, · · · ,un).

The bookmaker cannot control the number of bets placed onoutcome Ai directly.

However, the bookmaker can set the price of a bet placed on Aiand this in turn will affect the rate or intensity at which bets onAi are placed.

Generally, higher prices will result in a lower rate or intensityof bet arrivals.


BOOKMAKER’S REVENUE

Denote by Xu = (Xut )0≤t≤T the total revenue received by the

bookmaker.

Let Qu,i = (Qu,it )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 dependencyof Xu and Qu on bookmaker’s pricing policy u.

The relationship between Xu, Qu and u is

dXut =

n∑i=1

uit dQu,i

t .

Observe that Xu and Qu,i are non-decreasing processes for all i.


TWO ARRIVAL MODELSWe consider two models for Qu,i:

1. Continuous arrivals

Qu,it =

∫ t

0λi(Ps,us)ds + Qi

0.

2. Poisson arrivals

Qu,it =

∫ t

0dNu,i

s + Qi0, EtdNu,i

t = λi(Pt,ut)dt.

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).

We expect (1) pi ↑ ⇒ λi ↑ and (2) ui ↑ ⇒ λi ↓.


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

λi(pi,ui) :=pi

1− pi

1− ui

ui, λi(pi,ui) :=

log ui

log pi.

These examples have reasonable qualitative behavior.

(i) As the price uit of a bet on outcome Ai goes to zero, the

intensity of bets goes to infinity

limui→0

λi(pi,ui) =∞.

(ii) As the price uit of a bet on outcome Ai goes to one, the

intensity of bets goes to zero

limui→1

λi(pi,ui) = 0.

(iii) All fair bets uit = Pi

t have the same intensity

λi(pi, pi) = λi(qi, qi).


BOOKMAKER’S VALUE FUNCTIONLet Yu

T denote the bookmaker’s terminal wealth after payingout all winning bets. (Add Xu

0 −∑n

i=1 PiTQi

0, if non-zero)

YuT = Xu

T −n∑

i=1

PiTQu,i

T =

n∑i=1

∫ T

0

(ui

t − PiT

)dQu,i

t .

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

J(t, x, p, q; u) := E[U(YuT)|Xu

t = x,Pt = p,Qut = 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) := supu∈A(t,T)

J(t, x, p, q; u).

Admissible set A(t,T): non-anticipative and us ∈ [0, 1]n.


INFINITESIMAL GENERATOR

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

Lu :=

n∑i=1

λi(p,u)(ui∂x − ∂qi) + P, (1)

Lu :=

n∑i=1

λi(p,u)(θxuiθ

qi1 − 1) + P, (2)

where θqz is a shift operator of size z in the variable q.

Lu as defined in (1) is the generator of (Xu,P,Qu) assuming thedynamics of Qu are described by the continuous arrivals model.

Lu as defined in (2) is the generator of (Xu,P,Qu) assuming thedynamics of Qu are described by the Poisson arrivals.


PDE CHARACTERIZATION

Theorem 1

Let v be a real-valued function which is at least once differentiablewith respect to all arguments and satisfies

∂xv > 0, and ∂qiv < 0, ∀ i.

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

0 = ∂tv + supu∈A

Luv, v(T, x, p, q) = ET−

[U(x−

n∑i=1

qiPiT)].

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

u∗s = arg maxu∈A

Luv(s,X∗s ,Ps,Q∗s ).


OUTLINE

INTRODUCTION

Framework


Wealth Maximization



Assumptions(1) Qu is given by the continuous arrival model.

(2) Pt ≡ p ∈ (0, 1)n 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.

(4) λi = λi(ui) 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 onlydepends on its own price ui.


MAIN RESULTSDefine fi(x) := x · λ−1

i (x) for all x > 0 and i. Denote by f theconcave envelope of f .

Theorem 2

Given the previous assumptions, we have

V(t, x, p, q) = supu∈A

Et,x,p,qU

(x−

n∑i=1

qi1Ai + (T − t)n∑

i=1

fi(λi(ui))

− (T − t)n∑

i=1

λi(ui)1Ai

):= V,

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

Remark: We can define Λi := λi(ui) and optimize over Λi.


SKETCH

1. Express the bookmaker’s terminal wealth YuT using

functions fi, and define the “concave envelope version” YuT

by replacing fi with fi. (YuT ≥ Yu

T)2. Show that

supu∈A(t,T)

Et[U(YuT)] = sup

u∈AEt[U(Yu

T)].

Note: the r.h.s. is a constrained problem, namely, the priceprocess is a constant vector.

3. Show thatV(t, x, p, q) ≥ sup

u∈AEt[U(Yu

T)].


EXTRA REMARKS

I A sufficient condition: fi(λi(u∗i )) = fi(λi(u∗i )),where u∗ is the optimizer to V. If fi is concave itself, theabove automatically holds (e.g., λi(x) =

pi1−pi

1−xx ).

I If we further assume (1) Sets (Ai) form a partition of Ω; (2)λi(x) =

pi1−pi

1−xx ; and (3) U(x) = −e−γx, γ > 0, then we have

pi ·(

1(u∗i )2 − 1

)· g(t, pi, qi; u∗i ) =

∑j 6=i

pj · g(t, pj, qj; u∗j ),

where g(t, pi, qi; ui) := exp(γqi + γ(T − t) pi

1−pi

(1ui− 1))

.

We deduce∂u∗i∂qi

> 0 and observe by graph that∂u∗i∂t

> 0 .


OUTLINE

INTRODUCTION

Framework


Wealth Maximization


The Standing Assumption of this section is U(x) = x.Inferring the bookmaker is risk neutral.


METHOD I

Theorem 3

For both continuous and Poisson arrival models, we have

V(t, x, p, q) = x− p · q + Et

∫ T

tsupu∈A

n∑i=1

λi(Ps, u) · (ui − Pis)ds,

where p · q =n∑

i=1piqi.

Proof. By measurable selection theorems.

Transform a dynamic optimization problem over A(t,T) to astatic one overA.


Corollary 4

(i) If λi(p,u) =pi

1−pi

1−uiui

, we obtain

ui,∗t =

√Pi

t ∀ i = 1, · · · ,n.

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

V(t, x, p, q) = x− p · q + (T − t)n∑

i=1

pi

1− pi(1−√pi)

2 .

(ii) If λi(p,u) = log uilog pi

, we obtain ui,∗ as the unique solution on(e−1, 1) to the equation

ui,∗t (1 + log ui,∗

t ) = Pit ∀ i = 1, · · · ,n.


METHOD II (DPP)Assuming regularity, we compute

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

We then use the HJB equation in the PDE characterizationTheorem 1 to obtain the same results as in Theorem 3.

For instance, under the continuous arrival model withλi(p,u) =

pi1−pi

1−uiui

, we simplify the HJB into

∂xV −(

ui,∗s

)−2· ∂qiV = 0,

and obtain, for t ≤ s < T, that

ui,∗s =

√−∂qiV(s,X∗s ,Ps,Q∗s )

∂xV(s,X∗s ,Ps,Q∗s )≡√

Pis ∈ (0, 1).


METHOD IIIRecall Theorem 2 in Section Semi-Static: V(t, x, p, q) = V.

Corollary 5

Let the assumptions of Theorem 2 hold, we obtain

f ′i (Λ∗i ) = pi, (recall Λi = λi(ui)).

In particular, if λi(p,u) =pi

1−pi

1−uiui

, we obtain

Λ∗i =pi

1− pi

(1√pi− 1)

and u∗i =√

pi,

and if λi(p,u) = log uilog pi

,

pΛ∗i

i (1 + Λ∗i log pi) = pi and u∗i (1 + log u∗i ) = pi.


COMPARISON OF METHODS I-III

Prob. P Arrival Q Use Sol. u∗ ProblemI – both wealth X staticII – continuous general X dynamicIII constant continuous general X static

I “Model Generality”: I > II > III

I “Application Scope”: II = III > IMethod I only applies to the wealth max problem.

I “Solutions u∗”: II > I = IIIMethods I and III only characterize the value function.

I “Computational complexity”: I ≈ III < IIThe HJB in Method II is difficult to solve.


PROFITABILITY ANALYSIS

Purpose: study P(Y∗T > 0)

Assumptions: (1) λi(p,u) =pi

1−pi

1−uiui

; (2) Pt ≡ p = (p, 1− p);(3) two mutually exclusive sets A1 and A2; and (4) X0 = Qi

0 = 0.

Case 1: continuous arrival model

Y∗T(Heads) = ψ1(p) · T and Y∗T(Tails) = ψ1(1− p) · T,

where function ψ1 is defined by

ψ1(p) :=p

1− p

(2−

√p− 1√

p

)+

1− pp

(1−

√1− p

).


ψ1 is a decreasing function over (0, 1) with

limp→0

ψ1(p) =12

and limp→1

ψ1(p) = 0.

Conclusion: the bookmaker always makes profits by followingthe optimal price u∗.If p = 0.5 (the coin is fair), the bookmaker’s profit is a constantgiven by Y∗T = ψ1(0.5) · T ≈ 0.171573 · T.

0.0 0.2 0.4 0.6 0.8 1.0

0.1

0.2

0.3

0.4

0.5

Figure 1: Graph of ψ1 over (0, 1)


Case 2: Poisson arrival model

Y∗T(Heads) = (√

p− 1) ·Qu∗,1T +

√1− p ·Qu∗,2

T ,

Y∗T(Tails) =√

p ·Qu∗,1T + (

√1− p− 1) ·Qu∗,2

T ,

where Qu∗,iT (i = 1, 2) are independent Poisson r.v.’s with

expectations given by

λ∗i · T =

√pi(1−√pi)

1− pi· T, where p1 = p, p2 = 1− p.

We can express P(Y∗T > 0) as the sum of two infinite series.

If p = 0.5, we get P(Y∗T > 0) = 33.6747% (if T = 1); 54.4348% (ifT = 2); and 86.4919% (if T = 10).


OUTLINE

INTRODUCTION

Framework


Wealth Maximization



Assumptions

(1) The utility function is U(x) = −e−γx, where γ > 0.The bookmaker is risk averse.

(2) Pt ≡ p ∈ (0, 1)n 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

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

You may assign different parameters κi and βi for differentoutcomes Ai.


HEURISTIC DERIVATION

The HJB associated with the exponential utility max problem is

∂tV(t, x, p, q)+

n∑i=1

supui

λi(ui)[V(t, x+ui, p, q+ei)−V(t, x, p, q)] = 0,

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

a(q) := E exp

(γ

n∑i=1

qi1Ai

).


Theorem 6

The value function V is given by

V(t, x, p, q) = −e−γx [G(t, q)]−1/c ,

where c := βγ and function G is defined by

G(T, q) = [a(q)]−c, G(t, q) =

∞∑k=0

αk(q) · (T − t)k, t ∈ [0,T).

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

ui,∗s = −1

γlog

[β ·H(s,Q∗s )

(β + γ) ·H(s,Q∗s + ei)

], i = 1, · · · ,n,

where H(t, q) := [G(t, q)]−1/c.


REFERENCES IAdrian, 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 limitorder book. Quantitative Finance, 8(3):217–224.

Bayraktar, E. and Ludkovski, M. (2014). Liquidation in limit order bookswith controlled intensity. Mathematical Finance, 24(4):627–650.

Bayraktar, E. and Munk, A. (2017). High-roller impact: A large generalizedgame model of parimutuel wagering. Market Microstructure and Liquidity,3(01):1750006.

Cartea, A. and Jaimungal, S. (2015). Optimal execution with limit and marketorders. Quantitative Finance, 15(8):1279–1291.

Divos, P., Rollin, S. D. B., Bihari, Z., and Aste, T. (2018). Risk-neutral pricingand hedging of in-play football bets. Applied Mathematical Finance,25(4):315–335.

Gatheral, J. and Schied, A. (2013). Dynamical models of market impact andalgorithms for order execution. HANDBOOK ON SYSTEMIC RISK,Jean-Pierre Fouque, Joseph A. Langsam, eds, pages 579–599.

http://dx.doi.org/10.2139/ssrn.2844881


REFERENCES II

Ho, T. and Stoll, H. R. (1981). Optimal dealer pricing under transactions andreturn uncertainty. Journal of Financial Economics, 9(1):47–73.

Hodges, S., Lin, H., and Liu, L. (2013). Fixed odds bookmaking withstochastic betting demands. European Financial Management, 19(2):399–417.

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

THANK YOU!

optimal bookmaking - university of michigan

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