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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

Matt Lorig University of Washington Zhou Zhou University of Sydney (Ph.D. of Erhan from UM)

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

HIGHLIGHTS

◮ Propose a general framework for continuous-time betting markets ◮ Formula an optimal control problem for a bookmaker Control (strategy): price or odds of bet ◮ Two objectives: maximize profit or maximize utility of terminal wealth ◮ Different mathematical techniques ◮ 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

BETTING MARKET IS ENORMOUS!

◮ Zion Market Research: $104.31 (bn) sports betting in 2017, with annual growth rate 9% - 10% ◮ American Gaming Association (AGA): only $5 (bn) legally and at least $150 (bn) illegally in US ◮ 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) ◮ Asia-Pacific: most significant percentage of market shares population: +4 bn ◮ Sports betting is the biggest cake (+40%, still growing) NCAA, NFL, MLB, NBA, Soccer, Golf ... ◮ 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

◮ 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). ◮ American Gaming Association: 97% of the $10 (bn) betting

• n 2018 NCAA men’s basketball tournament was illegal

(3% with NV bookies). ◮ Supreme Court 2018 May ruling: the Professional and Amateur Sports Protection Act (PASPA) unconstitutional ◮ Prior to the PASPA ruling, sports betting was only legal in

• ne state (NV).

◮ 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 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, 2019

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

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

INTUITION ON SETUP

◮ 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.

◮ The bookmaker can set (control) the price (odds) of an

• utcome, say ui of Ai.

The bookmaker cannot directly control the number of bets placed on Ai, denoted by Qi; but the price ui apparently affects Qi (ui ↑ ⇒ Qi ↓). ◮ The bookmaker would like to see that, no matter which Ai

• ccurs, the revenues he collected are sufficient to pay off

the winning bets (ideally with leftovers). ◮ 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

A LESSON

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

COMPLEXITY OF DYNAMIC MARKETS

◮ Traditionally, bookmakers would only take bets prior to the start of a sports game. Hence, the probabilities of

• utcomes remain fairly static.

◮ We allow in-game betting, i.e., bookmakers take bets as the events occur. Now, the probabilities of outcomes evolve stochastically. ◮ 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 G¨

• tze 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)

◮ Hodges et al. (2013): horse race games with fixed winning probabilities (P(Ai) ≡ pi); # of bets Qi ∼ N (Normal);

• ne-period setup; look for u∗ to maximize utility

◮ 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 ◮ 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)

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

◮ Fix a filtered probability space (Ω, F, F = (Ft)0≤t≤T, P), where P as the real world or physical probability measure. ◮ Consider a finite number of outcomes Ai, i = 1, · · · , n, each

• f which finishes at T (Ai ∈ FT).

Note: Ai ∩ Aj = ∅ and ∪Ai = Ω are allowed. ◮ Denote Pi

t = Et[1Ai], Ft-conditional probability of Ai.

Note: Pi is a martingale.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

AN EXAMPLE

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

t : the number of goals scored by time t.

◮ Consider outcome A1, where A1 = {the game will finish with at least one goal}. ◮ We have P1

t = 1{Nµ

t ≥1} + 1{Nµ t =0}(1 − e−µ(T−t)).

◮ The dynamics of P1 can be easily deduced dP1

t = 1{P1

t−<1}e−µ(T−t)(dNµ

t − µdt).

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

STRUCTURE OF BETS

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, “payoff of a bet placed on Ai” = 1Ai = Pi

T.

ui = (ui

t)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 on

• utcome Ai directly.

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

• f bet arrivals.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

BOOKMAKER’S REVENUE

Denote by Xu = (Xu

t )0≤t≤T the total revenue received by the

bookmaker. Let Qu,i = (Qu,i

t )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

• f Xu and Qu on bookmaker’s pricing policy u.

The relationship between Xu, Qu and u is dXu

t = n

• i=1

ui

t dQu,i t .

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

Qu,i

t

= t λi(Ps, us)ds + Qi

0.

• 2. Poisson arrivals

Qu,i

t

= t dNu,i

s

+ Qi

0,

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 ↓.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

EXAMPLES OF RATE/INTENSITY FUNCTIONS

Examples 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 ui

t of a bet on outcome Ai goes to zero, the

intensity of bets goes to infinity lim

ui→0 λi(pi, ui) = ∞.

(ii) As the price ui

t of a bet on outcome Ai goes to one, the

intensity of bets goes to zero lim

ui→1 λi(pi, ui) = 0.

(iii) All fair bets ui

t = Pi t have the same intensity

λi(pi, pi) = λi(qi, qi).

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

BOOKMAKER’S VALUE FUNCTION

Let Yu

T denote the bookmaker’s terminal wealth after paying

• ut all winning bets. (Add Xu

0 − n i=1 Pi TQi 0, if non-zero)

Yu

T = Xu T − n

• i=1

Pi

TQu,i T = n

• i=1

T

• ui

t − Pi T

• dQu,i

t .

Suppose the bookmaker’s objective function J is of the form J(t, x, p, q; u) := E[U(Yu

T)|Xu t = x, Pt = p, Qu t = 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

u∈A(t,T)

J(t, x, p, q; u). Admissible set A(t, T): non-anticipative and us ∈ [0, 1]n.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

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)(θx

uiθqi 1 − 1) + P,

(2) where θq

z 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

Let v be a real-valued function which is at least once differentiable with 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 + sup

ˆ u∈A

Lˆ

uv,

v(T, x, p, q) = ET−

• U
• x −

n

• i=1

qiPi

T

• .

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

s = arg max ˆ u∈A

Lˆ

uv(s, X∗ s , Ps, Q∗ s).

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. (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 only depends on its own price ui.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

MAIN RESULTS

Define fi(x) := x · λ−1

i

(x) for all x > 0 and i. Denote by ˆ f the concave envelope of f.

Theorem 2

Given the previous assumptions, we have V(t, x, p, q) = sup

ˆ u∈A

Et,x,p,qU

• x −

n

• i=1

qi1Ai + (T − t)

n

• i=1

ˆ fi(λi(ˆ ui)) − (T − t)

n

• i=1

λi(ˆ ui)1Ai

• := ˆ

V, where A = [0, 1]n. This is a STATIC optimization problem. Remark: We can define Λi := λi(ˆ ui) and optimize over Λi.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

SKETCH

• 1. Express the bookmaker’s terminal wealth Yu

T using

functions fi, and define the “concave envelope version” ˆ Yu

T

by replacing fi with ˆ

• fi. (ˆ

Yu

T ≥ Yu T)

• 2. Show that

sup

u∈A(t,T)

Et[U(ˆ Yu

T)] = sup ˆ u∈A

Et[U(ˆ Yˆ

u T)].

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

• 3. Show that

V(t, x, p, q) ≥ sup

ˆ u∈A

Et[U(ˆ Yˆ

u T)].

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

EXTRA REMARKS

◮ A sufficient condition: fi(λi(ˆ u∗

i )) = ˆ

fi(λi(ˆ u∗

i )),

where ˆ u∗ is the optimizer to ˆ

• V. If fi is concave itself, the

above automatically holds (e.g., λi(x) =

pi 1−pi 1−x x ).

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

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

pi ·

• 1

(ˆ u∗

i )2 − 1

• · g(t, pi, qi; ˆ

u∗

i ) =

• j=i

pj · g(t, pj, qj; ˆ u∗

j ),

where g(t, pi, qi; ui) := exp

• γqi + γ(T − t)

pi 1−pi

• 1

ui − 1

• .

We deduce ∂ˆ u∗

i

∂qi > 0 and observe by graph that ∂ˆ u∗

i

∂t > 0 .

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 V(t, x, p, q) = x − p · q + Et T

t

sup

ˆ u∈A n

• i=1

λi(Ps, ˆ u) · (ˆ ui − Pi

s)ds,

where p · q =

n

• i=1

piqi.

• 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 λi(p, u) =

pi 1−pi 1−ui ui , 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 ui

log pi , we obtain ui,∗ as the unique solution on

(e−1, 1) to the equation ui,∗

t (1 + log ui,∗ t ) = Pi t

∀ i = 1, · · · , n.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

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 characterization Theorem 1 to obtain the same results as in Theorem 3. For instance, under the continuous arrival model with λi(p, u) =

pi 1−pi 1−ui ui , 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)

≡

• Pi

s ∈ (0, 1).

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

METHOD III

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

Corollary 5

Let the assumptions of Theorem 2 hold, we obtain ˆ f ′

i (Λ∗ i ) = pi,

(recall Λi = λi(ui)). In particular, if λi(p, u) =

pi 1−pi 1−ui ui , we obtain

Λ∗

i =

pi 1 − pi 1 √pi − 1

• and

ˆ u∗

i = √pi,

and if λi(p, u) = log ui

log pi ,

p

Λ∗

i

i

(1 + Λ∗

i log pi) = pi

and ˆ u∗

i (1 + log ˆ

u∗

i ) = 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

• dynamic

III constant continuous general X static ◮ “Model Generality”: I > II > III ◮ “Application Scope”: II = III > I Method I only applies to the wealth max problem. ◮ “Solutions u∗”: II > I = III Methods I and III only characterize the value function. ◮ “Computational complexity”: I ≈ III < II The HJB in Method II is difficult to solve.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

PROFITABILITY ANALYSIS

Purpose: study P(Y∗

T > 0)

Assumptions: (1) λi(p, u) =

pi 1−pi 1−ui ui ; (2) Pt ≡ p = (ˆ

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

0 = 0.

Case 1: continuous arrival model Y∗

T(Heads) = ψ1(ˆ

p) · T and Y∗

T(Tails) = ψ1(1 − ˆ

p) · T, where function ψ1 is defined by ψ1(ˆ p) := ˆ p 1 − ˆ p

• 2 −
• ˆ

p − 1 ˆ p

• + 1 − ˆ

p ˆ p

• 1 −
• 1 − ˆ

p

• .

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

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

ˆ p→0 ψ1(ˆ

p) = 1 2 and lim

ˆ p→1 ψ1(ˆ

p) = 0. Conclusion: the bookmaker always makes profits by following the optimal price u∗. If ˆ p = 0.5 (the coin is fair), the bookmaker’s profit is a constant given 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)

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

Case 2: Poisson arrival model Y∗

T(Heads) = (

• ˆ

p − 1) · Qu∗,1

T

+

• 1 − ˆ

p · Qu∗,2

T

, Y∗

T(Tails) =

• ˆ

p · Qu∗,1

T

+ (

• 1 − ˆ

p − 1) · Qu∗,2

T

, where Qu∗,i

T

(i = 1, 2) are independent Poisson r.v.’s with expectations given by λ∗

i · T =

√pi(1 − √pi) 1 − pi · T, where p1 = ˆ p, p2 = 1 − ˆ p. We can express P(Y∗

T > 0) as the sum of two infinite series.

If ˆ p = 0.5, we get P(Y∗

T > 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. (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 different

• utcomes Ai.

INTRODUCTION Framework Semi-Static Wealth Exp Utility References

HEURISTIC DERIVATION

The HJB associated with the exponential utility max problem is ∂tV(t, x, p, q)+

n

• i=1

sup

ui

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

• .

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 , where c := β

γ 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 γ log

• β · H(s, Q∗

s)

(β + γ) · H(s, Q∗

s + ei)

• ,

i = 1, · · · , n, 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

• rder 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
• rders. 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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