Optimal Bookmaking Bin Zou University of Connecticut - PowerPoint PPT Presentation

I NTRODUCTION 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

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

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References H IGHLIGHTS ◮ 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

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References OUTLINE I NTRODUCTION Framework Analysis of the Semi-static Setting Wealth Maximization Exponential Utility Maximization

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References B ETTING 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

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References R EGULATIONS 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 on 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 one state (NV). ◮ Now, the number is 13 and still counting, with a dozen more states in serious consideration (including MI).

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References S PORTS B ETTING IN 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/

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References I NTUITION ON SETUP ◮ One can bet on n outcomes of a game, A i , i = 1 , · · · , n , e.g., n = 2, A 1 = { win } and A 2 = { lose } . Note. ( A i ) do NOT need to be a partition of Ω . ◮ The bookmaker can set (control) the price (odds) of an outcome, say u i of A i . The bookmaker cannot directly control the number of bets placed on A i , denoted by Q i ; but the price u i apparently affects Q i ( u i ↑ ⇒ Q i ↓ ). ◮ The bookmaker would like to see that, no matter which A i occurs, the revenues he collected are sufficient to pay off the winning bets (ideally with leftovers). ◮ To balance the book, dynamically adjusting the price u i may be necessary. Question: Is there an optimal price u ∗ to the bookmaker?

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References A L ESSON 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.

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References C OMPLEXITY OF DYNAMIC MARKETS ◮ Traditionally, bookmakers would only take bets prior to the start of a sports game. Hence, the probabilities of outcomes 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¨ otze in World Cup 2014 Final (7 minutes to the end) nearly put the odds to 1.

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References L ITERATURE ( MORE RELATED ) ◮ Hodges et al. (2013): horse race games with fixed winning probabilities ( P ( A i ) ≡ p i ); # of bets Q i ∼ N (Normal); one-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

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References L ITERATURE ( 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)

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References OUTLINE I NTRODUCTION Framework Analysis of the Semi-static Setting Wealth Maximization Exponential Utility Maximization

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References P ROBABILITY P REPARATION ◮ Fix a filtered probability space (Ω , F , F = ( F t ) 0 ≤ t ≤ T , P ) , where P as the real world or physical probability measure. ◮ Consider a finite number of outcomes A i , i = 1 , · · · , n , each of which finishes at T ( A i ∈ F T ). Note: A i ∩ A j � = ∅ and ∪ A i � = Ω are allowed. ◮ Denote P i t = E t [ 1 A i ] , F t -conditional probability of A i . Note: P i is a martingale.

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References A N EXAMPLE ◮ Suppose the goals scored during a soccer game arrive as a Poisson process with intensity µ . Notation N µ t : the number of goals scored by time t . ◮ Consider outcome A 1 , where A 1 = { the game will finish with at least one goal } . ◮ We have P 1 t = 0 } ( 1 − e − µ ( T − t ) ) . t = 1 { N µ t ≥ 1 } + 1 { N µ ◮ The dynamics of P 1 can be easily deduced t − < 1 } e − µ ( T − t ) ( d N µ d P 1 t − µ d t ) . t = 1 { P 1

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References S TRUCTURE OF BETS We assume that a bet placed on outcome A i pays one unit of currency at time T if and only if ω ∈ A i ( A i occurs). Namely, “payoff of a bet placed on A i ” = 1 A i = P i T . u i = ( u i t ) 0 ≤ t < T denotes the price set by the bookmaker of a bet placed on A i . Let u = ( u 1 , · · · , u n ) . The bookmaker cannot control the number of bets placed on outcome A i directly. However, the bookmaker can set the price of a bet placed on A i and this in turn will affect the rate or intensity at which bets on A i are placed. Generally, higher prices will result in a lower rate or intensity of bet arrivals.

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References B OOKMAKER ’ S REVENUE Denote by X u = ( X u t ) 0 ≤ t ≤ T the total revenue received by the bookmaker. Let Q u , i = ( Q u , i t ) 0 ≤ t ≤ T be the total number of bets placed on a set of outcomes A i , given price u . Note that we have indicated with a superscript the dependency of X u and Q u on bookmaker’s pricing policy u . The relationship between X u , Q u and u is n � t d Q u , i d X u u i t = t . i = 1 Observe that X u and Q u , i are non-decreasing processes for all i .

I NTRODUCTION Framework Semi-Static Wealth Exp Utility References T WO ARRIVAL MODELS We consider two models for Q u , i : 1. Continuous arrivals � t Q u , i λ i ( P s , u s ) d s + Q i = 0 . t 0 2. Poisson arrivals � t Q u , i E t d N u , i d N u , i + Q i = 0 , = λ i ( P t , u t ) d t . t s t 0 We will refer to the function λ i as the rate or intensity function. In general, the function λ ( p , u ) could depend on vectors p and u prob. p = ( p 1 , p 2 , . . . , p n ) , price u = ( u 1 , u 2 , . . . , u n ) . We expect (1) p i ↑ ⇒ λ i ↑ and (2) u i ↑ ⇒ λ i ↓ .