Score-Difference Market Making Under Adverse Selection
Consider a "score-difference" market for a basketball game. Let $X$ be the final score difference (home minus away), modeled as $X \sim N(\mu, \sigma^2)$.
You are a market maker in a contract that pays $X$ at settlement.
- Under risk-neutral pricing, what is the fair midprice of this contract? How does it relate to $\mu$?
- You quote a symmetric spread of width $s$ around the mid, so your bid is $\mu - s/2$ and your offer is $\mu + s/2$. Suppose one unit of flow hits you on either side with equal probability. However, informed traders only trade when their private signal exceeds a threshold, effectively shifting the distribution of $X$ you face to $N(\mu + \delta, \sigma^2)$ when a buyer arrives and $N(\mu - \delta, \sigma^2)$ when a seller arrives. Ignoring inventory accumulation across games, derive your expected profit per game as a function of $s$, $\delta$, and $\sigma$.
- Explain qualitatively how you would choose $s$ to balance spread capture against adverse selection as $|\delta|$ grows.
Hints
- For a contract with a linear payoff, risk-neutral pricing is straightforward -- what is the expected value of $X$?
- When computing expected profit, think about what happens on each side of the market separately: what do you collect on the trade, and what is the true expected value of $X$ conditional on who is trading with you?
- The adverse selection cost is exactly $\delta$ per trade because informed flow shifts the conditional mean by $\pm \delta$ away from your midprice. Your net profit per trade is $s/2 - \delta$.
Worked Solution
How to Think About It: This is a classic market-making economics problem dressed up in a sports betting context. The contract is linear in $X$, so risk-neutral pricing is trivial -- the fair price is just the expected value. The interesting part is part (b): when you quote a spread, you earn $s/2$ on every fill, but you lose money to informed traders who know which way $X$ is shifted. The question is how much adverse selection eats into your spread capture.
Quick Sanity Checks: The midprice should just be $\mu$ (linear contract, risk-neutral means price = expectation). Profit should be positive when $s$ is large relative to $\delta$ and negative when $\delta$ dominates. As $\delta \to 0$ (no informed traders), profit should be $s/2$ per trade.
Part (a): Fair Midprice
Under risk-neutral pricing, the fair price of a contract paying $X$ at settlement is simply:
$V = E[X] = \mu$
The midprice equals the mean of the score-difference distribution. This is immediate because the payoff is linear in $X$ -- no convexity adjustments, no optionality. If you believe the home team wins by 5 points on average, the contract is worth 5.
Part (b): Expected Profit Per Game
You quote bid $= \mu - s/2$ and offer $= \mu + s/2$. With probability
When a buyer hits your offer: You sell one unit at $\mu + s/2$. But the buyer is informed, so the true distribution of $X$ given buyer arrival is $N(\mu + \delta, \sigma^2)$. Your expected P&L on this trade is:
$\text{P\&L}_{\text{sell}} = (\mu + s/2) - E[X \mid \text{buyer}] = (\mu + s/2) - (\mu + \delta) = s/2 - \delta$
When a seller hits your bid: You buy one unit at $\mu - s/2$. The true distribution given seller arrival is $N(\mu - \delta, \sigma^2)$. Your expected P&L is:
$\text{P\&L}_{\text{buy}} = E[X \mid \text{seller}] - (\mu - s/2) = (\mu - \delta) - (\mu - s/2) = s/2 - \delta$
Both sides give the same expected P&L (this is a consequence of the symmetric setup). Taking the expectation over buyer/seller arrival:
$E[\text{Profit per game}] = \frac{1}{2}(s/2 - \delta) + \frac{1}{2}(s/2 - \delta) = \frac{s}{2} - \delta$
The spread capture is $s/2$ per trade, and adverse selection costs $\delta$ per trade.
Part (c): Choosing $s$ as $|\delta|$ Grows
The expected profit is $s/2 - \delta$, so to remain profitable you need $s > 2\delta$. As $|\delta|$ grows:
- You must widen the spread to compensate for increased adverse selection. At minimum, $s > 2\delta$ to avoid losing money.
- But widening the spread has a cost: you trade less frequently. Uninformed flow (noise traders, hedgers) is sensitive to spread width. As $s$ grows, fewer uninformed counterparties are willing to cross your spread, so your volume drops.
- The optimal $s$ balances these two forces. You want $s$ large enough that $s/2 > \delta$ (positive expected profit per trade), but small enough to attract enough flow to make the business worthwhile.
- In practice, as $\delta$ gets very large relative to $\sigma$, the market becomes dominated by informed traders and it may not be profitable to make a market at all. This is the classic "market breakdown" result from adverse selection theory (Glosten-Milgrom).
- A more sophisticated approach would model the flow arrival rate as a decreasing function of $s$ and optimize $s$ to maximize total expected profit $= (\text{flow rate}(s)) \times (s/2 - \delta)$.
Answer: (a) The fair midprice is $\mu$. (b) Expected profit per game is $s/2 - \delta$, where $s/2$ is the half-spread capture and $\delta$ is the adverse selection cost. (c) Widen $s$ as $|\delta|$ grows to maintain profitability, but recognize that wider spreads reduce flow -- optimize the trade-off by maximizing $(\text{flow rate}) \times (s/2 - \delta)$, and exit the market if $\delta$ is too large for any spread to be profitable.
Intuition
This problem captures the fundamental tension in market making: you earn the spread on every trade, but you lose to people who know more than you. The score-difference setup makes it concrete -- if the home team's star player is injured (and you don't know yet), buyers of the away side are trading on real information, and the distribution of $X$ that hits your bid is worse than the unconditional one. Your spread is the buffer that protects you.
The key insight that carries over to real trading desks is that adverse selection is not about losing on every trade -- it is about the conditional expectation of the asset given that someone chose to trade with you. This is the "winner's curse" of market making. The Glosten-Milgrom framework formalizes this: the bid-ask spread exists precisely to compensate market makers for being picked off by informed traders. When information asymmetry grows (larger $\delta$), spreads must widen or markets break down entirely. This is why you see spreads blow out around earnings announcements, Fed decisions, or -- in this setting -- right before tip-off when late-breaking injury news drops.