The Option Principles of Fourth Down Calls
Malt Liquidity 118
In the past, I’ve talked about the “gamma sensitivity” of live football odds:
Think about what happens to an option as it approaches the money (a 50/50 whether it will expire in the money or not). The gamma of the option goes to infinity as the time to expiry goes to zero, as the tick right before settlement could be the tick that pushes the option ITM or OTM.
A money line bet can work the same way. “At the money” is the over/under, the spread to cover, or any other binary bet. Ideally, the bookie doesn’t want to take any risk on the event - the “spread” they charge is the vig, which is why an even bet pays -105 instead of 1:1. However, this is in an ideal world, and even in highly liquid bets - like the Super Bowl line - you aren’t going to ever be guaranteed an equally timed even amount of money flowing in on both sides. Thus the line shifts towards which side is being bet on more as a “dynamic hedge” of sorts to disincentivize the continued flow of people betting on one team rather than the other. The only difference is, unlike an order book, you can’t see the flow of bets unless you are the bookie itself.
But nowadays, there are live odds as well, which are far more interesting, because the flow in, say, a football game can only exist between plays. The live money line is ideally a 50/50 line that expires when the ball is snapped, which is within 40 seconds. The real time shifting of the line pre-snap is the same as gamma sensitivity of an ATM option right at expiry! So if a ton of flow all of a sudden comes in on the Chiefs after Mahomes get sacked on 3rd down, the live money line has to shift extremely rapidly to adjust. Thus, you might get better odds than the “true 50/50” as the bookie has had to impact the line based on the incoming bets. Sound familiar? It should, because that’s a fundamental movement of options.
But I was watching football this weekend and came up with an idea — what if we treated the game itself as an option?
Data analysis has certainly optimized sports. Standard statistical methods have long been used to predict and strategize in baseball, and the quant rush of the past decade brought this to many other sports, such as basketball, which has perfected the min-maxxing of expected value to the point that the game bores me to tears. Football hasn’t been resistant to this either — but one particular flaw I see comes with fourth down decisions specifically. Often, going for it in unintuitive situations comes with the justification of the “win probability” being higher if one decision is made rather than the other. This, at first sight, checks out — obviously if one action has a higher expected value than another, it makes sense to do it. But as any gambler would know, just because the count is good doesn’t mean you bet your entire bankroll on the next hand of blackjack — you need to manage the risk of ruin in the case the adverse outcome happens so you have enough money to win in the long run.
If you watched the Chargers-Raiders game this weekend, you might have noticed a particularly bizarre playcall where, up 24-17, they elected to go for a 4th & 1 on their own 34 yard line.
Assuredly, the “win probability” model they use probably spit out that going for it gave them the highest chance of winning, so the decision was made. But is that all there is to it? When they missed the conversion, look at what happened to the forward win probability:
A game which they had a 95%+ chance of winning suddenly started ratcheting down due to the opportunity afforded by missing this fourth down. Now, most people critiquing these decisions (and justifying them!) use post-hoc rationalization — “you wouldn’t have lost if you didn’t go for it”, “we won anyway”. This is wrong. However, there is a critique to be made here, deriving from near-expiry options trading.
What exactly is the game state of a football match? You have the score, the placement of the ball on the field, and the possession and timing of the game state. What if we combine these elements and think of them as a European (no early exercise) binary option expiring at the end of the game? Proper management of this position would not just be “maximize the odds it expires in the money”, but would also require taking into account how to manage the position should those odds shift very rapidly. Think back to the blackjack example from earlier — intuitively this should lead us to some sort of Kelly Criterion-esque thinking. It’s not “do I win if I convert this play”, but rather “how much am I affecting my ability going forward to still end up winning the game?”
Therefore, we need some sort of multi-step model in this situation. If I have a 95% chance of converting this 4th and 1 and winning the game, but in the 5% chance I don’t, my win probability drops to 60% with very little time remaining in the game, I have introduced the possibility of massive variance in the moneyness of my option. Whereas if I punt the ball with an 88% chance of winning the game going forward, subsequent plays might only take this down to 75%, slowly dwindling away my moneyness as the clock winds down. This is not an expected value calculation — rather, it is gamma management of the game state. Remember, gamma ramps up exponentially as the game expires in a tie game, similar to how rapidly options shift as they reach their expiry. The “delta sensitivity” here is what these win probability models are assessing — what is the rough odds that I expire in the money? — but I am positive that they’re not accounting for gamma when they make these decisions, particularly because it is individually dependent on the opponent, the conditions, and the game flow. You simply cannot use an all-encompassing expected value model to make individual decisions, because anything can happen in an n=1 circumstance; that’s the entire appeal of watching a game. I can predict with a 90% certainty what band Aaron Judge’s season stats will fall into, but I have no idea what he will do game to game.
Compare the Chargers decision to an infamous example of going for it in your own territory — when Bill Belichick went for it on 4th & 2 from his own 28 up 34-28 against the Colts.
Perhaps the win probability said this was the correct decision, but that’s not my concern here. What are the second order effects here applied in this individual circumstance?
The biggest argument for going for it here, beyond analytics, is that they’d be giving the ball to Peyton Manning, one of the greatest quarterbacks of all time. The offense was clearly functioning well given how the game had progressed — the Patriots were up 31-14 at the beginning of the quarter, but the Colts had stormed back to 34-28 in the situation above. Belichick’s gut decision showed an intuitive understanding of “football gamma” here — beyond the raw win probability, he figured that there wasn’t much difference between giving the ball to Manning in Colts territory vs Patriots territory in terms of how it would impact forward win probability, so he took the opportunity to win the game on his own terms. The moneyness, in a sense, would remain in a state of high flux either way, so all-inning made sense. Of course, they would miss the conversion and the Colts would win 35-34 — but the result doesn’t really matter here. It was probably the correct decision, but who knows? Being right doesn’t necessarily mean making money — being right doesn’t necessarily mean winning the game.
Meanwhile, when we look at the Chargers situation from this past weekend, consider that they were playing a backup rookie QB making his first start. Prior to getting the ball back, he had conducted two successful drives of 60 yards for touchdowns — seven other possessions had ended in short drives resulting in punts, or a turnover. Is giving the ball to Aidan O’Connell the same as giving the ball to Peyton Manning? If a coach isn’t taking this kind of context into decisions, he’s fundamentally misunderstanding how to use analytics relative to actual football knowledge. Of course, on the goal line, the Chargers intercepted a pass and won anyway. But does the option expiring in the money here justify the very rapid shift in moneyness taken? In a trading context, this can be assessed and incorporated in a strategic sense — we have well-defined risk and position-management metrics we can use. I think it could be in a sports context as well. There is room for debate here, but it’s not “the win probability said to do it” or “the Chargers got lucky and won anyway”. Rather, it takes the form of when is the risk justified and how close to the end of the game does it make sense to increase the size of the bets we take?
If you want a good way to practice trading zero day options without having to size up, live odds are your friend. Placing and managing bets on fluctuating moneylines and spreads is an incredibly similar low-stakes simulation of some of how zero-day trading works. Sports have become vehicles to push gambling in this day and age — maybe they should take some lessons from it as well. Live odds probably depict a game state much better than an independent win probability model because it reflects the real time market approximation of the game state as priced by a binary option.
If any reader is associated with the Dallas Cowboys, please forward this to Mike McCarthy — I swear, traders should make fourth down decisions, not head coaches, statisticians, or offensive coordinators. I’ll do it for a box seat, I swear!