Football Finance, Part 2
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The 2026 Men’s Football FIFA World Cup is now well underway, and what a busy tournament it has been! We have had plenty of VAR drama, surprise upsets from Cape Verde, and even bagpipes in Boston. But I am not here for a deep dive on the footie. I am, once again, here to talk maths.
In my last blog post, we looked at stocks and futures, and if you have not read it yet, I recommend taking a look now. For the sake of anyone needing a recap, futures are financial products in which the buyer agrees to pay the seller a fixed price at a specified day in the future. To see how we might estimate the price of a future, we took a sidestep into the world of sports betting, which is surprisingly good training for stock trading.
At this point, I do have to repeat the same disclaimers as before: Neither myself nor the HLFF endorses betting and none of this blog post should be interpreted as advice (financial, betting, or otherwise!).
Last month, we explored how stock market traders might predict the number of goals scored per minute during the Mexico vs South Africa match. We estimated it would be between 0.02277 and 0.03762, so, naturally, our first point of business is to learn what actually happened. Drum roll please!
The match, including injury time, lasted 101 minutes, which is exactly the value we used in our calculations, based on an average historic match time of 100 minutes and 36 seconds. So that is one tick already! Now for the number of goals scored, we estimated it to be between 2.3 and 3.8 on average. This is where, unfortunately, thing went awry. The final score was 2-0 to the hosts, Mexico, so 2 goals were scored in total.
Where did we go wrong? Well, predicting there will be between 2.3 and 3.8 goals means that there must be exactly 3 goals for our estimate to be correct – there is no wiggle room at all! Had we considered the variance of the number of goals scored, as well as the mean, we would have seen that 2 or 4 goals are also quite likely, and so we would have done better with an estimate of between 2 and 4 goals.
We can plot the number of goals per minute as the game progresses! This not only shows how close we were but also how big of an impact each goal had (I will not insult your intelligence and ask you to guess when the goals were scored).
This graph shows clearly that each individual minute of extra time does not have much impact on the goals scored per minute, but each individual goal really does!
A New Month, a New Game
Now it is a new month, the tournament has progressed, so let us mix things up and play a different game between us. We may as well go big, so let us gamble with what happens in the final of the tournament. Just as we did last month, before the match, you are going to pay an “entry fee” to play this game. Then at the end of the match I will give you your winnings, and you are hoping you win more than you paid!
We are not gambling on the goals per minute this month either. This month’s game has new rules. Once you have paid your entry fee (but before the start of the match), we toss a coin to choose one of the two teams. Then, I will agree to pay you the number of Euros equal to the percentage of time that team has the ball. This is equivalent to a futures contract, just like last week.
This seems like a fairly complicated thing to bet on, but we can actually immediately infer how much we can expect to win, on average. Because we toss a coin to pick the team, we are equally likely to choose the team with the higher, or lower percentage of possession. On average, the answer will be about 50% and you can expect to get €50. So, you definitely shouldn’t pay more than a €50 entry fee!
We can plot the amount of money you expect to profit (your winnings minus what you paid) vs the actual percentage possession on a graph. Let us say that you paid €K to play:
This month we are stepping things up a notch though. I will allow you to be the game master. When you are the game master, the game is the same but now I pay the entry fee, and after the match you pay me the percentage possession.
So, what entry fee will you charge me? This time, you should not charge me less than €50. We can plot your expected earnings as game master on a graph too:
This is now modelling you selling a future, whereas before we were specifically modelling you buying a future. If you have noticed the symmetry between these two graphs, that is no coincidence! Whatever profit the buyer makes, the seller loses and vice versa. The total money in the system stays the same, so at any spot, the profits of the two graphs always sum 0. Ergo, symmetry!
There’s Another Option…
By now, you may be getting to grips with this game. It is just a bet, really. So, this is where we shake things up a bit.
Let us suppose that today I take a deposit from you. In return, once the match is over, I will give you the opportunity to play the game for an entry fee of €50, but you can decide then, after having seen the score, if you will take me up on that offer. What deposit will you pay?
This suddenly feels a lot more complicated. If the possession ends up being less than 50%, you should not take up the opportunity to play. You will lose the deposit, but if you play then you will lose even more money. On the other hand, if the possession is more than 50%, you should choose to play the game – it will earn you some money – but you will not immediately recoup your deposit.
This is fiddly in words, so let us draw another graph!
The amount the initial line is below the axis corresponds to the deposit you paid. Just as before, you want to, on average, at least break even. So, you want to calculate the maximum deposit you can pay in whilst still, on average, breaking even. Tricky, huh?
As before, we can also swap things around and make you game master. But wait! That can mean three different things here:
- I take a deposit from you and at the end of the match you have the right (but not obligation) to make me play the game for €50.
- You take a deposit from me and at the end of the match I have the right (but not obligation) to play the game for €50.
- You take a deposit from me and at the end of the match I have the right (but not obligation) to make you play the game for €50.
Once again, let’s head over to the graphs to make sense of this!
For number 1, because you don’t have the obligation to make me play the game, once again, your losses are capped. But this time, you would prefer the possession percentage to be low. This graph looks as follows:
In numbers 2 and 3, now my losses are capped, so unfortunately for you, your profits are capped. In fact, if you think hard enough, you can see that 2 is the same as selling me the original game – it’s a total role reversal. So the graph of 2 will be the mirror image of our original graph, in the same way that our futures graphs were mirrored.
Similarly, the graph of 3 is the mirror of the graph of 1:
Back to Trading
So, what is the trading equivalent of this? In trading lingo, the right but not obligation to buy or sell a stock at a future, fixed time for a fixed price is called an ‘Option Contract,’ or just an ‘Option.’
The right to buy the stock is equivalent to our opportunity to play the game and is called a ‘call option’ (or, simply a ‘call’). The right to sell the stock is equivalent to the right to make someone else play the game and is called a ‘put option’ (or a ‘put’).
How best to price call and put options, is something both mathematicians in universities, and also quantitative traders in companies call hedge fund research. In the real world, you don’t just want to work out the average earnings either – it is also important to think about what you can persuade someone else to buy or sell you a contract for.
All methods to work out option pricing rely on mathematical models – simplifications of the real world to enable calculations. The most famous model for options pricing is the Black-Scholes model.
The Black-Scholes model considers the same variables we considered when pricing our futures contracts in last month’s blog:
- t: time
- S: the current price of the stock
- r: the interest rate.
But it also introduces a new variable, the volatility σ. This is a measure of how much the stock price varies in a given amount of time – it can be thought of as a measure of variance. Putting this all together Fischer Black, Myron Scholes, and Robert Merton (whose name is often left out of the title) derived the following partial differential equation for the price of an option, V:
The price at which you will buy or sell the stock, known as the ‘strike’ (and equivalent to our €50 above), is then used as a boundary condition when solving this equation, along with the time at which the trade will take place in the future (known as the expiry or, in our case, the end of the match).
And voila! We have a price for our option, albeit after a heavy amount of calculus.
More Than One Way to Be a Mathematician
Believe it or not, this is only dipping our toes into the world of financial mathematics. There are more arguments to be had over the best ways to price options. There are more complicated financial products out there. Many traders will also buy and sell calls and puts in specific combinations, to get a Franken-graph that earns them money under increasingly elaborate conditions.
One of the most exciting things about financial mathematics is that it isn’t just being done in academic institutions – real maths is being researched in the companies the world over, and this could provide lessons in how academia and industry can work together.
So good luck to all teams still in the World Cup, and I can’t wait for the final!
Stock markets are inflation bubbles like any other.
A carrot attracts bunnies, bunnies attract more bunnies, many bunnies attract many wolves. Since more and more bunnies have to fight over less and less carrot, it’s barely possible to distinguish a bunny from a wolf, which makes a lot of bunnies believe they are wolves.
When there’s not enough carrot, the wolves start selling fake carrots – IOUs, promises, just to attract more bunnies. They get the more successful the less actual carrot there is, as the bunnies get hungrier and hungrier, more desperate and blinded by wishful thinking.
So for the wolf, the game is about keeping the bunnies at the poker table, which means on the dinner table. For the average bunny, it’s about grabbing as much carrot as it can while it still can, then getting away while it’s still winning, before the carnage escalates and the bubble pops in blood and panic.
If you’re a farmer, you work on your farm and spend your evenings in the saloon. If you’re successful, you get hands to do all the work and spend all your time in the saloon. The more successful you are, the more surplus you produce, the more the bar grows into Vegas, the less time you spend at the farm, the less you care about it, the more your happy-go-lucky gambler’s decisions, driven by booze and cocaine and greed and imaginary gold rush, ruin it, the less carrot you bring to Vegas, the less carrot there is to win.
Money is an IOU from the Good Fairy. If no one wants to be the Good Fairy, only the weakest who can be forced to are, and the weakest at the bottom don’t produce enough carrot to feed an ever-growing bubble, getting stronger and stronger, hungrier and hungrier day by day – a cube goes vampire on a square, you need more and more square and less and less cube to make it grow, only the strongest survive, the winner takes it all, the cube becomes first a pyramid, then a witch’s hat, then a drawing pin, a golden winner tower with a broad area of loser slums around it.
If everyone just wants to make money, that’s what everyone does – you get inflation (lots of IOUs for nothing), debts (IOUs for IOUs for nothing) and exploding stock markets (IOUs for an infinity of IOUs for nothing). Which means, you see more and more desperation to sell people fake carrots with an ever bigger show, ever bigger promises. If they promise you infinity, they don’t bother about maths, they don’t give a fuck about reality, it’s just religion, they know they’ll never have to keep their promises, because you’ll be too dead to make them. It’s reverse quantification, countable flesh evaporates into a shapeless fog of infinite possibilities.
You see the middle class disappearing from the witch’s hat. You see the world’s largest economies telling their citizens to live off IOUs, because they can’t produce enough carrot to keep their countries in Vegas business. Everybody is grabbing their colt, accusing each other of cheating and preparing to knock over the table. The bubble is about to pop.
It’s up to the individual bunny to decide how much risks it’s going to take. But be cautious and be prepared to lose everything. Be aware that you’re an moron who can’t tell gambling from farming and your reaction to losing will be that of an moron.
Money does not exist. Any serious damage resulting from losing a game is done by people who think they’ve lost something that they never had, by violence in the physical world. Do not jump out of the window at Black Tuesday, just log out of your Windows. It’s just the Matrix, a fantasy world where planet Earth has turned into a desert and Mad Max and the zombies are fighting over the last oases. If you treat this fantasy like reality, you’ll make it reality within a few decades.
Do not bring a howitzer to a football game. Do not join forces with the other team to get rid of rules and referee, because you think he’s getting in the way. Yes he is, his job is to keep football football and block the escalation into war. If there’s a winner and the bubble collapses, just restart the game.
You gamble at your own risk. You are the only risk.