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!).

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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.

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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!

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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:

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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:

  1. 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.
  2. 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.
  3. 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!

Posted by Sophie Maclean

Sophie Maclean is a mathematician and maths communicator, currently studying for a PhD in Analytic Number Theory at Kings College London. She has previously worked as a Quantitative Trader and a Software Engineer, and now gives mathematics talks all over the UK (and Europe!). She is also a member of the team behind Chalkdust Magazine. You can follow her on Twitter at @sophietmacmaths

Source: Spektrum.de