This article is based on “The parable of the bookmaker” topic from “Financial Calculus: Introduction to Derivative Pricing” by Baxter and Rennie
Bookmaker is a person who sets odds, takes bets and pays out the winnings. But how do you set odds in an event – for example, a horse race between horse A and horse B?
Following the intuition:
Let’s say you look at the two horses, their training, diet and the choice of jockey and you take an educated guess of the winning chances of both the horses as below:
| Horse | Winning Probability |
|---|---|
| A | 25% |
| B | 75% |
So, you decide to keep the odds in the same ratio. 3-1 against for horse A and 3-1 on for horse B.
People come to you and place their bets. In total, people have bet a total of 5k on horse A and 10k on horse B. You have 15k in total.
So, there are two scenarios here: either horse A wins or the horse B wins. How much do you get to keep? Intuitively, 0 because you kept the same ratio as the winning probability. Let’s calculate:
| Winning Horse | Payout | Profit / Loss |
|---|---|---|
| A | 5k * 3 + 5k = 20k | -5k |
| B | 10k * (1/3) + 10k = 13333 | +1667 |
But, since B has three times the chance of winning compared to A, the expectation of Profit / Loss is 0. The problem with looking at long term is that you can surely go bankrupt if all the losses come at you first and then profits follow it.
In the long term, you are expected to not make any money as a book maker. You have to change the odds a little bit in order to make money. You reduce the payout, in that way, you lose less than 5k when horse A wins. In the long term, you are expected to make money. But wait, we can change the odds further.
What if you set odds in such a way that you make money in each game irrespective of horse A or B winning? This is where things get interesting.
You can calculate odds such that you make 1k$ in profit irrespective of horse A or horse B winning. In order to do that, you need to limit the payout to 14k$ in either cases.
| Winning Horse | Payout | Odds |
|---|---|---|
| A | 5k * = 14k | 9-5 against |
| B | 10k * = 14k | 5-2 on |
So, in order to make a profit of 1k irrespective of who is winning, you need to give the above Odds to betters.
Why is this important?
This exercise gives an important lesson to anyone starting out in quantitative finance. You are going to hear a lot of jargons from measure theory such as “Risk Neutral Measure” which sounds very fancy but at the crux of it, the concept remains the same as above. Its not enough to make profit only in the long term. You need to make sure shot profit in each game.