Suppose you’re in the derivatives business. You are interested in making a market on some events; say, whether or not your friend Jay will win tomorrow night’s poker game, or that the winning pot will be at least USD 100. Let’s examine some rules about how you should do business if you want this venture to succeed.
What do I mean by ‘make a market’? I mean that you will be willing to buy and sell units of a particular security that will be redeemable from the seller at some particular value after tomorrow’s poker game has ended (you will be making a simple prediction market, in other words). You can make bid offers to buy securities at some price, and ask offers to sell securities at some price.
To keep things simple let’s say you’re doing this gratis; society rewards you extrinsically for facilitating the market — your friends will give you free pizza at the game, maybe — so you won’t levy any transaction fees for making trades. Also scarcity isn’t a huge issue, so you’re willing to buy or sell any finite number of securities.
Consider the possible outcomes of the game (one and only one of which must occur):
- (A) Jay wins and the pot is at least USD 100.
- (B) Jay wins and the pot is less than USD 100.
- (C) Jay loses and the pot is at least USD 100.
- (D) Jay loses and the pot is less than USD 100.
The securities you are making a market on pay USD 1 if an event occurs, and USD 0 otherwise. So: if I buy 5 securities on outcome A from you, and outcome A occurs, I’ll be able to go to you and redeem my securities for a total of USD 5. Alternatively, if I sell you 5 securities on outcome A, and outcome A occurs, you’ll be able to come to me and redeem your securities for a total of USD 5.
Consider what that implies: as a market maker, you face the prospect of making hefty payments to customers who redeem valuable securities. For example, imagine the situation where you charge USD 0.50 for a security on outcome A, but outcome A is almost certain to occur in some sense (Jay is a beast when it comes to poker and a lot of high rollers are playing); if your customers exclusively load up on 100 cheap securities on outcome A, and outcome A occurs, then you stand to owe them a total payment of USD 100 against the USD 50 that they have paid for the securities. You thus have a heavy incentive to price your securities as accurately as possible — where ‘accurate’ means to minimize your expected loss.
It may always be the case, however, that it is difficult to price your securities accurately. For example, if some customer has more information than you (say, she privately knows that Jay is unusually bad at poker) then she potentially stands to profit from holding said information in lieu of your ignorance on the matter (and that of your prices). Such is life for a market maker. But there are particular prices you could offer — independent of any participant’s private information — that are plainly stupid or ruinous for you (a set of prices like this is sometimes called a Dutch book). Consider selling securities on outcome A for the price of USD -1; then anyone who buys one of these securities not only stands to redeem USD 1 in the event outcome A occurs, but also gains USD 1 simply from the act of buying the security in the first place.
Setting a negative price like this is irrational on your part; customers will realize an arbitrage opportunity on securities for outcome A and will happily buy as many as they can get their hands on, to your ruin. In other words — and to nobody’s surprise — by setting a negative price, you can be made a sure loser in the market.
There are other prices you should avoid setting as well, if you want to avoid arbitrage opportunities like these. For starters:
- For any outcome E, you must set the price of a security on E to be at least USD 0.
- For any certain outcome E, you must set the price of a security on E to be exactly USD 1.
The first condition rules out negative prices, and the second ensures that your books balance when it comes time to settle payment for securities on a certain event.
What’s more, the price that you set on any given security doesn’t exist in isolation. Given the outcomes A, B, C, and D listed previously, at least one must occur. So as per the second rule, the price of a synthetic derivative on the outcome “Jay wins or loses, and the pot is any value” must be set to USD 1. This places constraints on the prices that you can set for individual securities. It suffices that:
- For any countable set of mutually exclusive outcomes E1, E2, …, you must set the price of the security on outcome “E1 or E2 or..” to exactly the sum of the prices of the individual outcomes.
This eliminates the possibility that your customers will make you a certain loser by buying elaborate combinations of securities on different outcomes.
There are other rules that your prices must obey as well, but they fall out as
corollaries of these three. If you broke any of them you’d also be breaking
one of these.
It turns out that you cannot be made a sure loser if, and only if, your prices obey these three rules. That is:
- If your prices follow these rules, then you will offer customers no arbitrage opportunities.
- Any market absent of arbitrage opportunities must have prices that conform to these rules.
These prices are called coherent, and absence of coherence implies the existence of arbitrage opportunities for your customers.
But Why Male Models
The trick, of course, is that these prices correspond to probabilities, and the rules for avoiding arbitrage correspond to the standard Kolmogorov axioms of probability theory.
The consequence is that if your description of uncertain phenomena does not involve probability theory, or does not behave exactly like probability theory, then it is an incoherent representation of information you have about those phenomena.
As a result, probability theory should be your tool of choice when it comes to describing uncertain phenomena. Granted you may not have to worry about market making in return for pizza, but you’d like to be assured that there are no structural problems with your description.
This is a summary of the development of probability presented in Jay Kadane’s brilliant Principles of Uncertainty. The original argument was developed by de Finetti and Savage in the mid-20th century.
Kadane’s book makes for an exceptional read, and it’s free to boot. I recommend checking it out if it has flown under your radar.
An interesting characteristic of this development of probability is that there is no way to guarantee the nonexistence of arbitrage opportunities for a countably infinite number of purchased securities. That is: if you’re a market maker, you could be made a sure loser in the market when it came time for you to settle a countably infinite number of redemption claims. The quirk here is that you could also be made a sure winner as well; whether you win or lose with certainty depends on the order in which the claims are settled! (Fortunately this doesn’t tend to be an issue in practice.)
Thanks to Fredrik Olsen for reviewing a draft of this post.