Trading Insights: Exposing the St. Petersburg Paradox

The St. Petersburg Paradox is a famous problem in probability and economics. The problem states that a coin is repeatedly flipped. The bet starts with $2. Every time a tails shows up, the bet is doubled. The first time a heads shows up the game ends and you receive the current bet from the ‘house’. The question is, how much would you pay the house for such a game?

From a probability point of view, the expected payout is infinite, and it is not hard to see why: the payout is doubled each round. This leads us to the paradox: you should be willing to pay an infinite amount to play this game. Most people would refuse to.

I had seen this problem as an undergraduate and studied it using a purely probabilistic view. Attempts to explain it include that you need to be able to play rounds of the game (flipping the coin) extremely quickly so that you could get the right payout. Also, the house needs to have infinite money. Economics explained it using that laziest of academic crutches: utility theory.

I had no idea how far off pure probability was from trading until I was exposed to real traders as opposed to math professors (I was always clear on how divorced economics is from reality). I was taking an advanced derivatives course taught by a derivatives trader, Howie. At the end of the course he invited the students to dinner along with his boss, Richard, and a client.

As we munched on pizza slices at V & T, Richard and the client got into a discussion about the paradox. I will never forget it:

Client: Make me a market (i.e. tell me at what price you want to deal as house to me and at what price you want me as house to deal to you).

Richard: 4 / 8 (i.e. Richard will pay the client $4 to play the game with the client as house, and Richard demands $8 to act as house).

Client: 4 / 8?! That’s a 4 point spread! That’s not fair! (Client is complaining that Richard is making it too expensive to play).

Richard: Fine! I’ll deal to you at 4 / 8! (Richard here is calling out the client: If it is unfair to the client for Richard as house to quote a 4 point spread then it must be fair for the client to deal to Richard on a 4 point spread).

Client: Ummm… (Client doesn’t have the balls to act as house regardless of the spread).

Richard: Fine! I give you the option, for free, to decide who deals to who on a 4 point spread!

The last bit blew my mind. I immediately understood why Richard had the upper hand. He was pricing everything with a two way price, and then giving a two way price on that, and as a finale wrapping an option around it. This was a powerful lesson to me: all the math in the world isn’t going to help you if you don’t know how to trade the human element.