Trading 101: Understanding the Expected Value of Uncertain Bets

Industry veterans sometimes remark that successful gamblers tend to make good traders, and engineers tend to make lousy traders.

This is a gross generalisation, of course, but one reason is that trading, at the most fundamental level, is a game of pricing uncertain outcomes. This requires probabilistic thinking, and engineers tend to be trained to think deterministically.

Indeed, thinking probabilistically is hard for most people.

That’s why so much retail trading tends to involve buying when the news is good and the price is increasing, and selling when the news is bad and the price is decreasing.

That’s deterministic thinking, and we’re good at that because we evolved that way – there isn’t much need to think probabilistically when you are trying to avoid being eaten by a bear.

On the other hand, a more thoughtful assessment of probabilities is hard. If you like money, however, you’re going to need to learn to think probabilistically.

How do we “think probabilistically”?

Let’s go through the basics.

Imagine someone is offering you a game. You roll a dice. If you roll 5 or 6 then you get $1,000. If you roll any other number you get nothing.

What would you pay to play this dice game?

Well, this is a risky game, and most people don’t like playing games in which they can lose money. But, to start with, at least, let’s assume that you don’t mind taking risks and you are only interested in maximising the expected value of the bets you take.

First, you must work out the price at which you are indifferent to playing the game.

If you don’t mind taking the risk, then the price at which you are indifferent to playing the game is the price at which you break even over the long term playing it. You want to play the game if it’s cheaper than this value, and you want to pass if it’s more expensive.

How do I calculate that?

It’s simple (in this toy example, anyway.)

You write down all possible outcomes of the game, and for each possible outcome you write down:

• the probability of that outcome occurring
• the payout if that outcome occurs.

Then, you multiply these two figures for each possible outcome and sum them up.

That gives you the expected value of the game – which is also the price at which you should be indifferent to playing the game if you don’t mind taking on the risk.

Here it is for that game:

If you are just looking to maximise expected return and don’t mind taking on risk, then you should only play that game if you can play it for less than $333.3

But I don’t like taking on unnecessary risk!

Good. Me neither.

Of course, you are not indifferent to playing games in which you might lose a significant amount of money!

You probably won’t play this game, in which you can lose a reasonable amount of money, for $333, unless that’s a small amount of money to you and you can play the game repeatedly.

A 30 cent edge is just not worth the risk of losing $333, for most people.

This is where the idea of risk premium comes in.

Players of risky games with uncertain payoffs demand a premium for taking on that uncertainty. So maybe the price the aggregate market would be prepared to pay for this game would be less than $333. This results in positive expected value for those prepared to take on the risk.

How does trading differ from this toy example?

In trading, the outcomes, their probabilities and their payoff are subject to uncertainty.

In our toy example, we could calculate the “risk-neutral fair value” of the game exactly because we knew the probabilities and the payoffs.

In view-based trading, we have to forecast every one of these variables.

In derivatives pricing, we know the payoffs, but we have to forecast the probabilities.

This is hard, but just because something is hard doesn’t mean you shouldn’t do it.

And you really do need to get used to thinking in these terms if you want to be competitive in trading.

A nice side benefit of thinking this way is that if you understand this, you also understand pretty much all of derivative pricing (the important stuff anyway). The fair value of an option (in risk-neutral land) is simply the probability-weighted sum of all the payoffs at expiry. You can do this without understanding anything about BSM or understanding a single Greek letter.