Delta and Gamma Revisited… as Explained by Dice

Options pricesWhen I worked at my old bank job, I would constantly cajole my co-workers into betting on things. Anything, it didn’t really matter. Football games, non-farm payrolls, S&P closing price, anything. Our coffee machine was so useless we’d even bet on how many times we’d have to operate it before we got a proper coffee out of it (my employer was a real tightwad with “extras” like coffee machines). The idea was always the same – we tried to work out the expected value of a bet.

I like to use dice games to illustrate points about trading because of the similarities between rolling dice and entering the market. In both instances there is a huge element of chance. You need to be aware of the expected value of your bet, but that’s not all.

For example – Let’s say I have a die. I’m going to roll it, and if it’s a 1, 2, 3, or 4, I will give you $10. However, if it’s a 5 or a 6, you have to give me $10. This is a game you’d want to play, right?

Delta and gamma

The chances are clearly in your favor – two thirds of the time you will win $10, but only one third of the time will you lose $10. If you played the game with me 100,000 times, you would expect to end up with a profit of around $333,333. Not bad, right? This is definitely a game you would want to play.

Now let’s change the rules of the game. Instead of playing 100,000 times, we’ll play just once. But we’ll up the stakes a bit. If I roll a 1,2,3, or 4, then I will give you $1 million. However, if I roll a 5 or a 6, you have to give me $1 million. Do you still want to play?

Delta and gamma

Why not? The expected value of the bet is the same – according to the probabilities, if I play this game with a lot of people, the average person should still win $333,333. For each roll of the die, the chance of winning is still 4/6.

The DELTAS of the two games are the same – there is the same probability that you will win or lose before you start playing, and the expected profit is the same.

However, the GAMMAS of the games are different. In the first game, you can quit if you start to lose money. You can demand that we change the game. You can inspect the die if it seems like it’s biased. Even if the first ten rolls of the die go against you, your probability of making $333,333 from me hasn’t really changed very much. Your delta in the first game remains unchanged. It is a “low gamma” game.

In the second game, you can be wiped out in an instant. After a single roll, your probability of earning $333,333 (or more) changes hugely and instantly. You delta has moved massively. This is a “high gamma” game.

Just because an option has a 10 delta seven weeks before expiration does not mean that it has the same risk as a 10 delta option one week to expiration. Even though the probabilities are roughly the same, with seven weeks to expiration we have plenty of time to “inspect the die” if it starts rolling against us, we can simply stop playing altogether, or we can just play with another die by picking a different expiration month. With one week to expiration, one wrong roll of the die and you could lose a lot of money.

Play often, accept that you will lose often, but make sure the odds are in your favor.