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.

What is Option Delta?

Trading options without knowing your greeks is like flying a plane without looking at your flight instruments. Knowledge of your greeks is ABSOLUTELY NECESSARY to trade options. They are like the “vital signs” of your portfolio, that can tell you if your portfolio is healthy, or about to get sick. The more complex your option strategy, the more important it is to be able to read these signs – they’ll help you to extract more money from the market, or avoid losing money in the future.

There are a few greeks you need to know (mainly delta, gamma, vega, and theta), but we’ll start with delta.


Option Delta

Delta is the simplest greek to understand.

Delta is the expected change in price of an option when the underlying asset moves by $1

This means that if you have an option with a delta of 0.5, then for every $1 increase in the stock, the option should increase by $0.5. On the other hand, if your option had a delta of -0.5, then for every $1 increase in the stock, your option would LOSE $0.5.

Call options almost always have POSITIVE DELTA between zero and plus one. They INCREASE in value as the underlying asset goes up.

Put options almost always have NEGATIVE DELTA, between zero and minus one. They DECREASE in value when the underlying asset goes up.



As expiration nears, the delta of an in-the-money call will move towards 1, whereas the delta of an out-of-the-money call will move towards zero. This is because, as expiration nears, the in-the-money call is likely to be exercised and turned into stock, whereas the out-of-the-money call is unlikely to be exercised so is virtually worthless and won’t react to the stock’s price movement at all.

For puts options it is very similar. The delta of an in-the-money put will move towards -1, whereas the delta of an out-of-the-money put will move towards zero as it becomes more obvious that it will be worthless at expiration. This means we can think about delta in another way:

We can think about option delta as the probability that the option will end up in the money at expiration. For example, a delta of 0.5 means that there is about a 50% chance that call option will end up in-the-money at expiration. A delta of -0.2 would mean there is about a 20% chance of that put ending up in-the-money at expiration

Now while this is a useful way to think about delta, you should be aware that it is not a proper textbook definition – it is really a side-effect of the way delta is calculated.

The following table is real data taken from a broker that shows the deltas of individual calls and puts with various strike prices, when the stock is at 1160 (the blue highlighted row).

Screen Shot 2013-12-30 at 5.38.05 PM


In this example the underlying asset is the Russell 2000 index currently trading at 1160. The bright yellow line highlights an in-the-money call option and an out-of-the-money put option. The bright blue line highlights at-the-money call and put options. The bright green line highlights an out-of-the money call option and an in-the-money put option. Notice how the deltas of the calls decreases as the strike price increases, and how the deltas of the puts get more negative as strike price increases. The table shows very well how delta is affected by the how close the strike price is to the price of the underlying asset. Here is a graph of that data above that shows the same thing:


How stock volatility affects option delta

If we think about delta as the chance that the option will end up in-the-money at expiration, then clearly the volatility of the stock will affect an option’s delta. If the stock moves up or down by 50% or more every day, then there is plenty of chance for almost any option to end up in-the-money. Take a look at Tesla (TSLA). This is a stock that is very volatile, so even when the stock is at $200, there is still a moderate chance that within the next month it could fall to $100. This means the delta of a $100 strike put option might be around 0.1 or 0.2. Compare this to McDonalds (MCD), which is a very stable company whose stock hardly fell at all during the 2008 crisis. If the stock is trading at $100, there is only a very small chance that it will fall to, say, $50 within the next month. This means the delta of a $50 strike put option would be very close to zero.

If the volatility of a stock changes, it can change the deltas of the options. and cause you to make or lose money pretty quickly. When prices of options change due to changes in stock volatility, we call this VEGA RISK.


How time to expiration affects Delta

If we think about delta as a measure of the probability that the option will expire in-the-money, then it is common sense that if an option is way out-of-the-money and has very little time left until expiration, then it will have a delta close to zero, as there is very little time left for the stock to move a lot. If it is already way in-the-money and has very little time left until expiration, then it will have a delta close to 1 (if it is a call option) or -1 (if it is a put option). Essentially, at expiration, a call’s delta MUST be either zero (out of the money) or +/- 1 (in the money). Therefore, as time moves onward, deltas of options tend to gravitate towards these values. Take a look at this chart. With a lot of time left until expiration (171 days) the line is fairly flat. As time to expiration decreases, the line gets more curvy – i.e. the delta gravitates towards 1 or -1.

Delta affected by time

This results in something interesting around option expiration time – the deltas of at-the-money options tend to swing around wildly. Here’s an example:

Say you have a call option with a strike price of $25 and there is only one day left until expiration. The stock is also currently trading at $25, so the delta of the option is around 0.5. If the stock increases to $26, the probability of the call ending up in-the-money just increased a LOT simply because there is not much time left for the stock to go back down before the option expires. If you owned this call you would have suddenly made money on it. If you had sold it, you would suddenly have lost most of your money.

This is why trading options around expiration time can be pretty dangerous – small changes in the stock can cause very large changes in delta, and therefore very large changes in your profits.

This neatly introduces us to a second greek you need to know – GAMMA. Gamma is the rate of change of delta, i.e. how quickly delta changes. We don’t generally want delta to change too rapidly because it means we can lose (or make) money very quickly. As we approach expiration we say that our GAMMA RISK increases – our rate of change of delta increases which means that even though we’ve made money up until that point, we can lose it all in a matter of hours.