Delta and Gamma Revisited… as Explained by Dice

When 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?

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?

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.

Time Decay and the Gamma Slide – Why You Shouldn’t Let Your Options Expire Worthless

One of the toughest trades to make in an iron condor is the exit trade, but it’s also what tends to separate the winners and the losers.

The most intuitive thing to do is to let your options expire worthless. After all, this is why you started trading iron condors in the first place, right? To gain from the gradual erosion in value of out-of-the-money options.

Letting your options expire worthless is the most tempting thing to do. Not only do you gain the last few cents of the remaining value of the options, you also save on commissions as there is no commission charge for options that simply expire. However, over the long term it’s generally the wrong decision. Here’s why:

1 – The Gamma Slide

The “gamma slide” is not as much fun as it sounds. It’s a description of how your profit/loss chart looks at expiration compared to how it looks when you put the trade on. Here is the profit/loss chart of the put side of an iron condor that expires in 49 days. The white line is your profit/loss chart the day you put the trade on (today), while the red line is how the position looks on the expiration date (in 49 days time). As time progresses, the white line gradually moves until it becomes the red line.

Now imagine that the market starts moving down straight away, from 1740 down to 1680. It’s a big move, but these kind of moves can happen. It doesn’t matter how much time you have left until expiration, you are going to lose money on this move. However, if you have a lot of time left (the white line), the amount of money you lose will be smaller. If you have very little time left until expiration (e.g. if today is expiration day), then you will lose a LOT of money. Here is the same chart showing the move:

The red line shows a massive swing from profit to loss. The loss is large enough that it could wipe you out completely. The white line, however, shows only a moderate loss*. If you are on the white line, with lots of time left until expiration, you have plenty of time and capital to adjust your position. You live to fight, and profit, another day.

As a general rule, iron condor traders tend to like fairly flat profit/loss charts. However, as always, there is a tradeoff with collecting time premium.

2 – Theta decay changes with money-ness

Many option traders incorrectly state that the time decay of option value increases sharply right before expiration, i.e. that option value drops sharply in the final few days before expiration, like this:

ATM option theta decay**

What this chart shows is that the value of an option will decay fairly slowly for most of its life, and then rapidly decay in the final few days. However, this chart shows option value decay for at-the-money options only. Iron condor traders will rarely be short at-the-money options, as most iron condor strategies require some form of adjustment before the price approaches the short options.

For out-of-the-money options the value decay chart looks very different:

OTM option theta decay**

For out-of-the-money options, the value actually decays very slowly in the final few days. Iron condor traders generally trade out-of-the-money options, so the second theta decay chart is the one that is relevant to us.

Let’s look at some numbers from the second chart. First off, let’s realise that the option expires in 40 days. After this time, the option has a value of zero. However, after only 20 days, the option’s value has decreased from \$4 down to \$1. So, even though we’ve only been in the market for 50% of the time, we have collected 75% of the premium from this option. Instead of staying in this trade for another 20 days and collecting the rest of the premium, we should consider closing this trade and opening a new one. The risk/reward is greatest for us before the line starts to curve down towards zero.

So there you have it. The takeaway of this post is…

In terms of the risk/reward of an iron condor, it is more favorable to close out your soon-to-expire options and then open a new trade, rather than let your current options expire worthless

* I am ignoring the effect of the large increase in volatility for simplicity. With a shorter time to expiration, increases in volatility will cause you to lose money even faster, so the situation is even worse for soon-to-expire options than described here.

** These charts and numbers are not strictly accurate, and are instead meant to stylistically show the rough shape of the time decay of ATM vs OTM options.

Understanding Option Prices

Understanding option prices is important because your job, as an options trader, is to evaluate the price of an option and decide whether you should buy or sell it.

Too many people don’t really ‘get’ options prices, even though they trade them regularly. We should really understand expected return and theoretical value before we start thinking too much about the greeks. Here’s a quick intro to how and why option prices vary.

You already know some of the factors that must be included in an analysis of options prices. Here’s a short list of some of the most important factors:

1. The current price of the stock
2. The strike price of the option
3. The time left until the option expires
4. How quickly the price of the stock can change (the volatility)

Ideally, we’d know all of these inputs exactly, stick them into an equation, then get an answer for the true value of an option. We know the first three inputs exactly – they are all features of the option that we are considering trading. I’ve highlighted the last factor because it is the only thing we don’t know for sure when we trade an option. Volatility is the only thing we have to estimate.

For a discussion of volatility, let’s consider expected returns.

Let’s say you have a six-sided die. We all know that there is an equal probability of rolling any number between 1 and 6. Now let’s say I offer you the opportunity to roll the die and I will give you a dollar amount equal to whatever number come ups. If you roll a 1, you get \$1. If you roll a 2, you get \$2. And so on. How much would you pay to play this game with me?

Well, the first thing we need to think about is the expected return of the die. The average roll of the die will be (1+2+3+4+5+6)/6 = 3.5. So, on an average roll, I will give you \$3.5. For you to be profitable over the long term (assuming you love my game and want to keep playing it), the fee you should pay to play this game with me has to be under \$3.5. The theoretical value of the bet is \$3.5, so if you paid more than this, over the long run you would be a loser.

Now let’s do the same thing with a call option on a stock.

Let’s say there is a stock, that is currently trading at \$40. At expiration, the stock could be at \$20, \$30, \$40, \$50, or \$60. Each of these prices is equally likely (each one has a 20% chance). Now let’s say you buy a call with an exercise price of \$40. How much should this cost?

Well, you will only make money if the stock ends up at \$50, or \$60. If it ends up at \$20, \$30, or \$40, you get nothing. If it ends up at \$50, you make \$10, and if it ends up at \$60, you make \$20.

So, what is the AVERAGE return of the call option? We need to weigh the returns by their probabilities in order to find out, like this:

(20% x 0) + (20% x 0) + (20% x 0) + (20% x \$10) + (20% x \$20) = \$6

The average return of the call is \$6. This is its theoretical value, so you should always pay under \$6 for this call, in order to keep playing this game and make a profit over the long run.

Now obviously we simplified things by saying there was an equal chance that the stock would end up at each of these prices. That is never really the case. A more realistic situation would be if the stock had a higher probability of doing nothing, and a lower probability of moving significantly up or down, like this:

Now at the end of the period, the average stock price is the same (it’s still \$40, as it was in the first example). But because we’ve changed the probabilities, we’ve changed how much we should pay (on average) for this option. If we do the same calculation as before, but with the new probabilities, we find that the average return on the call option is now:

(10% x 0) + (20% x 0) + (40% x 0) + (20% x \$10) + (10% x \$20) = \$3

Wow. So even though the stock’s average ending price is still \$40, the price we should pay for the call has HALVED to \$3 from \$6.

The probability distribution (of where we expect the stock to finish at expiration) has a massive effect on how much we should pay for the option

So, the key input into option prices is the probability distribution. If we know the ‘real’ probability distribution, we will know the real price we should pay for the option. To consider the ‘real’ probability distribution of a stock, we can do a little thought experiment. Let’s say that each day, a stock has a 50/50 chance of going up or down by \$1. The stock’s movement is essentially a ‘random walk’. We cannot predict where a stock will end up in 30 days time, but we can predict a rough probability distribution of where the stock might end up. This video shows what happens if you go through a path many times where there are multiple 50/50 decisions:

Now that you’ve seen the video, imagine what would happen if the nails were much, much fatter. The distribution of balls at the bottom would be much, much wider. This is equivalent to a much more volatile stock.

The purple line is how it would look with normal sized nails (equivalent to a stock with low volatility), while the blue line is how the distribution would look with wider nails (equivalent to a stock with high volatility). Clearly, a call option with a strike price around 5 would be more valuable for the high volatility stock (the blue line). So, a clear conclusion is that higher volatility stocks will have higher priced options.

It seems that the probability distribution of a stock is roughly normally distributed. Is this a reasonable conclusion?

Not really.

The problem is that the normal distribution is symmetrical, and as you can see in the distribution above, this means that there is the possibility of a negative stock price. No matter how high we set the middle of the distribution, the chart will always say there is the possibility of a negative stock price. Clearly, we can’t get a negative stock price, so this cannot be right. We can make the situation better by saying that the normal distribution is the distribution of the percent return of a stock, instead of the stock price itself. At every moment in time, the price of a stock can go up or down by a given percent, and it is these percent changes that are actually normally distributed, not the price itself.

If the percent return of a stock is normally distributed, then mathematically the stock price of that stock must be lognormally distributed. The key features of a lognormal distribution is that it does not fall below zero (which is important, as stocks also do not fall below zero), and that it has a longer tail on the positive side, like this:

So if we want to think about the probability of a stock ending up within a particular range, we should allow that range to be slightly higher on the upside. If we include interest rates into our calculation (which I deliberately ignored when I listed the four factors that affect option prices at the start of this article), then we should assume an even greater upside bias to the stock price.

Knowing the theory behind the probability distribution of stock prices, and how this affects option prices, simply helps you understand options theory better.

The two conclusions you should make coming away from this article are as follows:

Probability theory shows us that the distribution of a stock’s price at expiration explains why options have different prices. I.e. volatility is the key ingredient in option pricing.

And….

The probability distribution of a stock’s price is NOT normally distributed. A better approximation is a lognormal distribution, because stocks can (in theory) go up an unlimited amount but can only go down to zero.

What is Option Delta?

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).

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.

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.