Meet the Greeks – Part 1

(Originally published in Active Trader magazine)

What a boring game all this would be if option prices showed no fluctuations! The price, or premium, of an option is constantly changing for whatever reason. As an options trader you must be prepared, armed with the knowledge of how an option price can react in certain scenarios. You can look at that from a reward or a risk perspective.

If you are relatively new to options, you may have heard the terms “delta” or “gamma” or several of the other Greek letters. These terms are used to measure how an option price can behave in certain environments.

This two part article (this being part one) seeks to explain “the Greeks” and more importantly give perspective to their relative importance in day to day option trading.

The Background

Those familiar with option pricing models such the Black-Scholes (B-S) Model will know certain inputs used to determine the theoretical or fair value of an option. These are:

Market price and strike price

  1. Time to expiry
  2. Volatility
  3. Option type (put or call)
  4. Interest rates
  5. Dividends or coupons
  6. American or European style option.

Now, if you picture yourself in any option position, which of the above factors can vary? That is, what factors can change so that the value of the option might change?

The answer:

  1. Market price
  2. Time to expiry
  3. Volatility
  4. Interest rates.

A change in any of these four factors will most likely change the value of the option price—or at least change its fair value.

The next obvious question is: ‘How significant are these four factors in relation to an option position?’

For example, if you have just bought a call option, what will your profit position look like if the market rallies 1 per cent, one day passes, volatility increases, and the Central Bank raises interest rates? To be an educated options trader, you need to know how these factors will affect you in real life.

That is where ‘the Greeks’ come in. They are measurements that describe the relative importance of these changing factors. These measurements are given names from the Greek alphabet given their initial derivation from the mathematics of calculus.

Interestingly, calculus is a branch of mathematics that is used to measure the rate of change of one factor, given a change in another. Throughout this article you will see the phrase ‘rate of change’ quite a lot. This is, in essence, the basis of this topic: analysing the rate of change of the option premium given a change in one of the above factors.

However, the mathematics is kept to a minimum since modern option software will easily handle the task. What you need, as an options trader, is an understanding of these concepts and the ability to apply them to your position. Rest assured, this is a formula-free zone.

Not all the following measurements will be applicable to your trading, but these are important concepts to learn and you are encouraged you to at least think about how you might apply them in real-time trading.

This month, we will only look at the concept of a change in the underlying market price and how this affects the fair value of an option position. Next week, we will look at the other factors.

The Delta

One could easily argue that delta is the most important and commonly used of the ‘Greek symbols’ in options trading. It is therefore prudent we spend a good deal of time here.

Suppose we have a 1 percent rally in our underlying asset. Think for a moment how the fair value of a deep in-the-money call option will change in relation to a far out-of-the-money call option.

One would think the fair price of the in-the-money option is more likely to move a greater amount than the price of the out-of-the-money call option. In fact, picture a call option way, way, way out-of-the-money. The price may not even move at all!

There is a way to measure the degree that an option price will move in relation to a movement in the underlying asset price. The measurement is called delta.

Measurement and Interpretation for Calls
The delta can be expressed in two ways:

  • As a percentage. For example, the delta of a $10 strike call option in Silver futures could be +26%. This means for every 10c rise in Silver, the fair value for the $10 call will increase by 2.6 cents.
  • As a decimal. In the above example, you would say the delta for the $10 call is +0.26. For every one unit rise in the Silver futures contract, the call option will increase 0.26 points. This is the more common method, but the interpretation is essentially the same.

Now, think about the range of values for the delta of a call option. Really far out-of-the-money call options would hardly move if the underlying asset were to increase by just a little. You could confidently say, therefore, that the delta for a way out-of-the-money call option would be close to zero. The further out-of-the-money you go, the closer the delta moves to zero.

What about in-the-money options? Well, assuming all other factors remain constant (especially volatility), it is fair to say that an option’s fair value would not increase at a faster rate than the underlying market. Right? Sure, if the other factors changed (interest rates, volatility, dividends and coupons), then it might be possible. But, holding all those factors steady, the option price would not increase at a faster rate than the underlying, given a solitary change in the underlying.

This means, the most the delta could be is +100 per cent, or +1.0. A delta of +1.0 means the fair value of the option will move ‘tick-for-tick’ with the underlying. A 5-cent increase in Silver will result in a 5-cent increase in the fair value of that particular call option.

Now, the deeper you go in-the-money, the closer the delta will be to +1.0.  That means that deep in-the-money call options tend to behave just like the underlying asset. (It’s an interesting thought when you want to hedge a position, but that is another story.)

A general rule of thumb is an at-the-money option will have a delta of close to +0.50.

Therefore, we have established that the delta ranges between zero and +1.0 for all call options. Another rule of thumb is, the longer an option has until expiry, the higher its delta will be, although the difference is never huge.

For example, an out-of-the-money call with 90 days left to live will have a higher delta that the same option with 10 days to live. The difference is may not be so big for at-the-money options, but can be significant as soon as you move in- or out-the-the-money.


Table 1:

This table shows option prices for the Silver futures, currently at 929.50 (quoted in cents) and the corresponding delta expressed as a percentage. Note each option has its own delta. In other words it has it’s own level of variability given a change in the market

Strike price

Option price

Delta

1150

21.20

22.2%

1100

28.80

28.0%

1050

39.10

31.4%

1000

52.70

43.4%

950

70.30

54.9%

900

93.90

67.1%

850

123.70

77.6%

800

158.20

84.9%

750

196.90

90.5%

Put Option Deltas

So far, we have talked about call options only. What about puts? Wouldn’t they just be the same? A $10 call should have the same delta as a $10 put right? Certainly not!

Think about it. If a market rallies ten cents, the fair value of our calls would increase by a multiple of the delta. Put options however would decrease in price. And yes, you guessed it; each put would decrease by a multiple of its very own delta.

The delta for a put option is a negative number. An at-the-money put delta will be close to –0.50 (where an at-the-money call delta will be close to +0.50). Out-of-the-money put deltas will approach zero, just like call options. In-the-money deltas for put options will approach –1.0 just like call deltas approached +1.0.

Delta facts:

· The delta on the underlying is always equal to +1.0, or +100 per cent. A one-point increase in the underlying means a one-point increase in the underlying.

· You can calculate the delta of a combined option and/or underlying position simply by adding up the individual deltas. For example: two long calls with 40% delta each gives a position delta of 80%; or a long call with a delta of 20% and a long put with a delta of -20% gives a position delta of zero.

· Shorting an option reverses the positive of negative sign of the position delta. For example going short a call with delta of 20% would give you a position delta of -20%. The same rules apply here when adding up multiple option positions.

· The terms delta positive, delta negative simply refer to the positive of negative bias of the option or option position. Delta neutral simply mean you have a delta of zero.

Delta and your option position
So from these basic rules, we can now apply the concept of delta to commonly know option strategies. Table 2 summarises each position:

Table 2:

Position

Description

Delta

Long call

Long call

Positive

Long put

Long put

Negative

Long straddle or

Long strangle

Long call and Long put

Neutral

Short straddle or

Short strangle

Short call and short put

Neutral

Bull call spread

Buy call, sell further out-of-the-money call

Positive

Bear put spread

Buy put sell further out-of-the-money call

Negative

Short Calendar spread

(Call options)

Sell short-term call option. Buy long-term call option.

Slightly positive

Short Calendar spread

(Put options)

Sell short-term put option. Buy long-term put option.

Slightly negative

You will notice that some of the more complex strategies, such as ratio spreads, have been left out. This is because the delta on these positions is not so clear-cut. It is very much dependent on your choice of strikes.

For example, in the case of a ratio call spread, if there is a large gap between the bought strike and the sold strike, then the sold component will have a small delta, relative to the bought component, thereby having little impact on the net delta. If the sold strike is close to the bought strike, it will have a greater impact and may even cause the net delta to be negative.

The Gamma
As discussed above, deltas range from zero to +1.0 for calls and zero and -1.0 for puts. The deeper the option is in-the-money, the closer the delta is to +1.0 or -1.0. What then happens to the delta of an option as the market moves in one way or another? Does it stay the same? Does it change?

Consider an at-the-money call option. Let’s say its delta is exactly +0.50. What would happen to the delta if the underlying market price started to fall and kept falling? Well, for a start, the at-the-money call will no longer be at-the-money. It will be out-of-the-money. We already understand that out-of-the-money options have deltas approaching zero.

So the delta of that call option would change. The further the market falls, the smaller the delta would get. Likewise, should the market rally, an at-the-money option would then become an in-the-money option. The deeper that call option goes in-the-money, the closer the delta will be to +1.0. In other words, the further an option is in-the-money, the more it will react to changes in the underlying market.

So it’s fair to say that, as the market moves up or down, the delta for each and every option changes. There is a measurement for the rate of change of the delta. It is called ‘gamma’. The gamma measures the rate of change of the delta given a change in the underlying price.

At first thought, the gamma does not seem like a very important measurement, but it really is. If the delta never changed throughout the life of your position, risk would be very easy to quantify. But, if a market starts to trend, your delta can change dramatically. And if the trend is not in the direction that you want it to be, you could be in trouble!

This point is best explained by example. In the mid-1980s a US-based clearing member collapsed after several traders with accounts at the firm took on positions where the gamma worked against them. These traders had built up large short positions in out-of-the-money gold call options. The deltas in these positions were small, since the strike prices were distant. That ordinarily meant small ticks up or down in the market that would not affect the options prices too much.

The gold market subsequently staged a strong rally. What were once options with very small deltas suddenly increased. As the market rose, not only did the price of the options increase, but the rate of change of the options also increased. The deltas got larger, hence, so did the losses. A few of the traders were unable to meet margin payments and the clearing firm collapsed. That is gamma!

So, what lesson could one learn from this example?

For a start, it is highly recommended that you understand what a market is capable of, before taking on any positions. Think about worse case scenarios in the market and how they may affect your chosen options position. It may pay to consider what some call ‘insurance’. In the above example, instead of selling call options naked, the traders could well have created a credit spread by buying a call strike even further out-of-the-money.

For example, they may have sold call options for 50¢ each. It might have been possible to go further one or two strikes higher and buy an equal number of calls for say 10¢ each. The net income received would have fallen from 50¢ to 40¢, but buying this call option would not only reduce the margin, it could have helped reduce losses as the market started to rally. The bought call will always have a smaller delta and gamma than that of the sold call, but it could still significantly reduce risk.

Can you see how adding such a seemingly small and insignificant component to a position can change your risk profile? That is the nature of trading options. You must always be prepared for the worst to happen—even if it means giving up some profit potential.

Facts about the gamma

· Gamma is normally expressed as the number of delta lost or gained per one point change in the underlying. Consider an option with a delta of 40% and a gamma of +2.0%. This means that for a one unit gain in the underlying, delta will increase from +40% to +42%.

· Like delta, the gamma of a total position is calculated by adding the sum of each component.

· Gamma is largest for at-the-money options (where deltas are close to ±50 per cent). Put another way, the delta of an option that is close to or at-the-money can rapidly change in response to a change in the underlying price.

· As the delta approaches zero or approaches ±1.0, the gamma gets smaller. Put another way, the delta tends not to change very much the further in-the-money or out-of-the-money you go.

· Modern options trading software, such as OptionVue5 pictured, should be able to display individual and position gammas.

· The delta of the underlying is always equal to one. It never changes. Therefore, the gamma of the underlying is always zero.

The best way to start thinking about gamma is not in its numerical form. Think about the concept itself, not its measurement. Think about your total delta when you buy some calls, or sell a strangle, or buy a bull call spread. Think about how small and large changes in the market will affect your delta and your profit and loss. A changing delta simply means a changing risk profile.

As an options trader, you must constantly re-evaluate your total risk and make sure it fits with your objectives. Do not be afraid to exit a position because the market has moved in such a way that your delta and gamma have moved against you.

Delta and Gamma in Perspective

So what is the big deal? What do we do with the delta and gamma? Is it worth all the bother? Most if not all options traders will tell you it is. Understanding your delta and gamma allows you to understand risk, because that is essentially what these things are: measurements of risk relative to market movements.

Even when you do not know what the actual numerical figures are, you should know if your delta is negative or positive and know if your delta is large or small and how that delta can change. It is simply measuring risk.

Some professional options traders would not be able to tell you what their exact delta and gamma are, but they do understand how movement in the market affects their positions.

The most important point from this article is you should be thinking in terms of risk and how your position changes relative to the market.

In the next article, we will look at how to measure the risk in an option given changes in other factors such as time and volatility. For an Active Trader, it is knowledge worth having.