Option Greeks

Performing mathematical calculations are significant to obtaining success with options trading. We have already discussed the Black-Scholes system and how it can help you make educated option decisions. As with stocks, the price of options is influenced by the market. Using the Black-Scholes pricing method, you can determine how specific market factors will affect an option. These market factors are referred to as the Greeks: Delta, Gamma, Vega, and Theta. The theories associated with options pricing are involved and are not discussed in detail here.

DELTA

In the Greek language, Delta is the fourth letter of the alphabet and in mathematical calculations means difference, or change. In regards to options, Delta is concerned with the following:

• The relationship between the underlying asset’s share price and the option’s theoretical value. Specifically, what affect does a change in the share price have on the option’s value?
• The likelihood that the option will become in-the-money by its expiration date.

The graph above shows what happens to the delta of both a Call and Put option as they move from not having any intrinsic value to being at-the-money, and eventually in-the-money. As they graph indicates, the deltas of a Call and Put are contrary to one another. While a Call option has a positive delta, a Put option has a negative one.

For example, if the price of stock XYZ increases by .75 and the value of its associated Call option increases by .25, the Delta is equal to 50% or .50. In this example, we can see that the option’s value is increasing at a pace that is slower than that of the stock’s price. The value of the option is not in line with the stock’s price until the Delta reaches 100% (1.00). At least this is the case for Call options, which span from zero percent to 100%. As you would imagine a Put option’s value may range from –100% to zero percent.

Call Options

As stated above, the delta of a Call option is a value between zero and 1. What this means is that with every increase of an underlying asset’s share price the value of the Call option increases according to the value of the option’s delta. This also works in the opposite direction. If the asset’s value decreases, so shall the value of the Call option (according to the delta amount).

The graph above shows the movement of the Call option’s delta as the stock price fluctuates. You should notice that when the delta shifts to a value of 1, the Call option’s value also shifts (one-for-one) in the same direction.

Put Options

The deltas of Put options are negative and have a value that ranges from -1 to zero. As the underlying asset’s share price goes up, the Put option’s value also decreases according to its resulting delta value. As the share price goes down, the Put option’s value increases according to its resulting delta value.

You can see from the graph above how exactly the Put option’s value changes with every change in the underlying asset’s share price. As the stock price changes, so does the option’s value.

GAMMA

Option Gamma refers to the movement of an option’s delta as its underlying stock shifts one point. Basically, Gamma indicates how the underlying asset’s share price influences the option’s delta. How is this information useful? When you know an option’s Gamma, you know the rate at which the delta of the option will change in relation to the share price. This indicates an option’s volatility in relation to fluctuations of the underlying asset’s share price, which ultimately translates to the amount of risk you can expect.

The graph above presents three exercise values and a comparison of how Gamma and an underlying assets share price changes for each value. As the graph shows, Gamma goes up as the option goes up. As you would expect, as the option shifts to a position of being out-of-the-money the Gamma goes down in value.

An important thought to remember is that the Gamma for Call and Put options is equivalent. Whether the option is a long Call or a long Put, the Gamma is a positive value. If the option is a short Call or a short Put, the Gamma is a negative value. Therefore, your expectations will change, depending on the Gamma. For any long option you have, you should hope that the underlying asset’s value shifts upward. In the converse, your anticipation for your short options is that you will be able to sell more as the delta shifts downward.

THETA

Since options have a specific life span, it makes sense that its value changes throughout that period of time. Option Theta tells you the extrinsic value or how much an option changes (loses) each day before its expiration date. Thus, Theta is the estimation of how much of the option’s value has decreased for any specific day it is being traded. Since Theta has a negative influence on an option’s value, it is always represented as a negative value.

For the sake of an example, assume you have an option with a value of 3.25 and a corresponding Theta of -.25. You can expect that tomorrow the option’s value will decline to 3.00 (3.25-.25) as long as the underlying asset’s price opens the same as the previous closing.

The graph above shows what happens to an out-of-the-money Call option as it continues toward its expiration date. The amount the option’s value declines each trading is ultimately what the Theta determines. As the expiration date becomes closer, the option’s value declines at a faster rate.

VEGA

The Option Vega is the delta in the option’s value as the implied volatility increases by one percent. For both Calls and Puts Option Vega has a positive value. For example, assume you have an option with an estimated value of 3.5 and a Vega of .25. If the volatility shifts upwards 1%, the value of the option increases to 3.75. As you can see, Vega is influenced by a downward or upward shift (depending on when it becomes at-the-money) in the value of the underlying asset’s price.

The graph above shows how Option Vega and an underlying asset’s share price changes for three exercise values. The graph looks very similar to the graph for Option Gamma. Vega goes up as the option’s value goes up and goes down as the option’s value goes down. Also like Gamma, the Vega of Call and Put options is equivalent.

RHO

From the explanation of the Black-Scholes system we saw the interest rate is a key factor for determining volatility. Among the Greeks, Option Rho shows how the option’s value responds to fluctuations of the interest rate. This amount is stated as the total amount the option’s value would vary at every one percent change in the interest rate. In comparison to the other Greeks, Option Rho is considered to have the least influence on an option’s value.

The graphs above show three dates within an option’s life span, a range of exercise values, and an initial value of 100. These graphs show that Rho increases substantially as an option’s value shifts to being in-the-money and decreases at a consistent rate as the option’s value becomes out-of-the-money. You should notice that Rho becomes larger when the option’s expiration date is farther in the future.

How does this all happen? Rho is dependent on interest rates, so the amount it will cost you to hold an option will fluctuate depending on which direction interest rates travel. You can expect to pay more for options that are farther in the future and that are already in-the-money. This fact varies for Call and Put options.

In effect, forward price and premium discounting are what cause interest rates to influence option values.