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In addition to the 2 ways that we've seen to interpret the delta, there is a third. The delta of a Call, can be thought of as the probability that the stock will be ITM at expiration. This is not mathematically precise, but it is widely used and is considered a decent estimate. So based on the fact that the Feb 85 Call has a delta of 90, we can think that there is a 90% chance that the stock will be above 85 at the expiration of the February options. This is a sometimes useful tool, but be careful when using this information, it's based on a model, it's based on assumptions, and it's not mathematically precise.

Gamma
Well, if you thought delta was fun, you're going to love gamma! Gamma represents the change in delta, given a one point change in the underlying stock. It's different from the other Greeks in that it represents change in another Greek (delta) as opposed to a change in the options value. For the mathematicians out there, while the other Greeks are partial derivatives of the options price with respect to the corresponding variable, gamma is the second partial derivative of an options price with respect to a change in the underlying stock. Looking at some examples will bring this discussion back into English.

Let's assume that XYZ stock is trading at $25. We would expect the April 25 Call to have a delta of about 50. If the gamma is equal to, let's say 6, then we would expect the delta of the Call to be 56 if the stock increases to $26 and 44 if the stock decreased to $24. Similarly, the April 25 Put would have a delta of -50 with the stock at $25. An increase to $26 would result in a delta of -50 + 6 = -44 and a decrease to $24 would result in a delta of -56. There are 3 interesting things to note from this.

First, the gammas of both Puts and Calls are positive, whereas the deltas of Calls are positive and Puts are negative. Second, the gammas of the corresponding Put and Call are equal. This is always the case and is a function of most option pricing models. Next, note that if gamma stayed at 6, the delta of the Call option would eventually exceed 100 if the stock price kept increasing. We'd have the same problem on the downside, if the stock price continued to decrease, eventually the delta would turn negative. So we can surmise that the gamma cannot be constant. In fact, gamma attains its maximum value when the option is ATM. As the option goes ITM or OTM, the gamma decreases to 0. Professional traders also look at the rate of change of the gamma, affectionately referred to as the "gamma of the gamma." Don't worry, I won't mention that again.

Note, that like delta, we can come up with a position gamma, simply by summing the gammas of each of the options in the position. So if we take last week's hypothetical example and add gammas, we get:

(click to enlarge)

The final points I want to make about gamma now is that stock has no gamma, ie, it is always 0. Also, while Puts and Calls always have positive gammas, when the options are short, their gammas will be negative. If you have software that is keeping track of your Greek positions, it will automatically take care of the positive and negative signs and give you a proper total. However, if you're keeping track of your gamma position by hand, it is imperative that the correct signs be used. Until it becomes second nature, perhaps this table will help to keep things straight:

Vega
Right away, we have a problem with this fake Greek. Vega may have been a Chevy back in the 70's, but was never an authentic Greek letter! However, it is thoroughly ingrained in the language of options, although the politically correct crowd is trying to change the name to either kappa or tau, both of which are authentic letters of the Greek alphabet. I'm old fashioned so we'll continue to use vega here, but if you see kappa or tau somewhere else, you'll know what they're talking about.

Vega is the change in an options value that results from a change in volatility. I know that we haven't talked much about volatility yet, but it's one of the most important variables in the determination of an options theoretical price and we will discuss it at length at a later date.

Vega is expressed in dollars per a 1 percentage point change in volatility. For example, assume the theoretical value of the XYZ March 25 Put is $4.00 with a volatility of 30%, and a vega of .25. If nothing else changes but the volatility increases to 31%, we would expect the value of the Put to increase to $4.25. Likewise, if the volatility decreased to 29%, we would expect the Put to decrease to $3.75.

Like gamma, the vegas of long Puts and Calls are positive and the vegas of short options are negative. Also like gamma, the vegas of a corresponding Put and Call will be equal. And again, like gamma, stock does not have any vega associated with it. Unlike gamma however, vegas don't change very much for a given volatility, unless the stock price moves a considerable distance. Since changes in volatility will be more pronounced when there is more time to expiration, vegas will decrease as the time to expiration approaches. Also, bear in mind that the vegas for the ATM options are usually greater than the vegas for ITM and OTM options.

Theta
Theta represents the change in an options value based on a one day decrease in the amount of time until expiration. It measures the rate of time decay of an option and is measured in dollars per day. Assume the XYZ April 35 Call has a theoretical value of $6.00 with 90 days to go until expiration and a theta of .06. Tomorrow with only 89 days to expiration, the Call will be worth $5.94, assuming nothing else changes. It seems rather obvious that both Puts and Calls have negative thetas. Just remember that time is an enemy of long options and a friend to short options. Not as obvious is the fact that the Put and Call thetas are different, ie, they decay at different rates. Calls typically decay faster than Puts and therefore have higher thetas. Also, long term options have very low thetas, while near term thetas are large.

Higher volatility options will have higher thetas than lower volatility options and generally the ATM options will be higher than either the OTM or ITM options. Like our friends gamma and vega, stock does not have any theta associated with it.

Rho
Alas, we have come to the last of our Greeks. Rho represents the change in an options theoretical value based on a change in the risk free rate of return. It is measured in dollars per 1 percentage point change in the risk free rate. Calls increase as the risk free rate increases, while Puts decrease. So rho for Calls is positive and negative for Puts. For example, assume the XYZ July 45 Put has a theoretical value of $5.70 and a rho of -.12. Then if nothing else changes, but the risk free rate goes to 6% the Put will be worth $5.58. Likewise, if the rate declines to 4%, the Put will increase in value to $5.82. In times of low interest rates or stable interest rates, rho was not considered one of the more important Greeks. However, if you're trading LEAPS or in a situation where the rate is changing often, then rho can be very important. In fact when I was trading on the floor of the Amex in the 1990's there was a time when interest rates were in the 12-13% range. Rho was very important.