By Stan Freifeld, formerly, a floor trader of the American Stock Exchange and a market maker for options*
Posted: Jan 2, 2009

It's times like these when knowing your risks (Greeks) could save the day. Well, prepared options traders should be able to weather the storms quite well and, in fact, look at the increased volatility as an opportunity. Stock traders just using options for increased leverage may experience a different fate.

Remember: There are stock traders, options traders and stock traders who think they are options traders. Each group has a clearly defined yet different mindset. Anyway, let’s get back on track.

In "The Greeks Are Coming, Part 1", we started our discussion of the Greeks with delta, probably the most well known and popular of the Greeks. Today, we'll define the others, and next time, we’ll go into more detail and see how to actually use them.

Gamma Γ

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 approximately 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 decreases 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. Third, 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, as with the 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 the previous article's hypothetical example and add gammas, we get:

The final points I want to make about gamma now is that stock has no gamma (i.e, 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 you use the correct signs. Until it becomes second nature, perhaps this table will help keep things straight:

Vega V

Right away, we have a problem with this fake Greek. The Vega may have been a Chevy back in the 70s, but it was never an authentic Greek letter! However thoroughly it is ingrained in the language of options, 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 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 that the theoretical value of the XYZ March 25 put is $4.00 with a volatility of 30 percent and a vega of .25. If nothing else changes, but the volatility increases to 31 percent, we would expect the value of the put to increase to $4.25. Likewise, if the volatility decreased to 29 percent, we would expect the put to decrease to $3.75.

As with gamma, the vegas of long puts and calls are positive, and the vegas of short options are negative. Also as with gamma, the vegas of a corresponding put and call will be equal. And, again as with gamma, stock does not have any vega associated with it.

Unlike gamma, however, vega doesn't change very much for a given volatility, unless the stock price moves a considerable distance. Because 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 Θ

Theta represents the change in an option's 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 of short ones.

Not as obvious is the fact that the put and call thetas are different (i.e., 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. As with our friends, gamma and vega, stock does not have any theta associated with it.

Rho Ρ

Alas, we have come to the last of our Greeks. Rho represents the change in an option’s theoretical value, based on a change in the risk-free rate of return. It is measured in dollars per a 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 that the XYZ July 45 put has a theoretical value of $5.70 and a rho of -.12. If nothing else changes, but the risk-free rate goes to 6 percent, the put will be worth $5.58.

Likewise, if the rate declines to 4 percent, the put will increase in value to $5.82. In times of low interest rates or stable interest rates, rho is not considered one of the more important Greeks.

However, if you’re trading LEAPS or in a situation where the rate is changing often, rho can be very important. In fact, when I was trading on the floor of the Amex in the 1990s, there was a time when interest rates were in the 12 to 13 percent range. Rho was very important.

*Reprinted (and modified) with permission from Online Trading Academy www.onlinetradingacademy.com. You can email Stan Freifeld at: SFreifeld@tradingacademy.com.