The Greeks

By Matt Krupski   ISSUE 611 | NOV 2007

I'm going to give an overview of some terminology related to options you may have heard of, but may be Greek to you. Understanding the "Greeks" gives you a better understanding of how an options value changes over time, and that information may help you make better trading decisions. I'll discuss four main Greeks: Delta, Gamma, Vega and Theta. There is one more, Rho, which relates changes in options values to changes in interest rates. I won't get into detail about Rho as its effect on option prices is typically small and is the least important Greek for the average investor to understand.

Delta

Delta is the most common Greek people refer to when discussing options pricing. It is defined as the sensitivity of an option's price to the change in the price of the underlying instrument it's based on. For example, if the S&P 500 index futures contract rises 10 points, delta tells us the change in the value of the S&P 500 index option.

Delta can also be thought of as the probability the option will finish in-the-money. When you have an option that's an in-the-money option, it has intrinsic value. If the option expired today, it wouldn't expire worthless. A call would have a strike price below the underlying price, and a put would have a price that's above. An option that has a 0.50 delta would have a 50 percent chance of finishing in the money, so it is right at-the-money. For example, if the S&P 500 index futures are trading at 1475, the 1475 S&P 500 put and call both have a delta of 0.50, and both are at-the-money.

In mathematical terms, delta is also the first derivative of the price of the option with respect to the underlying. Delta is banded between -1 and 1. If an option has a delta of 1, it acts like a long futures contract position. For every dollar the underlying futures contract goes up, the option's price increases by a dollar. This applies to calls, because the value of a call will increase as the chance of it finishing in the money increases. If an option has a delta of -1, it acts like a short futures contract position. If delta is -1, for every point the underlying goes up, the price of the option decreases by a dollar. This applies to puts. If you think about it logically, only puts can have a negative delta because the puts are less valuable as the market goes up. There is less chance of that option finishing in the money. So, puts have a negative delta ranging from -1 to 0, while calls have a positive delta of 0 to 1.

At-the-money options have a delta of 0.50 (for calls) or -0.50 (for puts). If the underlying instrument moves up one point, the call increases in value by 0.50, and the put decreases by 0.50. In-the-money options have higher deltas than out-of-the money options. That's because if an option is already in-the-money, it has a higher probability of finishing in-the-money at expiration than one that isn't. However, just because an option is in-the-money does not mean that it will have a delta of 1.0 or -1.0. If there is time left before expiration, then there is some probability that the option could still finish out-of-the money. So as the option appears more and more likely to finish in-the-money, its delta approaches 1 or -1.

Gamma

Gamma is the sensitivity of the change in the delta to the changes in the underlying. An option that's at-the-money has a 0.50 delta, as we discussed. As an option gets further into the money, its delta changes and approaches 1 or -1. The rate that its delta changes is the option's gamma.

At-the-money options have the greatest gamma. You can think of gamma in terms of extremes. Gammas are greater closer to expiration. With one minute left to expiration, say you have an option that is one tick out-of-the money. It will have a small delta. However, if the underlying moves a tick, and it is now in-the-money, the delta moved from nearly zero to almost 1, as it has a near 100 percent chance of finishing in-the- money. With three months left to expiration, a one-tick change won't have as much impact on the delta, as the probability of that option finishing in-the-money is lower with more time for the market to move before expiration. As the underlying moves away from the strike, the gamma decreases, and as the underlying moves toward it, gamma increases.

Let's say you have a call option, and the underlying goes up in price. Options that had been at-the-money will be now be slightly in-the-money, and the call with a 0.50 delta will move up, maybe to 0.60. The rate of that change is determined by its gamma. As the option gets more in-the-money, it starts trading like the future. In mathematical terms, the gamma is the second derivative in the change in the options price with respect to the underlying.

As the underlying goes up, the call values get more sensitive to the changes in the underlying, and the put values get less sensitive to changes in the underlying. That's because as the price goes up, the puts become further and further out-of-the money. On the flip side, as the price of the underlying declines, the value of the puts will become more sensitive to the price changes than the calls.

If you are long options (calls and/or puts) then you are long gamma. If you are long calls and the market is moving up, the value (and the delta) of the calls is increasing, so you are increasingly benefiting from the market's movement in your direction. The increasing positive deltas of your position correspond to an increase in long exposure. Similarly, if you are long puts and the market moves up, the deltas of the puts decreases, decreasing your negative delta, or short exposure. So, while you will be losing money on your long puts if the market moves upward, up movements gradually hurt you less and less. If you are short calls or puts, then you are short gamma. If you are short gamma then adverse movements in the underlying start to cause more and more losses. If you are short calls you are short delta, because you sold something with a positive delta. As the market moves up, the deltas of your short calls increases, increasing the size of your short exposure and increasing your losses. If you are short puts, you are long delta, because you sold something with negative exposure. However, as the market moves up, the deltas of your short puts decreases, thereby making you less long and you benefit from up movements less and less as the underlying increases.

Let's look at an example. You buy an at-the-money call with a value of 60. You know at-the-money calls have a delta of 0.50. The underlying instrument moves up a point. Your call value increases from 60 to 60.50, and it's now slightly in-the-money. If it had a gamma of 0.10, the option's new delta would be 0.60 and another one point move up would increase the value of the call by another 0.60, to 61.10.

Vega

Vega is a measure of the change of the value of the option due to a 1 percent change in implied volatility. When you think of implied volatility, remember that implied volatility means the volatility people think will happen between now and expiration, not what happened in the past. Implied volatility relates to expectations, not the realized volatility of the past. However, when the market has been volatile for some time, people anticipate that volatility will likely continue. So generally, implied and real volatility are fairly close in step. But there is a difference. For example, say a big economic number will be coming out the next trading day, and it's expected to have market implications. Implied volatility may increase, even though the market may not be moving today. People are anticipating a big price move after the release of the number tomorrow. The vega is a positive number for both calls and puts, because as implied volatility increases, the value of both calls and puts increase. If people are anticipating large price swings, the chances of any option finishing in-the-money would increase, so the value should increase too.

Theta

Theta is the measure of the drop in an options price due to the passage of time. This is also called time decay, and it's the erosion of the extrinsic value. Intrinsic value is how much an option is in-the-money, while the extrinsic is its time value. An options value is can always be expressed as intrinsic value plus extrinsic, or time value. At-the-money and out-of-the money options have an intrinsic value of zero by definition, so their value is purely time value. Theta is always negative for options. As a day or minute passes, the value of that option decreases. In-the-money options also have time value, because as long as there is time till expiration, there is a chance that the in-the-money option will finish even more in-the-money. The passage of time erodes the value for both puts and calls.

As a general rule, you can say that the higher an option's probability of finishing in-the-money, the more expensive the option. With 200 days before expiration, you have a lot of time on your side, so a passage of one day will not decrease the value of the option by very much. There is more time for something to move the market, and therefore a greater chance of an option finishing in-the-money. For each day that passes, you have less time to be right. Therefore, the value of the option decreases.

Theta does not reduce value at an even rate. It's highest for at-the-money options, and has the greatest effect as expiration approaches. The difference between 200 and 199 days to expiration isn't a big difference—there is still a lot of time. But as the days tick down and turn to minutes before expiration, theta will increase rapidly because there is less time for the market to make a move, and less time for you to be right and for your option to expire with value.

This is just an overview of one aspect of options, and I'd be happy to share my knowledge if you have further questions. There are a number of options strategies you can pursue in the futures markets, and I work with clients to tailor a unique strategy for their goals and risk tolerance.

Matt Krupski is a Senior Market Strategist with Lind Plus, Lind-Waldock's broker-assisted division. You can reach him at 877-847-3034 or via email at mkrupski@lind-waldock.com if you have questions on this topic, or to develop a trading strategy tailored to your specific needs and risk tolerance.

Kristina Zurla Landgraf is editor of Lind eWire. She can be reached by email at editor@lind-waldock.com.

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