## The Greeks: What They Are And How To Use Them

##### Delta

The delta of a call can range from 0.00 to 1.00; the delta of a put can range from 0.00 to –1.00. Long calls have positive delta; short calls have negative delta. Long puts have negative delta; short puts have positive delta. Long stock has positive delta; short stock has negative delta. The closer an option's delta is to 1.00 or –1.00, the more the price of the option responds like actual long or short stock when the stock price moves.

So, if the XYZ Aug 50 call has a value of $2.00 and a delta of +.45 with the price of XYZ at $48, if XYZ rises to $49, the value of the XYZ Aug 50 call will theoretically rise to $2.45. If XYZ falls to $47, the value of the XYZ Aug 50 call will theoretically drop to $1.55.

If the XYZ Aug 50 put has a value of $3.75 and a delta of -.55 with the price of XYZ at $48, if XYZ rises to $49, the value of the XYZ Aug 50 put will drop to $3.20. If XYZ falls to $47, the value of the XYZ Aug 50 put will rise to $4.30.

Now, these numbers assume that nothing else changes, such as a rise or fall in volatility or interest rates, or time passing. Changes in any one of these can change delta, even if the price of the stock doesn't change.

Note that the delta of the XYZ Aug 50 call is .45 and the delta of the Aug 50 put is -.55. The sum of their absolute values is 1.00 (|.45| + |-.55| = 1.00). This is true for every call and put at every strike. The intuition behind this is that long stock has a delta of +1.00. Synthetic long stock is long a call and short a put at the same strike in the same month. Therefore, the delta of a long call plus the delta of a short put must equal the delta of long stock. In the case of the XYZ Aug call and put, .45 + .55 = 1.00. Remember, a short put has a positive delta. (Note: delta can be calculated with different formulas, which won't be discussed here. Using the Black-Scholes model for European-style options, the sum of the absolute values of the call and put is 1.00. But using other models for American-style options and under certain circumstances, the sum of the absolute values of the call and put can be slightly less or slightly more than 1.00.)

You can add, subtract, and multiply deltas to calculate the delta of a position of options and stock. The position delta is a way to see the risk/reward characteristics of your position in terms of shares of stock. The calculation is very straightforward. Position delta = option theoretical delta * quantity of option contracts * number of shares of stock per option contract. (The number of shares of stock per option contract in the U.S. is usually 100 shares. But it can be more or less, due to stock splits or mergers.)

So, if you are long 5 of the XYZ Aug 50 calls, each with a delta of +.45, and short 100 shares of XYZ stock, you will have a position delta of +125. (Short 100 shares of stock = -100 deltas, long 5 calls with delta +.45, with 100 shares of stock per contract = +225. –100 + 225 = +125)

A way to interpret this delta is that if the price of XYZ rises $1, you will theoretically make $125. If XYZ falls $1, you will theoretically lose $125. IMPORTANT: These numbers are theoretical. In reality, delta is accurate for only very small changes in the stock price. Nevertheless, it is still a very useful tool for a $1.00 change, and is a good way to evaluate your risk.

An ATM option has a delta close to .50. The more ITM an option is, the closer its delta is to 1.00 (for calls) or –1.00 (for puts). The more OTM and option is, the closer its delta is to 0.00.

Delta is sensitive to changes in volatility and time to expiration. The delta of ATM options is relatively immune to changes in time and volatility. This means an option with 120 days to expiration and an option with 20 days to expiration both have deltas close to .50. But the more ITM or OTM an option is, the more sensitive its delta is to changes in volatility or time to expiration. Fewer days to expiration or a decrease in volatility push the deltas of ITM calls closer to 1.00 (-1.00 for puts) and the deltas of OTM options closer to 0.00. So an ITM option with 120 days to expiration and a delta of .80 could see its delta grow to .99 with only a couple days to expiration without the stock moving at all.

The delta of an option depends largely on the price of the stock relative to the strike price. Therefore, when the stock price changes, the delta of the option changes. That's why gamma is important.

##### Gamma

Gamma is an estimate of how much the delta of an option changes when the price of the stock moves $1.00. As a tool, gamma can tell you how "stable" your delta is. A big gamma means that your delta can start changing dramatically for even a small move in the stock price.

Long calls and long puts both always have positive gamma. Short calls and short puts both always have negative gamma. Stock has zero gamma because its delta is always 1.00 – it never changes. Positive gamma means that the delta of long calls will become more positive and move toward +1.00 when the stock prices rises, and less positive and move toward 0.00 when the stock price falls. It means that the delta of long puts will become more negative and move toward –1.00 when the stock price falls, and less negative and move toward 0.00 when the stock price rises. The reverse is true for short gamma.

For example, the XYZ Aug 50 call has a delta of +.45, and the XYZ Aug 50 put has a delta of -.55, with the price of XYZ at $48.00. The gamma for both the XYZ Aug 50 call and put is .07. If XYZ moves up $1.00 to $49.00, the delta of the XYZ Aug 50 call becomes +.52 (+.45 + ($1 * .07), and the delta of the XYZ Aug 50 put becomes -.48 (-.55 + ($1 * .07). If XYZ drops $1.00 to $47.00, the delta of the XYZ Aug 50 call becomes +.38 (+.45 + (-$1 * .07), and the delta of the XYZ Aug 50 put becomes -.62 (-.55 + (-$1 * .07).

Position gamma measures how much the delta of a position changes when the stock price moves $1.00. Position gamma is calculated much in the same way as position delta.

Just as delta changes, so does gamma. If you were to look at a graph of gamma versus the strike prices of the options, it would look like a hill, the top of which is very near the ATM strike. Gamma is highest for ATM options, and is progressively lower as options are ITM and OTM. This means that the delta of ATM options changes the most when the stock price moves up or down. Let's look at a deep ITM call option (delta near 1.00), an ATM call option (delta near .50), and an OTM call option (delta near .10). If the stock rises, the value of the ITM call will increase the most because it acts most like stock. Even though the ITM call has positive gamma, its delta really doesn't get much closer to 1.00 than before the stock rose. The value of the OTM call will also increase, and its delta will probably increase as well, but it will still be a long way from 1.00. The value of the ATM option increases, and its delta changes the most. That is, its delta is moving closer to 1.00 much quicker than the delta of the OTM call. Practically speaking, the ATM call can provide a good balance of potential profit if the stock rises versus loss if the stock falls. The OTM call will not make as much money if the stock rises, and the ITM will lose more money if the stock falls.

Judging how gamma changes as time passes and volatility changes depends on whether the option is ITM, ATM or OTM. Time passing or a decrease in volatility acts as if it's "pulling up" the top of the hill on the graph of gamma, and making the slope away from the top steeper. What happens is that the ATM gamma increases, but the ITM and OTM gamma decreases. The gamma of ATM options is higher when either volatility is lower or there are fewer days to expiration. But if an option is sufficiently OTM or ITM, the gamma is also lower when volatility is lower or there are fewer days to expiration.

What this all means to the option trader is that a position with positive gamma is relatively safe, that is, it will generate the deltas that benefit from an up or down move in the stock. But a position with negative gamma can be dangerous. It will generate deltas that will hurt you in an up or down move in the stock. But all positions that have negative gamma are not all dangerous. For example, a short straddle and a long ATM butterfly both have negative gamma. But the short straddle presents unlimited risk if the stock price moves up or down. The long ATM butterfly will lose money if the stock price moves up or down, but the losses are limited to the total cost of the butterfly.

Gamma is a good reason to look at a profit/loss graph of your position over a wide range of possible stock prices.

##### Theta

Theta, a.k.a. time decay, is an estimate of how much the theoretical value of an option decreases when 1 day passes and there is no move in either the stock price or volatility. Theta is used to estimate how much an option's extrinsic value is whittled away by the always-constant passage of time. The theta for a call and put at the same strike price and the same expiration month are not equal. Without going into detail, the difference in theta between calls and puts depends on the cost of carry for the underlying stock. When the cost of carry for the stock is positive (i.e. dividend yield is less than the interest rate) theta for the call is higher than the put. When the cost of carry for the stock is negative (i.e. dividend yield is greater than the interest rate) theta for the call is lower than the put.

Long calls and long puts always have negative theta. Short calls and short puts always have positive theta. Stock has zero theta – its value is not eroded by time. All other things being equal, an option with more days to expiration will have more extrinsic value than an option with fewer days to expiration. The difference between the extrinsic value of the option with more days to expiration and the option with fewer days to expiration is due to theta. Therefore, it makes sense that long options have negative theta and short options have positive theta. If options are continuously losing their extrinsic value, a long option position will lose money because of theta, while a short option position will make money because of theta.

But theta doesn't reduce an option's value in an even rate. Theta has much more impact on an option with fewer days to expiration than an option with more days to expiration. For example, the XYZ Oct 75 put is worth $3.00, has 20 days until expiration and has a theta of -.15. The XYZ Dec 75 put is worth $4.75, has 80 days until expiration and has a theta of -.03. If one day passes, and the price of XYZ stock doesn't change, and there is no change in the implied volatility of either option, the value of the XYZ Oct 75 put will drop by $0.15 to $2.85, and the value of the XYZ Dec 75 put will drop by $0.03 to $4.72.

Theta is highest for ATM options, and is progressively lower as options are ITM and OTM. This makes sense because ATM options have the highest extrinsic value, so they have more extrinsic value to lose over time than an ITM or OTM option. The theta of options is higher when either volatility is lower or there are fewer days to expiration. If you think about gamma in relation to theta, a position of long options that has the highest positive gamma also has the highest negative theta. There is a trade-off between gamma and theta. Think of long gamma as the stuff that provides the power to a position to make money if the stock price starts to move big (think of a long straddle). But theta is the price you pay for all that power. The longer the stock price does not move big, the more theta will hurt your position.

Position theta measures how much the value of a position changes when one day passes. Position theta is calculated much in the same way as position delta, but instead of using the number of shares of stock per option contract, theta uses the dollar value of 1 point for the option contract. (The dollar value of 1 point in an option contract for U.S. equities is usually $100, but can be different due to stock splits.)

##### Vega

Vega (the only greek that isn't represented by a real Greek letter) is an estimate of how much the theoretical value of an option changes when volatility changes 1.00%. Higher volatility means higher option prices. The reason for this is that higher volatility means a greater price swings in the stock price, which translates into a greater likelihood for an option to make money by expiration.

Long calls and long puts both always have positive vega. Short calls and short puts both always have negative vega. Stock has zero vega – it's value is not affected by volatility. Positive vega means that the value of an option position increases when volatility increases, and decreases when volatility decreases. Negative vega means that the value of an option position decreases when volatility increases, and increases when volatility decreases.

Let's look at the XYZ Aug 50 call again. It has a value of $2.00 and a vega of +.20 with the volatility of XYZ stock at 30.00%. If the volatility of XYZ rises to 31.00%, the value of the XYZ Aug 50 call will rise to $2.20. If the volatility of XYZ falls to 29.00%, the value of the XYZ Aug 50 call will drop to $1.80.

Vega is highest for ATM options, and is progressively lower as options are ITM and OTM. This means that the value of ATM options changes the most when the volatility changes. The vega of ATM options is higher when either volatility is higher or there are more days to expiration.

Position vega measures how much the value of a position changes when volatility changes 1.00%. Position vega is calculated much in the same way as position theta.

##### Rho

Rho is an estimate of how much the theoretical value of an option changes when interest rates move 1.00%. The rho for a call and put at the same strike price and the same expiration month are not equal. Rho is one of the least used greeks. When interest rates in an economy are relatively stable, the chance that the value of an option position will change dramatically because of a drop or rise in interest rates is pretty low. Nevertheless, we'll describe it here for your edification.

Long calls and short puts have positive rho. Short calls and long puts have negative rho. How does this happen? The cost to hold a stock position is built into the value of an option. It all has to do with the idea of an option being a substitute of sorts for a stock position. For example, if you think the stock of XYZ is going to rise, you could buy 100 shares of XYZ for $4800, or you could buy 2 of the XYZ Aug 50 calls for $400. (2 XYZ Aug 50 calls would give me a position delta of +90 — pretty close to the XYZ stock position delta of +100.) As you can see, you would have to spend about 12X the amount spent on the options that you would spend on the stock. That means that you would have to borrow money or take cash out of an interest-bearing account to buy the stock. That interest cost is built into the call option's value.

The more expensive it is to hold a stock position, the more expensive the call option. An increase in interest rates increases the value of calls and decreases the value of puts. A decrease in interest rates decreases the value of calls and increases the value of puts.

Back to the XYZ Aug 50 calls. They have a value of $2.00 and a rho of +.02 with XYZ at $48.00 and interest rates at 5.00%. If interest rates increase to 6.00%, the value of the XYZ Aug 50 calls would increase to $2.02. If interest rates decrease to 4.00%, the value of the XYZ Aug 50 calls would decrease to $1.98.