# The Greeks: Delta

*Understanding how options behave means knowing about the various options “Greeks.” Here, we take a look at the most important of the measurements, delta.*

If you want to understand options, you have to understand the “Greeks.” An option lives for a short time in an environment of constantly changing factors that affect its value: the price of the underlying, the volatility of the underlying, interest rates, its own implied volatility, etc. The Greeks are simply numbers that tell you how sensitive an option is to changes in that environment.

**Delta**

If you choose to understand only one Greek, it should be delta, since delta measures the sensitivity of the option value to the factor most changeable in the option’s environment: the price of the underlying.

A call option that is far out-of-the-money is not very sensitive to changes in the price of the underlying. As an example, consider a call option with a strike price of 60 on a stock whose current price is 30. For this call option to become worth anything, the stock has to more than double before expiration.

This option may be worth $0.50 or so, depending upon the volatility of the stock and the time left before expiration. If the volatility is high it may be worth more, because high volatility will give the option a better chance of reaching 60 before expiration. If the volatility is low, there may be only a one in a million chance that the stock will trade above 60 before expiration, and the option value may be even lower than $0.50.

If the stock moves from 30 to 31, its chances of exceeding 60 before expiration have improved a bit, but not by much. And the option value increases, but again not by much. For this option, the value is not very sensitive to movement in the price of the underlying. This option has a delta that is positive but very small, nearly zero.

On the other hand, consider the same call option when the price of the stock is 90. This option now has an intrinsic value of $30, and a $1 increase in the price of the underlying will cause the intrinsic value of the option to increase to $31. The value of the option is at least the intrinsic value, but in this case it is probably not much more, and the value moves roughly one-for-one with the price of the underlying; it will increase (or decrease) about $1 for each $1 increase (or decrease) in the price of the stock. This option has a delta almost equal to 1.

What about an at-the-money option? Suppose the stock price is $30, and it increases to $31. The call option has now become in-the-money, but only slightly. As a rule of thumb, you won’t be far off if you use a delta of .5 for an an-the-money option. That is, the value will increase by $0.50 for each $1 increase in the price of the underlying.

A call delta is sometimes quoted as a number between 0 and 100. A stock option is an option on 100 shares of the underlying stock. Suppose you see a delta of 57. This means that the option in this environment (at this stock price, this time to expiration, this interest rate and volatility) behaves like 57 shares of the stock, meaning that your profit and loss on a one-option position will be similar to the profit and loss if you instead held 57 shares of the stock.

**Put deltas**

A put’s value decreases when the underlying price increases, and increases when the underlying price decreases, a negative relationship. A put delta is always negative. A deep in-the-money put has a delta of almost -1, which means that for each $1 increase in the price of the underlying the put value increases by -$1–i.e., it decreases. And if the underlying increases by -$1, the put value increases by -1 * -$1 = $1. An out-of-the money put has a delta close to zero but on the negative side of zero. And an at-the-money put has a delta around -0.5, meaning that (for equities) the put behaves about like a short position of 50 shares of the underlying stock.

**Putting delta to use**

One use is to see how much you expect the option price to increase for a small change in the price of the underlying. Suppose you are long an option with a delta of .4. If the stock price increases by $2, you would expect the option value (and price) to increase by about .4 * $2 = $0.80; if the stock price decreased by $3 you would expect the option value to decline by about .4 * $3 = $1.20. Because option prices typically follow value, the price should do about the same.

Another use is to determine the sensitivity of a position consisting of several options to movement of the underlying. Suppose you are long a slightly out-of-the-money call with delta equal to 0.4, and also long an in-the-money call with delta equal to 0.8. Your “portfolio” behaves like a portfolio consisting of a long position of 40 shares of the stock and a long position in 80 shares of the stock, which is to say like a portfolio consisting of a long position in 120 shares of the stock. The delta of your combined position is 0.4 + 0.8 = 1.2. Deltas add.

This is one of the principal uses of delta: You can add the deltas of the various options in your position to see the equivalent position in the underlying at this price level of the underlying (delta’s change with the price of the underlying, as you may have noted above).

**What is “delta neutral?”**

A spread is delta neutral if the combined delta of the two options in the spread is zero. Let’s look again at the above two calls, one with delta equal to 0.4 and the other with delta equal to 0.8. The first behaves like 40 shares of the underlying, so a position consisting of two of the behaves like 80 shares of the underlying, just like one of the second call. So if you buy two of the first and sell one of the second, you have a position that behaves like a position that is simultaneously long 80 shares of the stock and short 80 shares of the stock– i.e., a neutral position. Note also that if you sell two of the first call and buy one of the second, your combined position is also delta neutral.

Delta neutral positions are used to eliminate the effect of the underlying so the position gains or loses solely on the basis of the behavior of the options themselves. A typical delta neutral position is one which is long a number of underpriced options and short another number of overpriced or less-underpriced options, with the numbers chosen so that the equivalent stock of the long position is equal to the equivalent stock of the short position.

Suppose you have the following information on two calls on the same underlying:

Â | Delta | Price | Value |
---|---|---|---|

Call A | 3.00 | 6.00 | |

Call B | 5.00 | 4.50 |

Call A is underpriced and call B is overpriced, so you would like to purchase call A and sell call B. To make this position delta neutral you need the equivalent stock of the long position to equal the equivalent stock of the short position. Each A in the position behaves like 50 shares of stock, and each B in the position behaves like 75 shares of stock. So if you bought 3 of A and sold 2 of B short, the combined position would behave like 3 * 50 = 150 shares of stock long and 2 * 75 = 150 shares of stock short–it would be delta neutral.