The delta, derived from a theoretical pricing model like Black-Scholes, is a number between 0 and +/-100 that has a variety of different uses and interpretations including:
- A hedge ratio
- Change in price of an option given a $1 change in the underlying stock
- The probability that an option will finish in-the-money
The table below shows call and put deltas over a range of strike prices. Note that the at-the-money 105 strike has 48 and -52 deltas for calls and puts respectively. Deep in-the-money deltas approach +/-100 while far out-of-the-money deltas approach 0.
Stock Price: $104
37 days to expiration
| Option | Delta | Option | Delta |
| 70 Call | 100 | 70 Put | -1 |
| 75 Call | 97 | 75 Put | -3 |
| 80 Call | 91 | 80 Put | -9 |
| 85 Call | 85 | 85 Put | -15 |
| 90 Call | 78 | 90 Put | -21 |
| 95 Call | 68 | 95 Put | -32 |
| 100 Call | 59 | 100 Put | -40 |
| 105 Call | 48 | 105 Put | -52 |
| 110 Call | 37 | 110 Put | -63 |
| 115 Call | 27 | 115 Put | -73 |
| 120 Call | 18 | 120 Put | -82 |
| 125 Call | 10 | 125 Put | -90 |
| 130 Call | 6 | 130 Put | -93 |
| 135 Call | 2 | 135 Put | -99 |
Hedge Ratio
Traders who use the delta as a hedge ratio do so to know how many shares of stock to buy or sell in order to establish a theoretically riskless hedge. By doing so, they are able to establish a position that should make money regardless of market direction.
Before establishing a hedge, it's important to remember the following:
- The delta of a stock position is always in a 1:1 ratio with the number of shares (in other words, 100 shares of stock have a 100 delta)
- Calls have a positive delta
- Puts have a negative delta
| Position | Delta |
| Long 100 shares of stock | + 100 |
| Short 100 shares of stock | - 100 |
| Long 1 call (45 delta) | + 45 |
| Short 1 call (55 delta) | - 55 |
| Long 1 put (70 delta) | - 70 |
| Short 1 put (60 delta) | + 60 |
Like other arbitrage strategies, delta neutral strategies are often used to capitalize on price discrepancies in the market. If the delta of a call option is 45 and the trader wants to stay delta neutral, it will be necessary to sell 45 shares of the underlying stock for every call contract purchased. Similarly, if the calls are sold, stock will have to be purchased to create a neutral hedge. As the stock price moves, the option delta also changes. For this reason, it may be necessary to adjust the position by buying or selling stock to remain delta neutral.
| Initial Position | Delta |
| Stock Price: $35 |
|
| Long 1 40 call | 38 |
| Sell 38 shares @ $35 | -38 |
| Position Deltas | 0 |
With a stock trading at $35, let's imagine that your theoretical pricing model shows 40 calls offered $0.75 below their theoretical value. Since each contract represents 100 shares, this adds up to a theoretically riskless profit of $75 per contract. To lock in this profit, you would buy the 40 calls (with a 38 delta) and sell 38 shares of stock for every 40 call you purchased. In this way, you establish a delta neutral position.
Stock Price: $39
| Position | Delta |
| Long 1 40 call | 49 |
| Short 38 shares @ $35 | -38 |
| Position Deltas | +11 |
| Adjustment |
|
| Sell 11 shares @ $39 | -11 |
| Adjusted Delta | 0 |
Later, if the stock jumps to $39 and the 40 call delta increases to 49, it will be necessary to sell an additional 11 shares of stock to remain delta neutral.
Stock Price: $30
| Position | Delta |
| Long 1 40 call | + 25 |
| Short 38 shares @ $35 | - 38 |
| Short 11 shares @ $39 (from adjustment) | - 11 |
| Delta | - 24 |
| Adjustment |
|
| Buy 24 shares @ $30 | + 24 |
| Adjusted Delta | 0 |
If the stock plummets from $39 to $30 and the delta of the 40 call drops to 25, it will be necessary to buy 24 shares (49 - 25) to remain delta neutral.
Throughout the life of the position, ongoing adjustments may be necessary to maintain the risk neutral position. At expiration, any out-of-the-money options expire worthless, any in-the-money options are sold (or exercised), and any long or short stock position is liquidated. At that point, the net result of all the trades should approximate the $75 profit per contract predicted by the model. This is the essence of how delta neutral trading works.
Now, let's examine how traders use delta to measure the change in price of an option as the underlying moves.
The Delta as a Measure of Changing Option Prices
One of the other common uses of the delta is as a measure of the change in an option's value given a $1 change in the underlying. For example, imagine that a stock trading at $75 has at the money options with the following prices and deltas.
Stock Price: $75
| Option | Price | Delta |
| 75 Call | $ 5 | + 52 |
| 75 Put | $ 4.75 | - 48 |
If volatility and all other factors remain the same and the stock price rises to $76, the price of the options will change by the amount of the deltas. More specifically, the call price will increase by $0.52 and the put price will decrease (because of the negative delta) by $0.48. Thus, the new option prices will be $5.52 and $4.27 respectively.
It is important to note that the delta of the 75 call and 75 put changes as the stock price moves. If you think in the case of an large price move, $10 for example, this makes sense. If the 75 put delta didn't change, a $10 price increase would imply that the put value would drop by $4.80 ($10 per share x -.48) bringing the value to -$0.05. That, however, is impossible because options never have negative prices. With the stock at $85, the 75 put will be worth significantly less than $4.75, but it will still have a positive value.
Deep In- and Out-of-the-Money Options
While the deltas of at-the-money options tend to hover near 50, the deltas of deep in- and out-of-money options tend to approach +/-100 and 0 respectively.
Using the example above, a 50 call might be considered deep in-the-money with the stock at $75. As such, it's value would consist primarily of its $25 of intrinsic value. For this reason, deep in-the-money options tend to move in tandem with the underlying stock. For example, a 100 delta option implies a $1 move in the price of the option for every $1 move in the underlying stock. Therefore, if the stock price dropped from $75 to $73, the 50 call would drop from $25 to $23.
Deep out-of-the-money options, the 50 put for example, have deltas that approach 0. In other words, since the option has no intrinsic value and as little as 1/16 of time value, it would take more than a $1 move in the stock to have an impact on the value of the put. In this case, it might take a $5 drop in the stock price to get the 50 put as high as 1/8.
With these examples in mind, it will be easy to understand the third interpretation that views the delta as a probability.
The Delta as Probability
Although purists might argue that the delta was not intended as a probability, there are many who view the delta as the likelihood that an option will finish in-the-money.
Consider the following option chain where deltas have been substituted for prices:
Stock Price: $104
37 days to expiration
| Option | Delta | Option | Delta |
| 70 Call | 100 | 70 Put | -1 |
| 75 Call | 97 | 75 Put | -3 |
| 80 Call | 91 | 80 Put | -9 |
| 85 Call | 85 | 85 Put | -15 |
| 90 Call | 78 | 90 Put | -21 |
| 95 Call | 68 | 95 Put | -32 |
| 100 Call | 59 | 100 Put | -40 |
| 105 Call | 48 | 105 Put | -52 |
| 110 Call | 37 | 110 Put | -63 |
| 115 Call | 27 | 115 Put | -73 |
| 120 Call | 18 | 120 Put | -82 |
| 125 Call | 10 | 125 Put | -90 |
| 130 Call | 6 | 130 Put | -93 |
| 135 Call | 2 | 135 Put | -99 |
First, let's look at the at-the-money options. In this case, the closest strike to $104 is the 105 strike. Here, we see that the 105 calls have a + 48 delta while the 100 puts have a -52 delta. When viewing delta as a probability, it doesn't matter whether the value is positive or negative. Only the number is important. Thus, the 52 delta of the put can be interpreted as a 52% probability the option will finish in-the-money. Considering the option is already $1 in-the-money with the stock at $104, it makes sense that the option would have a slightly better than even chance of finishing in-the-money. Similarly, the 105 call has a slightly less than even chance of finishing in-the-money. More precisely, the probability is 48%.
At every strike, the sum of the call and put deltas--all taken as a positive number--add up to approximately 100.
Deep In- and Out-of-the-Money Deltas
Looking at the 135 strike, we see the call and put deltas at 2 and 99 respectively. With the stock at $104, this can be interpreted to mean there is a 99% probability the 135 put will finish in the money. At the same time, there remains an outside probability (roughly 2%) the stock will rally above 135 so the 135 calls finish in the money. Not great odds no matter how you look at it.
How Deltas Behave Closer to Expiration
The closer the options get to expiration, the more the deltas tend to approach 0 and +/-100. Using the example above, if we fast forward from 37 until expiration to just 9 days, the deltas for each strike are markedly different. For the sake of comparison, we'll assume the stock price didn't move during the 28 days.
Stock Price: $104
| Days to Expiration |
| Days to Expiration |
| 37 | 9 |
| 37 | 9 |
| Option | Delta | Delta | Option | Delta | Delta |
| 70 Call | 100 | 100 | 70 Put | -1 | 0 |
| 75 Call | 97 | 100 | 75 Put | -3 | 0 |
| 80 Call | 91 | 100 | 80 Put | -9 | 0 |
| 85 Call | 85 | 99 | 85 Put | -15 | -1 |
| 90 Call | 78 | 95 | 90 Put | -21 | -5 |
| 95 Call | 68 | 87 | 95 Put | -32 | -13 |
| 100 Call | 59 | 71 | 100 Put | -40 | -29 |
| 105 Call | 48 | 48 | 105 Put | -52 | -52 |
| 110 Call | 37 | 26 | 110 Put | -63 | -74 |
| 115 Call | 27 | 10 | 115 Put | -73 | -89 |
| 120 Call | 18 | 3 | 120 Put | -82 | -97 |
| 125 Call | 10 | 1 | 125 Put | -90 | -99 |
| 130 Call | 6 | 0 | 130 Put | -93 | -100 |
| 135 Call | 2 | 0 | 135 Put | -99 | -100 |
As you can see, the further the option is out-of-the-money, the more its delta approaches 0 or +/- 100. Looking at the at-the-money 105 strike, we see that the deltas remain exactly the same. However, just one strike away, the 100 calls gain 12 deltas, while the 100 puts lose 11 deltas. Similarly, the out-of-the-money 110 calls lose 11 deltas. In other words, with only 9 days remaining until expiration, the probability that the 110 calls would finish in-the-money is only 26%. Just 28 days earlier, the same option had a 37% probability of finishing in-the-money.
A few strikes away, the difference is even more pronounced. The 125 calls which once had a 10% probability of finishing in-the-money now have only a 1% probability of doing so. Conversely, the 125 put now has a 99% probability of finishing in the money whereas before the probability was only 90%.
Pin Risk
It sometimes happens that the stock price at expiration is exactly the same as one of the strike prices. In the example above, if the stock closed at $105 on expiration, the 105 calls and puts would technically have a 50 delta up until the moment of expiration because the stock's next move, theoretically, has an equal probability of being up or down.
If it becomes apparent the stock will settle on a particular strike price, traders generally get out of the position if they are short options at the strike because they have no way to know how many contracts on which they will be assigned. This uncertainty is known as pin risk because they may find themselves unexpectedly short or long if they receive an assignment notice and the stock moves sharply against them.
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