A Different Way To Think About Delta

So far probably you have been given the textbook definition of delta. But here’s another useful way to think about delta: the probability an option will wind up in-the-money at expiration. Technically, this is not a valid definition because the actual math behind delta is not an advanced probability calculation. However, delta is frequently used synonymously with probability in the options world.

In casual conversation, it is customary to drop the decimal point in the delta figure, as in, “My option has a 60 delta.” Usually, an at-the-money call option will have a delta of about .50, or “50 delta.” That’s because there should be a 50/50 chance the option winds up in- or out-of-the-money at expiration. Now let’s look at how delta begins to change as an option gets further in- or out-of-the-money.

As an option gets further in-the-money, the probability it will be in-the –money at expiration increases as well. So the option’s delta will increase. As an option gets further out-of-the-money, the probability it will be in-the-money at expiration decreases. So the option’s delta will decrease.

Imagine you own a call option on stock XYZ with a strike price of 50, and 30 days prior to expiration the stock price is exactly 50. Since it’s an at-the-money option, the delta should be about .50. For sake of example, let’s say the option is worth 2. So in theory, if the stock goes up to 51, the option price should go up from 2 to 2.50. What, then, if the stock continues to go up from 51 to 52? There is now a higher probability that the option will end up in-the-money at expiration. So what will happen to delta? If you said, “Delta will increase,” you’re absolutely correct.

If the stock price goes up from 51 to 52, the option price might go up from 2.50 to 3.10. That’s a 0.60 move for a 1 movement in the stock. So delta has increased from .50 to .60 (3.10 – 2.50 = 0.60) as the stock got further in-the-money. 

On the other hand, what if the stock drops from 50 to 49? The option price might go down from 2 to 1.50, again reflecting the 0.50 delta of at-the-money options (2 – 1.50 = 0.50). But if the stock keeps going down to $48, the option might go down from 1.50 to 1.10. So delta in this case would have gone down to 0.40 (1.50 – 1.10 = 0.40). This decrease in delta reflects the lower probability the option will end up in-the-money at expiration. 

Like stock price, time until expiration will affect the probability that options will finish in- or out-of-the-money. That’s because as expiration approaches, the stock will have less time to move above or below the strike price for your option.

Because probabilities are changing as expiration approaches, delta will react differently to changes in the stock price. If calls are in-the-money just prior to expiration, the delta will approach 1 and the option will move penny-for-penny with the stock. In-the-money puts will approach -1 as expiration nears.

If options are out-of-the-money, they will approach 0 more rapidly than they would further out in time and stop reacting altogether to movement in the stock.

Imagine stock XYZ is at 50, with your 50 strike call option only one day from expiration. Again, the delta should be about .50, since there’s theoretically a 50/50 chance of the stock moving in either direction. But what will happen if the stock goes up to 51?

Think about it. If there’s only one day until expiration and the option is one point in-the-money, what’s the probability the option will still be at least 0.01 in-the-money by tomorrow? It’s pretty high, right? Of course it is. So delta will increase accordingly, making a dramatic move from .50 to about .90. Conversely, if stock XYZ drops from 50 to 49 just one day before the option expires, the delta might change from .50 to .10, reflecting the much lower probability that the option will finish in-the-money.

So as expiration approaches, changes in the stock value will cause more dramatic changes in delta, due to increased or decreased probability of finishing in-the-money.

Happy Trading!!!

Cheers.