Monday, March 2, 2009


Delta, the fourth letter of the Greek alphabet, is one of the most commonly used Greeks. Delta can be used as follows:

1. Rate of Change
2. Equivalent Underlying Position/Hedge Ratio
3. Probability of an option expiring in-the-money

Rate of Change:
This is the characteristic of Delta, most people are familiar with. Delta measures an option’s sensitivity to movements in the stock price. More specifically it measures the change in an options value given a $1 increase in the underlying. Bullish positions have positive delta, bearish positions have negative delta. Remember the following table:

Positive Delta = Long stock, Long Calls, Short Puts
Negative Delta = Short stock, Long Puts, Short Calls

Delta ranges between 0 and 100. However, most option chains list delta on a per share basis, thus a positive 50 delta will be listed as .50 and a negative 50 delta will be listed as -.50.

In-The-Money options generally have delta greater than 50
At-The-Money options generally have delta about equal to 50
Out-of-the-Money options generally have delta less than 50

Knowing the relationship between delta and strike price, if we want to buy an option with a higher delta, we buy deeper in-the-money. Conversely, if we want to buy an option with a lower delta, we buy further out-of-the-money.

Understanding delta as the rate of change enables us to forecast our potential risk and reward given a certain price movement in the underlying stock. Let’s look at an example:
Suppose stock XYZ is currently trading at $50 and we buy a one month 45 strike call option with a delta of 60. Let’s further assume that our target on the stock is $55 and our stop loss is below $48. Using this target and stop, we can see that our potential reward is $5 (55-50), and our potential risk is $2 (50-48). The 60 delta implies that we make $60 for every $1 move in the underlying. Therefore, our potential reward on the option is $300 ($60 x 5) and our potential risk is $120 ($60 x 2).

Equivalent Underlying Position:

Equivalent Underlying Position: Suppose I own a call option with a delta of 50. If the stock increases $1, I will profit $50, and it the stock decreases by $1 I will lose $50. Therefore, my call option is the equivalent of 50 shares of the underlying.

Long Call with delta of 50 = 50 shares of stock

Suppose I own a put option with a delta of -50. If the stock rises $1, I will lose $50, and if it falls $1 I will profit $50. Therefore, my put option is the equivalent of shorting 50 shares of the underlying

Long Put with delta of -50 = short 50 shares of stock

Understanding this correlation should help you see how options can be used as a substitute for purchasing shares of stock. Furthermore, it helps to illustrate how options can be combined with stock to create delta neutral positions.
Suppose I had on the following positions in my portfolio:

Long call on XYZ with delta = +50
Short call with XYZ with delta = -75
Long 25 shares of XYZ

Combining these three positions on XYZ gives me a net delta of “0”. Although the call spread (long & short calls) has a net delta of -25, it is hedged by the long 25 shares of stock. Thus, theoretically whether XYZ rises or falls $1, I shouldn’t make or lose any money due to delta.

Hedge Ratio:

Hedge ratio simply refers to how many shares of the underlying you would have to buy/short to delta hedge an existing option position. As we’ve already established delta measures an option’s sensitivity to a $1 increase in the stock price. If you have a large delta position, then you have large exposure to price movement in the underlying. Conversely, if you have a small delta position, you have minimal exposure to price movement in the underlying. Suppose my delta position was “0”. Would my position be sensitive to movement in the stock price? Not really. In other words, whether the stock rises or falls, because my position is delta neutral I shouldn’t be making or losing any money. This can be quite beneficial on trades that are theta positive because you can somewhat eliminate your exposure to stock movement and still make money due to time decay.

Let me give you a quick example: Suppose I have an Iron Condor on stock XYZ. An Iron Condor is a positive theta trade that looks to exploit market stagnation and decreasing volatility. At trade inception the condor is usually delta neutral, but over time if the stock rises or falls too much, the trade can become delta positive or negative. If I wanted to, I could buy or short shares of stock every time my position delta got too positive or negative (such as 100 or -100), to adjust the position back to neutral.

One of the caveats with delta hedging is that delta is in a constant state of flux. In other words, it continually changes, requiring me to constantly monitor the position and readjust (e.g. buy or short more shares of stock) to stay delta neutral.

Probability of an option expiring in-the-money:

Delta calculates the probability of an option expiring in-the-money. For example, AAPL is currently trading at $89. The one month 80 strike call option has a Delta of 79. This means that there is a 79% probability that the 80 strike call will be in-the-money at expiration. Put another way, there is a 79% probability that the stock price will be above $80 at expiration. Now, we can use a little arithmetic to calculate the probability of an option expiring out-of-the-money. We can all agree that there is a 100% probability of the stock price residing somewhere at options expiration. If there is a 79% chance the stock will be above $80 at expiration, then it stands to reason that there is a 21% chance that the stock will reside below $80 at expiration. If the 80 strike call has a delta of 79, then the 80 strike put should have a delta close to 21. Generally the same strike call and put deltas should add up to 100. The formula for calculating the probability of an option expiring out-of-the-money is: 1 minus Delta.


KrengelFamily said...

This is a great post......i really like it. Its a lot of different ways of interpreting Delta that I was unaware of. My question is why does it imply the chances of the option expiring in/out of the money? I like the concept and think its great to determine trades, but am confused as to why Delta means that it has a ..% chance of expiring out of the money etc. Can you elaborate more on this?

Tyler said...

Thanks for the question- I'll elaborate on delta in a future post (hopefully in the next day or two). My answer may be a little lengthy, so I'd rather do it in a post rather than the comments section.

Chandra Singh said...

Are you sure short puts have +ve delta, as you have mentioned at the beginning of the blog? I think, short puts should have -ve delta.

Great work, by the way.

Tyler Craig said...


Thanks for stopping by. Let me see if I can shed some insight on your question:

Positive delta is synonymous with bullish, meaning that I want the stock to rise.

Negative delta is synonymous with bearish, meaning that I want the stock to fall.

If you BUY a put option, you want the stock to drop in value. Thus buying puts gets you negative delta. If you SHORT a put option, you're on the flip side of that equation. You want the stock to rise in value. Thus shorting puts gets you a positive delta.

Remember, there is a person on either side of the trade. It's a good intellectual exercise to think of it as a zero sum game. Let's say that I sell you a put option. Since you bought the put, you want the stock to drop. I on the other hand DON'T want the stock to drop. If the stock drops too much, you profit, I lose. IF the stock doesn't drop or rises, I profit and you lose.

So, we don't state that trading puts gets you negative delta. Rather we say BUYING puts gets you negative delta, SHORTING puts gets you positive delta.

Hope that helps-

Muhammad Amir said...

They asked me what it was, I told them fuckers it was ketchup
Nutty like my Chex mix, trading binary options

Changis said...

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I am now subscribed to your RSS feed and look forward to continual improvement in my options trading knowledge with your posts.
Thank you and keep blogging.

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