Thursday, March 5, 2009

Delta and Probabilities

I had a really good question in response to my post on delta that I wanted to highlight today.

"This is a great post......i really like it. It’s 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 it’s 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?"

Elaborating on delta and probabilities requires delving into a bit of option theory. I’m not an expert on statistics by any stretch, but hopefully the following explanation helps.

Option pricing is based on the likelihood of an event occurring (such as the stock reaching a certain price).

When we discuss the odds of the stock reaching a certain price, we use terms such as likely, unlikely, probable, improbable, possible, etc… The one problem with this is it’s not specific enough. In other words, it’s not enough to say it’s likely or unlikely for stock XYZ to reach $100 by expiration…. We need to be able to quantify how likely or unlikely it is. When pricing an option, the Black-Scholes Model assumes that stock prices are random and that there is just as much of a chance of the stock rising as there is falling.

In other words, if stock XYZ is trading at $100, then there is a 50% chance of us being above $100 in a month and a 50% chance of us being below $100 in a month. Thus the 100 call or put (at-the-money), should have a delta around 50. When explaining this, some option theory books show a picture of a bell curve to illustrate the likely outcomes of random events. Some brokers, such as Optionsxpress, display bell curves within their analytical tools.

Now, assuming there is a 50% chance of the stock residing above $100 at expiration, what is the probability of residing above $90? Well, it would have to be greater than 50%, so let’s say 65%. Thus, its delta should be around 65. What about being above $80 at expiration? Well it’s going to be higher than 65%, so let’s say 75%. Thus the 80 call option will have a delta of around 75. Hopefully you get the picture, the deeper in-the-money we go, the higher the delta, reflecting the higher probability of remaining above these strike prices.

What if we look at out-of-the-money calls? If the stock has a 50% chance of residing above $100, then it stands to reason that there is less of a chance of residing above 110, and even less of a chance of residing above 120 in a month. Thus, the delta of the 110 call will be less than 50 (let’s say 30) and the delta of the 120 would be even lower (such as 20).

If you want to read more on options theory, I'd recommend Option Volatility & Pricing by Sheldon Natenberg.

7 comments:

KrengelFamily said...

Thanks for the response...I really appreciate it. One thought to see if I understand everything. I understand that the model is unbias in that it looks at the current price and makes the chances based upon equal percent chance it moves up or down.

My thought then is this. When looking at these percentages (Delta's) can we take into account trends to adjust our outlook on the Delta. For example if the stock is at 100 then the 100 Strike has a delta of around .5, but if the stock has been trading very bullishly can we assume the actual percentage is more like 60% or whatever based upon trend? And of course the reverse in a bearish trending stock. Also, could we assume that a stock that has been flat lining more accurately follows the Delta due to it's neutral position?

Just wondering if I'm thinking along the right lines. To me at least this added implication of making mental trend adjustments could be a good tool for determing potential returns etc. Let me know what you think.

Tyler said...

When looking at these percentages (Deltas) can we take into account trends to adjust our outlook on the Delta?

Not really… We do consider trends in determining our outlook in the stock and whether or not we should place a trade. But, we don’t really have an outlook on Delta. Delta is what it is- an unbiased number based on the laws of probability. If I entered a bullish trade on a bullish stock, I’m pretty much betting the stock has a better chance of moving one way than another. In saying so, we don’t really have to recalculate “new” deltas to convey my bias.

Could we assume that a stock that has been flat lining more accurately follows the Delta due to its neutral position?

Not necessarily… As I mentioned in the blog, the Blakck Scholes model assumes that stock prices are random and that there is just as much of the stock rising as falling. IF that is truly the case, then over the long run the distribution of prices should be similar to the bell curve that we’ve mentioned. Whether the stock is currently in an uptrend, downtrend, or sideways – it shouldn’t matter.

Let me give you an example of when I look at Delta for a bear call spread.
1. Analyze price chart of stock to ensure I’m neutral to bearish
2. Analyze implied volatility to see if it is mid/low/high
3. Assuming number 1 & 2 support doing a bear spread, I may then look at delta to help determine which strike prices I should use in my bear spread
4. Given the strikes I’ve chosen, and probability of profit implied by their corresponding deltas, I then make sure I have an adequate reward to risk ratio.

Wayne said...

In the paragraph above the Trade & Probability Calculator chart you wrote this sentence - "In other words, if stock XYZ is trading at $100, then there is a 50% chance of us being above $50 in a month and a 50% chance of us being below $100 in a month." Did you mean to say - there is a 50% chance of us being above $100, not $50?

Tyler Craig said...

Quite right Wayne! My mistake. It should be fixed now. Thanks for the heads up.

Suneet said...

@Krengel: I can get the drift what you are hinting at. But as explained by Tyler, all the models assume lognormal distribution for stock prices and assume that the markets are efficient (all the information from trends and other indicators have already been captured by the current price). All the remaining bias to probability of up/down move is just a view, and hence doesn't affect the pricing of option.

Tyler Craig said...

Suneet,

Very well said. Thanks for the additional insight to Krengel's question.

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