Using Option Delta to Choose Covered Call Strikes by Probability

# Calculating a Covered Call Strike with N% Probability that Shares Won't be Called Away


# Calculating a Covered Call Strike with N% Probability that Shares Won't be Called Away












I'm starting to experiment with covered call strategies, and I'm trying to find the right strike price to sell for my covered calls such that I can maximize premium while being generally "confident" that the strike price won't hit and I can just bank the premium. "Confidence" would hopefully be an adjustable parameter in my calculations; I.e., maybe this week I'm willing to sell with 70% confidence whereas next week I'm more conservative and willing to sell calls with 95% confidence that shares won't be called away. My plan is to write weekly covered calls against steady/boring unsexy stocks.

I downloaded weekly adjusted close data and calculated the weekly absolute price move and the percentage move from one week to the next for the last 2 years. That became my sample set.

Here are some things I considered:

- I started by trying to calculate the normal distribution (sigma) for all of the absolute price changes over my sample. I was thinking that if I could settle on a price that's 2-sigma higher than the current price, then I'd have 95% confidence that my shares wouldn't be called away. But this is a pretty rudimentary approach, and potentially skews more to the positive side than negative (or vice-versa) depending on recent stock movement. I don't think this is a good approach.

- So then I tried separating the positive moves and negative moves, and doing a normal distribution over those samples. I was thinking If I set my strike price at `current price + average positive move + sigma` then I'd get a 68% confidence that my underlying would stay below the strike price. I think that method skews artificially high though, just based on where the majority of the distribution occurs if graphed on a number line.

- I also considered using a percentile of price moves over that sample and basing strategy on that, i.e., just choose the positive move that occurs at the 85th percentile and set the strike at `current price + 85th percentile price`, but... I don't think that's a statistically sound approach either.

So I've done some homework, but I'm pretty sure I'm not on the right track. And statistics are not my strong suit.

Is there a well-defined way to choose strike prices that meet a statistical confidence threshold like what I'm describing?

## Answer by amdopt (score 4, accepted)

https://quant.stackexchange.com/a/55189

Using an option's delta could be a quick and easy way to back into the probability that you are seeking. For example, if you are looking for an option that has a 70% chance of expiring worthless, you would look for an option with a delta of 1 - .70 or .30.

There are lots of resources regarding this technique and the potential inaccuracies of it. I don't know of any statistical technique that will be any better or worse for this purpose so please check the links below and look around for more information before using something like this to make a decision.

From Wikipedia: As a proxy for probability Main article: Moneyness

> The (absolute value of) Delta is close to, but not identical with, the percent moneyness of an option, i.e., the implied probability that the option will expire in-the-money (if the market moves under Brownian motion in the risk-neutral measure).[5] For this reason some option traders use the absolute value of delta as an approximation for percent moneyness. For example, if an out-of-the-money call option has a delta of 0.15, the trader might estimate that the option has approximately a 15% chance of expiring in-the-money. Similarly, if a put contract has a delta of −0.25, the trader might expect the option to have a 25% probability of expiring in-the-money. At-the-money calls and puts have a delta of approximately 0.5 and −0.5 respectively with a slight bias towards higher deltas for ATM calls. The actual probability of an option finishing in the money is its dual delta, which is the first derivative of option price with respect to strike.[6]

A note of caution from volcube.com regarding this in practice:

> This probability is highly theoretical. It is not a FACT about the options that will always be true. All it means is that if every assumption in the pricing model that has been used to formulate the delta turns out to be true, then the delta can be interpreted as the probability of expiring in-the-money, in some cases. This is very unlikely to be the case consistently or even frequently. Volatility can be higher or lower than expected. Interest rates can move. Indeed, for some options where cost of carry or dividends are relevant, this interpretation of delta is even more precarious. Nevertheless, as a rule of thumb, option delta as the probability of expiring in-the-money is undoubtedly useful to know.

More info on interpreting delta in this way can be found here: Macroption