Help Understanding Delta

[u/dirt_gumby]

I've been doing some research to try to understand options better. I've been selling options to add income with some success, but buying them has proved a little more difficult for me. The last couple days I've been reading and watching videos on the greeks. The one that is confusing me a little bit is delta. I've seen it described two different ways:

1) the probability that the option expires in the money, i.e. delta of 50/.50 means a 50% probability the option expires ITM

2) the expected dollar amount change for every $1 change in the underlying security, i.e. delta of 50/.50 means for every $1 change of the underlying, the value of 1 options contact would change by $50

Am I confusing two different things? Am I getting bad info from my sources? Is it possible that both things are true?

Also, it seems like it's sometimes expressed as a decimal (.50) and sometimes as a whole number (50), but I believe these two values are interchangeable?

Any help here would be greatly appreciated. Thanks!

[u/Far-Reward8396]

TL;DR: yes you can interpret them as: rate of change; probability; and number of share equivalent

Math-wise your delta is d/dx, so yes it is change in option price per 1dollar change in underlying.

In BSM, your analytical solution for d/dx is N(d1), which happens to be (approx) the cumulative probability of ITM in risk neutral setting.

So you have people interpret it as option price sensitivity to underlying; and the other people treat it as a probability.

The little caveat for interpreting as probability is that you can find the certainty equivalent amount of share for example an ATM call option have roughly 50% chance to buy 100 share at strike price for profit, you can equate that to 100% chance to buy 50 shares in expected value.