probability of in the money question

The delta of an ATM option is barely affected by change in pricing variables. It's like a flat lining EKG. Da-dead at ~50.

The delta of an option that is not ATM is obviously affected by underlying price change, but the passage of time and change in implied volatility also affects its value. The change is NON linear.

If you want to demonstrate this, use a pricing model and make up the numbers that result in a 90 put with a delta of 15 with the stock at 100. Then modify only price (drop it to 95). The delta might double to to 30 or so, doubling the probability (delta is an approximation of the probability of an option finishing ITM).

 

I agree with a lot of points made here.

So in short I’d have to say the answer to the original question would be no. There’s no linear equation that would work. It could be close but there’s many variables. By the time a security moves 10 points it would effect IV, the move would take time which would effect the theta, and gamma would differ, which all would effect delta. Delta is the price which to expect the option to move if the underlying moves one point(also % of expiring ATM or ITM). To complicate matters even more many of the equations used in the other Greeks that effect delta are non linear to begin with and they’re all changing in that move.

 

Good morning, all!
There is some confusion about what the OP requested/stated, in evidence by the responses. So, a general clarifier:

Linearity: VERY LITTLE in life is linear in toto. And this is *especially* true in anything connected to a Normal/logNormal distribution, as with markets/options.
However, what the OP asked was whether it reasonable to assume as a starting point, that a shift in a market price would produce a like shift in the associated option strike P(ITM)s. And the answer is yes. Why? The P(ITM) [or the associated deltas, for those using that proxy] is about distance from the underlying, not the starting point. A linear adjustment does not change the distance. It just shifts the distribution up or down.

Market, Time, Vol: These are the big three of those things that will affect option prices, to wit: delta, theta, and vega. And as the OP surmised, any circumstance sufficient to produce a $5 shift in a $100 stock is likely to produce a shift in IV as well -- whether an increase of fear, or a decrease from fears-allayed. And as the OP's question was kind-of snapshot oriented, it is fair to include theta-risk in there, too, to complete the picture: theta's bleed-out of value ticks on with every tick-of-the-clock, regardless of the market move towards/against our position. In a snapshot, it's commonly negligible, but for those watching today's expiring options from 1400hrs through market close, Mistress Theta (with whip in hand) will stomp all through their inventory -- and there will certainly be option values decreasing from theta-burn even as the market approaches them.

So, "linearity" of a shift? Yes. Great starting point. But vol shifts may counter or conjoin, dependent on the stimulus, and time (as time grows short) may work a hand in there, too.

 

Mr Gamma (who stands tallest when closest to the money) would highly disagree with you.

 

I think that you misunderstood what I wrote. For an ATM option, delta changes very little if other variables (other than price) change. Away from the money, the other variables come into play. Yabba gabba gamma doo

 

Is this well-established truth? If so, this is a very easy and useful axiom to remember.

 

I'm too new to have a theory, LOL.... I'm a beginner. I'm on about page 3 of a 1000 page book with this stuff, metaphorically speaking.

 

It's an approximation. Google for articles about it as well as POT calculators.

 

Also keep in mind that the probability of an OTM option becoming ITM increases if you are short the option and decreases if you are long the option. This is know as Murphy's Law.