Probability of a stock to follow a specific pathway

[W-M-A]

Numerical example:

How can I calculate the probability of the complex event below to occur within a particular duration of time. If the event does not occur it is classed a failure.


The current price of a stock called XYZ is $100 (i.e delta of +-0.5 for an $100 strike call/put)


First Condition:

I need the stock to touch $90. Before touching the $90 any price may be touched that is less than $110. If $110 or higher is touched before the $90 that is a failure.


NOTE: When XYZ at the start was $100; A $90 put strike had a delta of -0.27 and A $110 call strike had a delta of +0.27.



Second Condition:

If the first Condition is satisfied(i.e $90 had just been touched without touching $110 first), then now XYZ can touch any price, even $110 or above but the new restriction is that it can not touch $80.


NOTE: Because price just touched $90 deltas have changed. However at the start when XYZ was at $100 the $80 put strike would have a delta of -0.21.


Delta is approximately twice the probability of touch. I need an approximate answer to what the probability of the stock to follow the complex part is going to be?


Thanks for your contributions.

What is your purpose, trying to arb some issuer that you have access to that you can trade with, or are you trying to price and quote them yourself?

Knock-ins can be priced from knock-outs and vice-versa. For example, the combination of a down-and-out call and a down-and-in call creates a standard European call, so the value of a down-and-in can be obtained by subtracting the value of a down-and-out from the value of a standard European call.

However calculating the risk neutral probability is somewhat more complicated, so my question before pointing you to some material or even try to expand it with you is... Answer honestly how comfortable are you with Ito's calculus and applying it to multiple Brownian processes with correlation?

I make use of these instruments from time to time as a hedge instrument when I have a large short position in the options and don't want to hedge only with the underlying. Even-though the hedge is not complete once they have been knocked out, but I somehow manage to tell myself that if the price have fallen so much that it has knocked out my hedge then the need to hedge against high prices diminishes...