pevece


Total Posts: 6 
Joined: Jul 2021 


Hello people, can anyone help me understand the question below?
Question: If a stock price is currently at 100$, what is the probability of the stock touching 85$(The 85$ is currently at a delta of 0.27 with the stock currently at 100$)? The answer to the question is 2*0.27=0.54 with the assumption that the probability of touch is approximate twice the delta.
Restrictions: However, a restriction is added to reduce the probability of the event happening: The stock can’t touch the 115$ price or above first before touching the 85$.
The 115$ is also at a delta of 0.27 with the stock currently at 100$. After 85$ is touched the stock can now touch any other price with no restriction.
So simply put the stock can touch any price below 115$. But before it can touch the 115$ price or above, it must have touched the 85$ price first.
Please can you help me with this question using probability? Thanks for your contributions in advance. 



nikol


Total Posts: 1366 
Joined: Jun 2005 


First hint: you forgot to mention maturity ot any kind of horizon, like number of steps (i see minimum 2), because you put some conditions here. 
... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c) 

pevece


Total Posts: 6 
Joined: Jul 2021 


Maturity can be of any value. Let's say with 30 days to expiration, stock XYZ is currently trading at $100, Its $115 strike call has a delta of 0.27 and its $85 strike put has a delta of 0.27.
As for conditions, it is one actually. The probability that the stock would touch $90. But you know the stock theoretically can rise to as much as $900 or more before going down to touch the $90. So, my condition is strictly limiting the rise to <$115 before touching 90$. After the $90 price is touched nothing else matters about the stock path.
So by that approximate formula: the probability of touching $90 is 2*0.27=0.54
But if the condition is added that the $115 price can not be touched first before $90, I think the probability should be less than 0.54.
I don't know if my reply made the question clear? 



nikol


Total Posts: 1366 
Joined: Jun 2005 


Second hint/question: "assumption that the probability of touch is approximate twice the delta" How can you get downtouch probability at 54%? It would then mean that uptouch probability is also 54%, right? Does it imply 108% total probability? How can that be?
Something is wrong here.
I presume, we are within BlackScholes framework.
This condition is 2steps."The stock can’t touch the 115$ price or above first before touching the 85$".
By the way, downbarrier = 85$ or 90$ ?? 
... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c) 

pevece


Total Posts: 6 
Joined: Jul 2021 


Yes, my mistake the down barrier is 85, not 90.
Yes actually you're right it is indeed a second condition.
So since my assumption is wrong, how can I solve it instead? 



nikol


Total Posts: 1366 
Joined: Jun 2005 


I didn't solved it, even didn't look into solution. I just help you to formulate the problem so you can do it yourself.
Try 1,2 step binomial model, plugin your touch probabilities, derive deltas and use them too. It will help you to calibrate up/down probabilities and you will get an answer. 
... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c) 

pevece


Total Posts: 6 
Joined: Jul 2021 


I would do that. Thank you. @nikol 



nikol


Total Posts: 1366 
Joined: Jun 2005 


Put an answer here. I m curious. Others might be too. 
... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c) 
