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pevece


Total Posts: 6
Joined: Jul 2021
 
Posted: 2021-07-09 17:38
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: 1385
Joined: Jun 2005
 
Posted: 2021-07-09 20:28
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
 
Posted: 2021-07-09 20:56
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: 1385
Joined: Jun 2005
 
Posted: 2021-07-10 21:49
Second hint/question:
"assumption that the probability of touch is approximate twice the delta"
How can you get down-touch probability at 54%? It would then mean that up-touch probability is also 54%, right? Does it imply 108% total probability? How can that be?

Something is wrong here.

I presume, we are within Black-Scholes framework.

This condition is 2-steps."The stock can’t touch the 115$ price or above first before touching the 85$".

By the way, down-barrier = 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
 
Posted: 2021-07-16 17:32
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: 1385
Joined: Jun 2005
 
Posted: 2021-07-16 18:43
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, plug-in 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
 
Posted: 2021-07-17 10:30
I would do that. Thank you. @nikol

nikol


Total Posts: 1385
Joined: Jun 2005
 
Posted: 2021-07-17 17:45
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)
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