Forums > Basics > probability of touch with restrictions

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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: 1457 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=0.54But 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: 1457 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: 1457 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: 1457 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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