Expected touch time

M

MrAgi1

A rough guide(or approximation) to the probability of touch(POT) for a strike is the delta of its option x 2. For example, a 30 delta OTM option has approximately 60% POT.

Question: Is there also a simple rough formula to calculate the expected touch time given that we already know the probability of touch?

There is a concept called hitting time for a random walk. It estimates the time for a random walk to reach a point. It’s kind of related to my question(i.e assuming the market is random walk). However, I have not seen it’s application to stocks and other underlying assets.

Application: Knowing the expected hit/touch time would be useful. For example: Assume I sold a 45 day - 16 delta strangle. Except for usual market conditions, I should not expect the market to breach my strikes within the first 2 days. Based of the probability of touch the formula can be used to estimate a hitting time like after 50% of option life is gone(22 days).

Of course it’s just an estimate and the market can be irrational, but it can help a trader plan their trade adjustments and exits properly.
 
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The first example is for calculating p for stock going from 100 to <= 85.
The 2nd example is the same, but now for >= 85
The 3rd and 4th are similar for from 100 to 115.

One can calc the p if the option params are given (like DTE (here 60d), IV (here 50), etc.).
Of course p is from 0 to 1; multiply by 100 to make it a percent number.

But I doubt one can calc the said "expected touch time".
OTOH, maybe by multiplying t (ie. DTE) by p.

Code: 
Sx=85.000000 S=100.000000 s=0.500000 t=0.164384 r=0.000000 q=0.000000 u=0.000000 st=0.202721 ut=0.000000
S_at_-1SD[p=0.158655]=81.650584 S_at_0SD[p=0.5]=100.000000 S_at_+1SD[p=0.841345]=122.473098
z=-0.801686842122 p(z)=0.211367064473 --> p["<=85.0000"]=0.211367064473 pRest=0.788632935527

Sx=85.000000 S=100.000000 s=0.500000 t=0.164384 r=0.000000 q=0.000000 u=0.000000 st=0.202721 ut=0.000000
S_at_-1SD[p=0.158655]=81.650584 S_at_0SD[p=0.5]=100.000000 S_at_+1SD[p=0.841345]=122.473098
z=-0.801686842122 p(z)=0.211367064473 --> p[">=85.0000"]=0.788632935527 pRest=0.211367064473

Sx=115.000000 S=100.000000 s=0.500000 t=0.164384 r=0.000000 q=0.000000 u=0.000000 st=0.202721 ut=0.000000
S_at_-1SD[p=0.158655]=81.650584 S_at_0SD[p=0.5]=100.000000 S_at_+1SD[p=0.841345]=122.473098
z=0.689429290346 p(z)=0.754723421816 --> p["<=115.0000"]=0.754723421816 pRest=0.245276578184

Sx=115.000000 S=100.000000 s=0.500000 t=0.164384 r=0.000000 q=0.000000 u=0.000000 st=0.202721 ut=0.000000
S_at_-1SD[p=0.158655]=81.650584 S_at_0SD[p=0.5]=100.000000 S_at_+1SD[p=0.841345]=122.473098
z=0.689429290346 p(z)=0.754723421816 --> p[">=115.0000"]=0.245276578184 pRest=0.754723421816
 
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earth_imperator said:
The first example is for calculating p for stock going from 100 to <= 85.
The 2nd example is the same, but now for >= 85
The 3rd and 4th are similar for from 100 to 115.

One can calc the p if the option params are given (like DTE (here 60d), IV (here 50), etc.).
Of course p is from 0 to 1; multiply by 100 to make it a percent number.

But I doubt one can calc the said "expected touch time".
OTOH, maybe by multiplying t (ie. DTE) by p.

Code: 
Sx=85.000000 S=100.000000 s=0.500000 t=0.164384 r=0.000000 q=0.000000 u=0.000000 st=0.202721 ut=0.000000
S_at_-1SD[p=0.158655]=81.650584 S_at_0SD[p=0.5]=100.000000 S_at_+1SD[p=0.841345]=122.473098
z=-0.801686842122 p(z)=0.211367064473 --> p["<=85.0000"]=0.211367064473 pRest=0.788632935527

Sx=85.000000 S=100.000000 s=0.500000 t=0.164384 r=0.000000 q=0.000000 u=0.000000 st=0.202721 ut=0.000000
S_at_-1SD[p=0.158655]=81.650584 S_at_0SD[p=0.5]=100.000000 S_at_+1SD[p=0.841345]=122.473098
z=-0.801686842122 p(z)=0.211367064473 --> p[">=85.0000"]=0.788632935527 pRest=0.211367064473

Sx=115.000000 S=100.000000 s=0.500000 t=0.164384 r=0.000000 q=0.000000 u=0.000000 st=0.202721 ut=0.000000
S_at_-1SD[p=0.158655]=81.650584 S_at_0SD[p=0.5]=100.000000 S_at_+1SD[p=0.841345]=122.473098
z=0.689429290346 p(z)=0.754723421816 --> p["<=115.0000"]=0.754723421816 pRest=0.245276578184

Sx=115.000000 S=100.000000 s=0.500000 t=0.164384 r=0.000000 q=0.000000 u=0.000000 st=0.202721 ut=0.000000
S_at_-1SD[p=0.158655]=81.650584 S_at_0SD[p=0.5]=100.000000 S_at_+1SD[p=0.841345]=122.473098
z=0.689429290346 p(z)=0.754723421816 --> p[">=115.0000"]=0.245276578184 pRest=0.754723421816
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@earth_imperator Ok, that example is from? The code is for what language? I am not sure I understand.
 
Kevin Schmit said:
Quiz: Probability and ITM?
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@Kevin Schmit Just going back through my previous threads. I noticed if only I had paid careful attention to you answer a year ago, I would not even be making this thread. Research papers on that FPTD is a bit heavy to understand(for my level:D) and not as many videos or simple articles on the topic. It seems not to be a much discussed subject. However, it’s nonetheless informative, thank you.
 
stochastix said:
First Return Time or First Passage Time. Search those terms and you will find literature related to your query
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@stochastix The concept is very heavy(for my current level of understanding). I have a basic idea of the concept(with markov chain). The subject is very vast and used in different fields but what I am interested in is it’s application to stock prices. I can’t find any useful information on its application to stock prices.
 
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