Expected touch time

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.

 

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

 

Quiz: Probability and ITM?

 

First Return Time or First Passage Time. Search those terms and you will find literature related to your query

 

Thanks. Noted, I will.

 

Yes, stocks aren't brownian motion but same concept applies if u have a good model

 

@earth_imperator Ok, that example is from? The code is for what language? I am not sure I understand.

 

@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) 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.

 

It's just the result of calculations. It was intended for the case that you maybe want to verify your own values.
It's some advanced maths: using lognormal first calc the z value, then the p for that z...

 

@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.