TLDR version: first-hitting-time-in-years = log(K / S)^2 / (v * 0.6745)^2 That is an approximation of the median hitting time for GBM, the mean will be to the left of that. But median is, given the jump component of stock price innovations, probably close to the right number. Here is an approximation of the first hitting time density and cumulative density: Code: gbmFPTD <- function(S,K,T,v) { a <- log(K/S) / v dens <- (a / sqrt(2 * pi * T**3)) * exp(-a**2 / (2 * T)) return(dens) } gbmFPTCD <- function(S,K,T,v) { a <- log(K/S) / v cdens <- 2 * pnorm(-a / sqrt(t)) return(cdens) } The expression above is approximately the pdf integrated out to .5 probability mass. Edit: you can also easily incorporate first hitting time into CRR or other binomial pricers. And note that the cdf in the code block above is the familiar two times dual-delta without the discounting back to the present.
For those who may be interested in this topic in the future: The following attached excel calculator(I found online) may be useful and also the attached link. http://marcoagd.usuarios.rdc.puc-rio.br/hittingt.html#appendix There are some errors in the article but overall it would be helpful. Thanks especially to all for their contributions.
The concept it quite math heavy for my level of understanding, I was able to grasp a little bit of it. Even though the concept relates to what I am looking for, the expected hit time is not exactly what I wanted. Instead for Example: Assuming I give the model the following input values; volatility=30%, days(time frame/range)=30, probability of touching barrier b above spot=5%, no drift and assuming interest rate/yield is 0%. Assume we conducted 100,000 simulations of possible paths. question(using above inputs): For the few times that b is breached out of the 100,000 simulations, what is the expected average time left out of the 30 days?
I think the following screenshot from the above link explains the research problem in options maths (barrier options etc.) behind this topic good:
Man, this is so simple, IMO: Just run a GBM simulation using user defined params (initial spot, volatility, barrier spot etc.) for a timespan of say 10 years, say in daily intervalls (shorter is possible too, just takes more CPU time), and note after what time such a hit or cross occurs the first time, then stop this run. Then repeat this 100,000 times or even more, and each time collect the hit time in a table/vector. Now it's possible to compute or visiualize the average or median time for the barrier hit. Q.E.D. If you pay me say $10k I can do such a research for you, even a multithreaded ultra-fast solution in C++
Programming and financial modeling is your bread and butter I assume. So this things would be of course very easy for you. For me I am new to those concept but I am learning though. Unfortunately, most of those things are not as intuitive to me. I would look at simulation thing, thanks for that suggestion.
It’s similar. It does not explain exactly what I want(or maybe I just don’t understand it). Also I don’t understand why if drift is zero in that excel sheet for instance, the expected hit time goes to infinity no matter how close the barrier is to the current price and no matter how high volatility is.
Sorry, can't tell as I haven't tried out the XLS. If you post the formula in the XLS then maybe someone can tell more... As an alternative or workaround or "kludge" you can set the drift to the minimum allowed by this formula, ie. try 0.00001 or so... FYI: also the BSM formula has such a limit: for example IV cannot be 0, nor t.