# Binomial Tree for CDF

I'm tasked with solving an optimal stopping problem relating to stochastic process representing a firms profit namely $$X_t = X_0 + \mu t + \sigma Wt$$ where $$X_0, \mu$$ and $$\sigma$$ are constants.

specifically I need to find $$V(x,\mu) = \sup_\tau E^x [\int_{0}^{\tau} e^-rs Xs \,ds]$$ which after some calculations reduces to $$F(x) - \inf_\tau [ E^x [ e^-r\tau F(x_\tau) ]]$$

F(x) can now be found using Monte Carlo methods, however working out $$\inf_\tau [ E^x [ e^-r\tau F(x_\tau) ]$$ is proving to be very tricky. So far I've been trying to approximate it using a binomial tree and the Nelson Ramaswamy scheme ( I've succesfully approximated $$X_s$$ using Nelson Ramaswamy and have been able to work out $$\inf_\tau [ E^x [ e^-r\tau X_s ]$$ using binomial trees much in the same way as one would for an American option). However I've been struggling to approximate the cdf of $$X_s$$ using binomial trees and so have struggled computing $$\inf_\tau [ E^x [ e^-r\tau F(x_\tau) ]$$

Any help would be greatly appreciated!

• Hi: I have no experience with this kind of thing so take the following with a grain of salt. I assume that when you run an MC iteration, you obtain the value of $x_{\tau}$. So, given the value of $x_{\tau}$ for many MC iterations, you should be able to build an empirical CDF, $F(x_{\tau}$)$. That's the first step. But then how you integrate that empirical CDF is the next mystery ? I'm not clear on that. Jul 9, 2022 at 13:57 • Thanks for you're suggestion @markleeds , Apologies I made a bit of a typo including the integral where there shouldn't be one (that I've now edited) . I've managed to do as suggested, now my$F(x_\tau)\$ is no longer stochastic and my answer is still not quiet correct.
– lt12
Jul 9, 2022 at 14:33
• Hi It12: Intuitively ( but possibly wrong ) is that the expectation of F(x) should be 0.5 since you can sample from it as a uniform(0,1). I don't know if that helps and I honestly don't feel confident saying too much because I have no experience with this sort of simulation. Hopefully someone else can add to our discussion in a meaningful way. Jul 9, 2022 at 21:49