Binomial Trees for Optimal Stopping with a Diffusion Process

Summary

The document poses an optimal stopping problem for a firm's profit modeled as a Brownian process with drift and volatility. The value is expressed as accumulated discounted profit up to a stopping time, then rearranged into a term involving an expected discounted function of the process at stopping. The author can estimate the function with Monte Carlo methods but is seeking a way to handle the stopping component.

A Nelson–Ramaswamy binomial approximation is used to represent the diffusion, and a tree approach has been applied to a related stopping problem with the process value as the payoff, analogous to valuing an American option. The unresolved issue is how to approximate the cumulative distribution function of the process on the tree so that the transformed stopping objective can be evaluated. No solution or numerical evidence is provided, so the document frames a computational question rather than a demonstrated method.

Key ideas

• The problem seeks an optimal stopping rule for discounted profit from a drifted Brownian process.
• The value expression separates accumulated profit from an expected payoff evaluated at the stopping time.
• A Nelson–Ramaswamy binomial tree can approximate the underlying diffusion for stopping calculations.
• The stated difficulty is approximating the process cumulative distribution function within the tree.
• The document presents an open numerical question without a completed solution.

Tags

# Binomial Tree for CDF


# 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!

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