Locating an American Put’s Early Exercise Boundary with LSM

Summary

The note explains how to estimate an American put’s early exercise boundary when using Longstaff–Schwartz least squares Monte Carlo (LSM). Its central method is to compare the option value at neighboring underlying-price nodes with the value from exercising immediately. A node where the values are equal and an adjacent node where continuation is more valuable bracket the boundary, giving an interval for its location.

To refine the estimate, increase the number of price points in the grid and narrow that interval. The answer cautions that finding the boundary this way is slow for a vanilla option and may be unreliable for options generally. It provides no numerical example or empirical validation, so the result should be understood as a grid-based approximation rather than an exact boundary. The guidance also applies to standard grid schemes, not only to an LSM solver.

Key ideas

• Compare immediate exercise value with the option value at underlying-price nodes to locate the boundary.
• Find neighboring nodes where the values meet and where continuation value is higher to bracket the boundary.
• Use a finer price grid to narrow the interval around the estimated boundary.
• The approach can be slow and unreliable, so treat the result as an approximation.

Tags

# Estimating early exercise boundary for American put


# Estimating early exercise boundary for American put












I am trying to estimate the early exercise boundary for an American put option. I can find the put value through the Longstaff-Schwartz LSM method.

How do I obtain the early exercise boundary within the same LSM framework?

## Answer by Brian B (score 4, accepted)

https://quant.stackexchange.com/a/8657

For a vanilla option, this is a very slow way to get the boundary, and it's somewhat unreliable for any option.

In either a more standard grid scheme or in a LS solver, you obtain the boundary by finding two nodes such that one of them has option value equal to early exercise value, and its neighbor has option value above early exercise value.

This gives you an interval of underlying prices between which the boundary must lie. To narrow down that interval, you must increase the number of points in your grid.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.