# Longstaff-Schwartz (Least Squares Monte Carlo) applied to American Options

I'm working on an implementation in R of Longstaff & Schwartz method from the this 2001 article. I've managed to build code that replicates their prices in table 1 (p. 127), but only for the ones with volatility .2. For the .4 ones, my code estimates prices below theirs.

So, first I would like to ask what precisely the dependent variable in each of the LSM steps is? LS state: 'regress the discounted values of $C(\omega, s; t_{K-1}, T)$ on...' where the $C$ notation indicates 'path of cashflows'. I've taken this to mean that, at each step the dependent variable along each path is the exercise value of the option (either terminal or early if a pervious step has indicated early exercise) discounted to the current period. Is this accurate?

Second, does anyone know of a publicly available implementation of LSM for American options that I could check my work against?

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## 1 Answer

With respect to your first question: Yes. The regression has to determine the conditional expectation of the continuation value, i.e., the (discounted) value of the future cash flows including the exercise criteria(s) you have determined for the remaining future exercise times, conditional to the assumption that you did not exercise at or prior the current date. These are determined in a backward algorithm, going backward in time.

With respect to your second question: An open source Java implementation of the backward algorithm and the least square regression can be found at http://www.finmath.net/java/ - see also http://www.finmath.net/topics/bermudanoptionmontecarlo/

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