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Longstaff–Schwartz Regression Targets for American Option Exercise

Article Quant Q&A · Author: Andrew Christianson
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Summary

The document discusses implementing the Longstaff–Schwartz least-squares Monte Carlo method for valuing American options. The author reports matching prices in one volatility case from the paper’s first table but obtaining lower estimates in a higher volatility case, and asks what the regression target should represent.

The response confirms that the target is the discounted value of future cash flows along each path, incorporating exercise decisions at later dates already determined by the backward algorithm. The regression estimates the continuation value conditional on not having exercised by the current date. An open source Java implementation is also suggested as a comparison point. The exchange clarifies the regression concept, but gives no diagnosis for the pricing discrepancy or detailed implementation guidance.

Key ideas

  • Longstaff–Schwartz estimates continuation value by regressing discounted future cash flows on state variables.
  • The regression target includes future exercise choices determined at later dates.
  • The method proceeds backward through the exercise dates.
  • An open source implementation can help compare an independent implementation.

Tags

OptionsDerivatives pricingMachine learningBacktesting
Full text
# Longstaff-Schwartz (Least Squares Monte Carlo) applied to American Options


# 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?

## Answer by Christian Fries (score 4, accepted)

https://quant.stackexchange.com/a/7798

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/

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.