Least-Squares Monte Carlo: Multivariate Evidence and Convergence Theory

Summary

The exchange discusses whether least-squares Monte Carlo (LSM) for Bermudan options has numerical support and theoretical guarantees beyond a limited proof based on two exercise dates and a Markovian state on positive values. Mark Joshi replies that substantial further work exists, pointing to a study of practical policy iteration for obtaining rapid, tight bounds on Bermudan exotic derivatives through Monte Carlo simulation.

For theory, the answer cites earlier work by Carriere, which predates Longstaff and Schwartz and motivates describing the approach more generally as least squares. The post supplies pointers rather than explaining the algorithms, assumptions, numerical results, or proof statements in detail. It therefore signals a broader literature but does not itself establish how multivariate performance or convergence depends on basis choice, state dimension, or simulation design.

Key ideas

• The discussion asks about numerical evidence and proofs for LSM beyond a restrictive early result.
• A cited study examines practical policy iteration for bounding Bermudan exotic derivative values.
• The response points to Carriere’s earlier theoretical work as relevant to least-squares methods.
• The document references further reading but does not summarize its methods, assumptions, or findings.

Tags

# Least-Square Monte Carlo in multiple variable


# Least-Square Monte Carlo in multiple variable












The paper by Longstaff-Schwatz on Least Square Monte Carlo offers very little proof. The only proof they have given assumed the option can only be exercised at two different time point and the price dynamics is supported on $(0,\infty)$ and is Markovian.

I have two questions

- Are there further numerical studies on LSM for multiple variables?

- Has people proven made any more progress proof-wise? (or do practitioners simply not care and cross their fingers?)

## Answer by Mark Joshi (score 2)

https://quant.stackexchange.com/a/21658

there has been a huge amount of work on this. In terms of numerical studies, see my paper

Beveridge, Christopher and Joshi, Mark S. and Tang, Robert, Practical Policy Iteration: Generic Methods for Obtaining Rapid and Tight Bounds for Bermudan Exotic Derivatives Using Monte Carlo Simulation (January 23, 2009). Available at SSRN: http://ssrn.com/abstract=1331904 or http://dx.doi.org/10.2139/ssrn.1331904

For a proof

https://ecommons.cornell.edu/bitstream/handle/1813/9176/TR001296.pdf

Carriere's work preceded Longstaff Schwartz and it is probably better to call it least-squares.

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