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

  1. Are there further numerical studies on LSM for multiple variables?

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

  • 1
    $\begingroup$ It can be used for multiple time points. Philip Protter (check his web site) has a paper for the convergence. $\endgroup$
    – Gordon
    Nov 9, 2015 at 15:41
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    $\begingroup$ The Protter paper is available at ecommons.cornell.edu/bitstream/handle/1813/9176/…. $\endgroup$
    – Gordon
    Nov 9, 2015 at 18:17
  • $\begingroup$ @Gordon thanks a lot. I went to look at it when I saw your first comment. $\endgroup$
    – Lost1
    Nov 9, 2015 at 23:51

1 Answer 1


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


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

  • $\begingroup$ one more question if you do not mind, Mark. Are there papers on Americn options for SV model? Or from experience, do you know have intuition about which set of basis functions would be a good starting place for stochastic volatility models such as Bates model or BNS? $\endgroup$
    – Lost1
    Nov 10, 2015 at 14:19
  • $\begingroup$ well you can add the current instantaneous vol and its square to the basis functions $\endgroup$
    – Mark Joshi
    Nov 10, 2015 at 19:38
  • $\begingroup$ so you recommend trying polynomial ones rather than fancy ones? also, what about cross terms? $\endgroup$
    – Lost1
    Nov 12, 2015 at 0:35
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    $\begingroup$ I prefer more to go for enhancements than complicated basis functions. eg multiple regression see my Kooderive paper $\endgroup$
    – Mark Joshi
    Nov 12, 2015 at 1:56

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