# Longstaff Schwartz algorithm

I am new in finance, I have implemented the Longstaff Schwartz algorithm for pricing american otion - one asset (dimension = 1).

My questions :

Does this algorithm still efficient for a high dimension ? What are the other drawbacks of this algorithm ?

• LSMC algorithm was designed for and is usually used for $d>1$. (I think this design goal is mentioned in the LS paper IIRC). For $d=1$ there are many better non-MC algorithms, so we would not normally use LSMC in this case. – Alex C May 12 '19 at 17:36

As Alex C says in the comments, Longstaff and Schwarz did consider multiple factors and mention it as one of the advantages (page 114 in the journal):

By its nature, simulation is a promising alternative to traditional finite difference and binomial techniques and has many advantages as a framework for valuing, risk managing, and optimally exercising American options. For example, simulation is readily applied when the value of the option depends on multiple factors.

Emphasis mine. Disadvantages that spring to mind are that the method

• Is a Monte Carlo method which implies that results will have a simulation error;
• It is a priori unclear what kind of basis function work best and how many are needed for performing the least squares regression with success.

On the upside, the diagnostic test described on pages 127 and 128 can be used when testing whether the parameters one chooses work well for the problem at hand.

• Thanks for your answers, what about the rate of convergence when the dimension is very high? – Yass Abbah May 13 '19 at 12:13
• Check out section 8.1, it should work also when the number of dimensions become large, not sure what you mean by 'very high'. – Bob Jansen May 13 '19 at 17:21