The Cost of Accuracy in the Least Squares Monte Carlo Approach
Gilles B. Desvilles
Conservatoire National des Arts et Métiers
December 7, 2010
Bankers, Markets and Investors, November 2011
This article follows in the footsteps of Longstaff and Schwartz’ seminal article about the use of regressions to model expectations in the valuation of American options with Monte Carlo simulation. The article repeats the original American put pricing in order to check for estimation accuracy and computation speed.
In addition the article investigates the use of the control variate technique in order to accelerate the Least Squares Monte Carlo simulation, and implements a way to get the delta sensitivity without much raising the response time.
However the results underline what is believed to be the main impediment of the approach: the cost of accuracy.
Performed in dimension one on a standard computer the simulations lead to conclude that pricing an option à la Longstaff Schwartz is not advised when the option is simple enough to be valued with a recombining binomial tree. Indeed the response times of the binomial pricing are incomparably shorter. Moreover the standard error proposed by the method under study is not reliable both in theory and in practice. There remains a mere conjecture according to which when increasing significantly the number of trajectories then convergence to the true price is reached and the estimated standard error is negligible. But, due to the involved pathwise regressions, such an increase would lengthen considerably the response time.
Finally hope comes from computer improvements, especially in the memory field. In the least resource-consuming cases running the simulation with much more trajectories on a recent computer ends up yielding the true prices with no surrounding uncertainty and in a reasonable time. Hence, for similar pricings, one can expect to rely on the estimated standard error to tell when the simulation has converged.
Number of Pages in PDF File: 14
Keywords: Least Squares Monte Carlo, Berry-Essien, derivation under summation sign, control variate technique
JEL Classification: G12, G13
Date posted: April 4, 2009 ; Last revised: January 18, 2011