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Original Articles

An efficient implementation of a least squares Monte Carlo method for valuing American-style options

Pages 1024-1039 | Received 26 Aug 2008, Accepted 24 Nov 2008, Published online: 01 Jun 2009
 

Abstract

Several methods for valuing high-dimensional American-style options were proposed in the last years. Longstaff and Schwartz (LS) have suggested a regression-based Monte Carlo approach, namely the least squares Monte Carlo method. This article is devoted to an efficient implementation of this algorithm. First, we suggest a code for faster runs. Regression-based Monte Carlo methods are sensitive to the choice of basis functions for pricing high-dimensional American-style options and, like all Monte Carlo methods, to the underlying random number generator. For this reason, we secondly propose an optimal selection of basis functions and a random number generator to guarantee stable results. Our basis depends on the payoff of the high-dimensional option and consists of only three functions. We give a guideline for an efficient option price calculation of high-dimensional American-style options with the LS algorithm, and we test it in examples with up to 10 dimensions.

2000 AMS Subject Classifications :

Notes

This conditional expectation is the orthogonal projection of the continuation value onto a subspace .

An implementation in C++ can be found in Citation38.

See Citation33. We use the following linear congruential method for generating a seed vector, , i=1, …, 623, a=214, 013, b=25, 31, 011, M=4, 294, 967, 296.

See Citation28.

Beginning with 1, growing by 100,000 for each run.

The basis suggested in Citation31 consists of a constant, the first five Hermite polynomials in X 1, X 2, X 3, X 4, X 5, , , X 1 X 2, X 2 X 3, and where X 1 is the highest value, and so on of a five-dimensional vector.

We refer to Citation21 for an overview of these sequences and especially to Citation10 Citation26 Citation38 for the Sobol sequence.

The initial vectors of the process have entries , , and for all d.

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