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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.
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.