With Monte Carlo simulation, a single path cannot settle an exercise decision. On each exercise date, LSM regresses the future cash flows of the in-the-money paths on a function of the underlying asset's price, and uses the fitted value as the estimate of the continuation value in the exercise decision1.
It is used for Bermudan options, and with finely spaced exercise dates it also approximates an American option that can be exercised continuously2.
Notes
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Longstaff and Schwartz (2001), pages 122–123 (Section 2.2). Lyuu's lecture notes, Pages 927–928, also use information across paths through least squares. ↩
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The same paper, page 121 (Section 2.1). Continuous exercise can be approximated by making the number of exercise dates sufficiently large. ↩