Monte Carlo and American options are natural enemies: simulation runs forward in time, but the exercise decision requires a backward comparison — at every date, is the immediate payoff worth more than the (unknown!) value of continuing? The missing object is a conditional expectation, and the resolution — estimate it by least-squares regression across the simulated paths — is one of the most influential ideas in computational finance (Longstaff & Schwartz 2001). This note develops the algorithm in full, with the implementation details (ITM-only regression, basis choice, SVD, cash-flow bookkeeping) that separate a working implementation from a subtly wrong one. The theory of stopping times and the dynamic programming recursion is in Note 2, American options and early exercise; the hand-worked 8-path example is in Note 2, exam templates and worked examples.
Learning objectives. By the end of this note you should be able to:
- explain precisely why American + MC is hard, and what object the regression estimates;
- write the full regression algorithm and the Longstaff–Schwartz cash-flow variant;
- justify the ITM-only restriction and the special treatment of ;
- discuss basis functions, overfitting and the SVD solve;
- state the direction of the bias and why the cost becomes dimension-dependent.
In this lesson
- 5.1 Why American + MC is hard
- 5.2 Regression algorithm
- 5.3 Implementation details
- 5.4 Worked example
The full lesson, with worked examples and exercise solutions, is for members.
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