The ultimate goal of this note is to provide a background for, and describe, the Longstaff-Schwartz LSM (Least-Squares Monte Carlo) algorithm for evaluating derivatives with early (American/Bermudan) exercise. The reason for writing yet another paper on the matter is that this method is often presented in a complicated context, which makes the reading difficult, and obfuscates the rather simple idea of the algorithm. For example, the presentation in (Longstaff & Schwartz, 2001) comprises of a little example, and applications of the algorithm to more complex derivatives, without actually spelling out the algorithm. Indeed, Section 2.2, promisingly entitled “LSM algorithm”, elaborates more on which regression method should be used for approximating the expected value from continuation rather than the actual algorithm. An informal description of the method is arguably given in a short paragraph, which nevertheless fails to lay down the steps used in the backward traversal algorithm. Perhaps the authors thought that the actual steps of the algorithm are too simple, but wouldn’t that be a compelling reason to write them down? In addition to introducing unnecessary regression details, the presentation in (Brigo & Mercurio, 2006) brings in another complication: the algorithm is formalized in the context of a path-dependent payoff function, leading to a formalism arguably hard to follow. This paper aims at revealing that neither the regression method used for approximating the value from continuation, nor the path dependence (or not) of the derivative payoff are relevant to the method, from the algorithmic and conceptual prospective. They are implementation details that should be left at the latitude of the quant developer. Besides the presentation of the LSM algorithm, this note has a hidden agenda: to provide an introduction to the basic notions and concepts that can be heard in any discussion about derivatives with an early exercise provision. In Section 1 we introduced stopping times and stopped processes, optimal exercise strategies and exercise boundaries. Many statements are left without proof, or even justification at times, to allow the attention stay focused on the intuition behind these notions rather than be distracted by an exhaustive presentation of the framework in all its complexity. In the course of this section, we hope to have simplified even further the rather clear presentation in (Shreve, 2004), source which we recommend to be kept at hand. Incidentally, we hope to have clarified a rather ambiguous construct, which can be found in Shreve’s first book, Chapter 4 (page 98), in the definition of stopped process: the lattice operator ∧ (min) is rather ambiguously used over time steps and stopping times, despite of the fact that the stopping times are random variables over a tree’s paths (or equivalently, leaves) and not on the internal tree nodes. This remark will be mentioned in Section 2.2 in more detail. In addition, an equivalent definition of stopping times, in filtration terms and with a rather non-trivial justification, is presented in the second part of Section 2.1. Section 3 fulfills the mandate of this note: we revisit the example in (Longstaff & Schwartz, 2001) with additional details and hopefully in a more organized manner, and we spell out the mere one-page-short LSM algorithm.
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Oct 1, 2024
I.M Van Ruijven + 4
Open Access