Regression methods for pricing complex American-style options

TLDR

The paper proposes a simulation‑based approximate dynamic programming approach to price complex American‑style options in high‑dimensional state spaces. The method employs a finitely parameterized family of value functions, applying a variant of value iteration and evaluating them on a finite set of representative states, including a single time‑state‑dependent function variant. The study finds that arbitrary selection of representative states can cause exponential error growth, whereas sampling them from the risk‑neutral distribution keeps the approximation error bounded.

Abstract

We introduce and analyze a simulation-based approximate dynamic programming method for pricing complex American-style options, with a possibly high-dimensional underlying state space. We work within a finitely parameterized family of approximate value functions, and introduce a variant of value iteration, adapted to this parametric setting. We also introduce a related method which uses a single (parameterized) value function, which is a function of the time-state pair, as opposed to using a separate (independently parameterized) value function for each time. Our methods involve the evaluation of value functions at a finite set, consisting of "representative" elements of the state space. We show that with an arbitrary choice of this set, the approximation error can grow exponentially with the time horizon (time to expiration). On the other hand, if representative states are chosen by simulating the state process using the underlying risk-neutral probability distribution, then the approximation error remains bounded.