# Implied Stopping Rules for American Basket Options from Markovian Projection

## TL;DR

This paper introduces a low-dimensional Markovian projection approach to efficiently approximate the pricing of high-dimensional American basket options, providing bounds and near-optimal exercise strategies with minimal error.

## Contribution

It proposes a novel method using Markovian projection and PDE techniques to price high-dimensional American basket options efficiently and accurately.

## Key findings

- Feasible for baskets with up to fifty assets.
- Achieves option price errors of only a few percent.
- Provides both lower and upper bounds for the option price.

## Abstract

This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and the Black-Scholes models. In high dimensions, nonlinear partial differential equation methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. It is shown that the ability to approximate the original value function by a lower-dimensional approximation is a feature of the dynamics of the system and is unaffected by the path-dependent nature of the American basket option. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early-exercise boundary of the option by solving a Hamilton-Jacobi-Bellman partial differential equation in the projected, low-dimensional space. The resulting near-optimal early-exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow to assess the accuracy of the proposed pricing method. Indeed, our approximate early-exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to fifty. In these examples, the resulting option price relative errors are only of the order of few percent.

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Source: https://tomesphere.com/paper/1705.00558