An analysis of least squares regression and neural networks approximation for the pricing of swing options

Abstract Least squares regression was first introduced for the pricing of american-style options, but it has since been expanded to include swing options pricing. The swing options price may be viewed as a solution to a Backward Dynamic Programming Principle (BDPP), which involves a conditional expectation known as the continuation value. The approximation of the continuation value using least squares regression involves two levels of approximation. First, the continuation value is replaced by an orthogonal projection over a subspace spanned by a finite set of $m$ squared-integrable functions yielding a first approximation $V^m$ of the swing value function. In this paper, we prove that, with well-chosen regression functions, $V^m$ converges to the swing actual price $V$ as $m \to + \infty$. A similar result is proved when classic regression functions are replaced by neural networks. For both methods (least squares regression and neural networks), we analyze the second level of approximation involving practical computation of the swing price using Monte Carlo simulations and yielding an approximation $V^{m, N}$ (where $N$ denotes the Monte Carlo sample size). Especially, we prove that $V^{m, N} \to V^m$ as $N \to + \infty$ for both methods and using a Hilbert basis assumption in the least squares regression. Besides, a convergence rate of order $\mathcal{O}\big(\frac{1}{\sqrt{N}} \big)$ is proved in the least squares case.