Mathematical Finance (q-fin.MF)

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, 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 (regression 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 the regression functions are replaced by neural networks. For both methods (least squares or 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 Hilbert basis 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. Several convergence results in this paper are based on the continuity of the swing value function with respect to cumulative consumption, which is also proved in the paper and has not been yet explored in the literature before for the best of our knowledge.

Abstract: We present simple general conditions on the acceptance sets under which their induced monetary risk and deviation measures are comonotonic additive. We show that acceptance sets induce comonotonic additive risk measures if and only if the acceptance sets and their complements are stable under convex combinations of comonotonic random variables. A generalization of this result applies to risk measures that are additive for random variables with \textit{a priori} specified dependence structures, e.g., perfectly correlated, uncorrelated, or independent random variables.