Optimal exercise of swing contracts in energy markets: an integral constrained stochastic optimal control problem

TL;DR

This paper models the valuation of swing contracts in energy markets as a stochastic control problem, solving a Hamilton-Jacobi-Bellman equation with boundary conditions, including cases with strict constraints using a penalty approach.

Contribution

It introduces a novel approach to handle strict constraints in swing contract valuation via a penalty method and viscosity solutions of HJB equations.

Findings

Characterizes swing contract value as a viscosity solution of HJB equations.

Develops a penalty method for strict constraints in stochastic control.

Provides a unique polynomial growth solution with boundary conditions.

Abstract

We characterize the value of swing contracts in continuous time as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation with suitable boundary conditions. The case of contracts with penalties is straightforward, and in that case only a terminal condition is needed. Conversely, the case of contracts with strict constraints gives rise to a stochastic control problem with a nonstandard state constraint. We approach this problem by a penalty method: we consider a general constrained problem and approximate the value function with a sequence of value functions of appropriate unconstrained problems with a penalization term in the objective functional. Coming back to the case of swing contracts with strict constraints, we finally characterize the value function as the unique viscosity solution with polynomial growth of the Hamilton-Jacobi-Bellman equation subject to appropriate boundary conditions.