Optimal Exercise of Swing Contracts in Energy Markets: An Integral Constrained Stochastic Optimal Control Problem

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 HJB equation subject to appropriate boundary conditions.

Utility indifference pricing and hedging for structured contracts in energy markets

“…with terminal condition v(T, x, z; q) = qΦ(p(T, x), z). Moreover, one can weaken the boundedness of µ F and require only linear growth in x uniformly in t. This result extends to our setting previous ones in [3,11,16,22,45], which were obtained for particular types of structured contracts, e.g., swings and virtual storages, and without trading in forward contracts.…”

“…Its buying UIP will be characterized as the difference between the two log-value functions of the agent (with and without the contract), that can be obtained as the unique viscosity solutions of a suitable HJB equations. Our results are consistent with the ones in [3,11,16,22,45], in the case of complete market models. Moreover, the shape of such HJB equation gives reasonable candidates for the optimal withdrawal strategy of the structured product, as well as for the related hedging strategy.…”

“…for constants C > 0 and 0 ≤ m < M (see [3,11] and references therein). We will focus on the latter case, i.e., a non-zero penalty function Φ(P T , Z u T ) without any other contraints on the admissible controls.…”

A First‐order Bspde for Swing Option Pricing: Classical Solutions

“…Moreover, the global constraint (which is a finite fuel constraint) imposes that the total volume spent by the holder is bounded by one. We refer to Keppo () for the modeling of swing options as continuous‐time optimal control problems, and note that the above problem and related problems were recently investigated in a Markovian diffusion setting by Benth, Lempa, and Nilssen (), Dokuchaev (), and Basei, Cesaroni, and Vargiolu ().…”

“…Taking into account that this BSPDE is a non‐Markovian version of a Hamilton–Jacobi–Bellman (HJB) equation, we think that existence of a classical solution is a striking feature. Indeed, recent studies of the corresponding HJB equation for stochastic control problems with integral constraints in the Markovian diffusion case such as Basei et al () only discuss the HJB equation in the framework of viscosity solutions.…”