Joint client selection and contract design for a risk-averse commodity broker in a two-echelon supply chain

We study an expected payoff maximization problem for a risk-sensitive broker aiming to evaluate the merits of designing and underwriting an option contract on a traded commodity with geometric Brownian motion (GBM) spot price trajectories. Candidate firms for whom the contract would mitigate the commodity’s price risk, each face Poisson demands that are currently the broker’s responsibility to satisfy. Subject to a variance risk budget and a robustness requirement, the broker’s objective is jointly to (1) choose a so-called trigger price function that will fundamentally define the option contract, and (2) select a value-maximizing set of client firms to whom the broker will offer the contract. We reformulate the problem as a bilevel program whose continuous relaxation we transform into a single-level, univariate problem with a convenient property that makes it amenable to line search methods. The optimal solution for that single-level problem is then raw material for constructing the optimal solution for the original problem. Our theoretical and experimental findings indicate that the contract’s optimal value, and optimal trigger price function are both strictly monotone increasing in a cost parameter in the model, as well as in the GBM’s volatility coefficient. The findings also show that those two quantities are strictly monotone decreasing in the GBM’s drift coefficient. We conclude with a benchmarking sensitivity study which uses real-world data to study the implications of violating a certain constraint which implicitly bounds the optimal trigger price.