Problem Statement and Computational Complexity

In order to find possibly optimal combinations between spot, forward, and European straddle option contracts, it is proposed to embed the three-moment utility function as formulated in Definition 12.9 in a single-period stochastic combinatorial optimization problem (SCOP) framework with linear constraints. Decisions are made solely at t = 0 and cannot be revised in subsequent periods. The decision horizon is finite and set to T = 1. Due to the current popularity of multiperiod financial models, a single-period description of the problem might be considered as disputable. In a dynamic modeling context, decisions are optimized in stages, because uncertain information is not revealed all at once. In our context this would require building an integrated dynamic simulation model that additionally describes the future development of the forward rates as well as the option premiums over the next 12 months in a stochastic manner. Obviously, such an approach would be much more difficult to model than our single-factor approach, contain significantly higher model risk (three risk factors plus interactions), and would require more computational resources. It is therefore doubtful if such a model would indeed be advantageous, especially if we consider the underlying decision context. In practice, firms’ currency hedging decisions are hardly enforced by speculative behavior towards future price developments of hedging vehicles. Instead, they rather concentrate on reward/risk estimates that depend solely on the underlying risk factor, i.e., the exchange rate in our case. Any further assumptions or risks are usually avoided. Hence, although a multiperiod decomposition of reward and risk poses an interesting theoretical problem, we argue that it is less relevant in practice. Therefore, only forward rates and option prices as known in t = 0 are relevant.