Financial boundary conditions in a continuous model with discrete-delay for pricing commodity futures and its application to the gold market

In this work, we approach the solution of a differential problem for pricing commodity futures when the spot price follows a stochastic diffusion process with memory, that is, it depends on two discrete times: the present instant and a delayed one. In this kind of models, a closed-form solution is not feasible to obtain and, in most of the cases, numerical methods should be applied. To this end, it is normal to introduce a bounded domain for the state variable, so suitable boundary conditions have to be established. The conditions based on mathematical reasons often introduce difficulties in the boundary and poor accuracy. Here, we propose new nonstandard boundary conditions based on some financial reasons and then, we face the numerical solution of the problem that arises. Some experiments are presented which show that the drawbacks in the behavior of the solutions are overcome, providing more accurate futures prices. This new procedure is implemented in order to obtain a more precise valuation of gold futures contracts traded on the Commodity Exchange Inc. (US).