We study forward prices and prices of European call options, which are written on a renewable resource. The price of this resource is assumed to follow the inverse of a geometric mean reverting process. We assume that the resource is not tradable, until the option matures at time T and study the dynamics of the forward prices of the resource. In contrast to Black (1976) we show that forward prices do not evolve according to a geometric Brownian motion, but follow a more complex process. Even though, we are able to compute forward prices in closed form. For the case of an option we show that the Black (1976) formula needs to be adapted in such a way, that the normal distribution is replaced by a reciprocal T-distribution, to get at least a very good approximation of the true option price. We include numerical evidence to strengthen our result. We also include an analytic expression for the true option price in terms of an integral representation. Finally, we derive pricing formulas for options written on forward contracts, and show how forwards contracts can be hedged under the assumption that there is a spanning asset.

Christian-Oliver Ewald is an Associate Professor in Applied Mathematics at the University of Sydney. His main research fields are Mathematical Finance and Economics. He is also affiliated to the Center of Dynamic Macroeconomic Analysis at the University of St. Andrews (UK).

Slides: Christian-Oliver Ewald, Options on Renewable Resource: A New Version of the Black (1976) Pricing Formula for Commodity Options

Paper: Christian-Oliver Ewald, (SSRN) Options on Renewable Resource: A New Version of the Black (1976) Pricing Formula for Commodity Options