A multi-factor jump-diffusion model for commodities†

Abstract

In this paper we develop an arbitrage-free model for the pricing of commodity derivatives. The model generates futures (or forward) commodity prices consistent with any initial term structure. The model is consistent with mean reversion in commodity prices and also generates stochastic convenience yields. Our model is a multi-factor jump-diffusion model, one specification of which allows the prices of long-dated futures contracts to jump by smaller magnitudes than short-dated futures contracts, which, to our knowledge, is a feature that has not previously appeared in the literature, in spite of it being in line with stylised empirical observations (especially for energy-related commodities). Our model also allows for stochastic interest rates. The model produces semi-analytic solutions for standard European options, which enable option prices to be evaluated in typically about 1/50th of a second (depending upon parameter values and the required accuracy). This opens the possibility to calibrate the model parameters by deriving implied parameters from the market prices of options. We perform such a calibration on crude oil options and show that, allowing long-dated futures contracts to jump by smaller magnitudes than short-dated contracts, gives a greatly enhanced fit. †This is a condensed and enhanced version of a previous working paper (Crosby 2005) entitled “Commodities: a simple multi-factor jump-diffusion model”.