Abstract
This paper proves that the prices of options on forwards in commodity markets converge to the Black-76 formula when the short-term variations of the logarithmic spot price are a stationary Ornstein-Uhlenbeck process and the long-term variations are following a drifted Brownian motion. The convergence rate is exponential in the speed of mean reversion and time to delivery of the underlying forward from the exercise time of the option. This can be applied to energy markets like electricity and gas to argue for the use of Black-76 in pricing of options, although the spot prices may show large spikes. Furthermore, we prove that the quadratic hedging strategy converges in a similar fashion to the delta-hedge in the Black-76 model. The results are illustrated with a numerical example of relevance to energy markets.
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