The purpose of this paper is to obtain a fractional Black-Scholes formula for the price of an option for every, a fractional Black-Scholes equation and a risk-neutral valuation theorem if the underlying is driven by a fractional Brownian motion ( ). For this purpose we will first prove some results regarding the quasi-conditional expectation, especially the behavior to a Girsanov transform. We will also compare our results with the classical results based on the standard Brownian motion and we conclude that in the case of the fractional Brownian motion the price of the option no longer depends only on t T − .

In this paper, we propose an easy-to-use yet comprehensive model for a system of cointegrated commodity prices. While retaining the exponential affine structure of previous approaches, our model allows for an arbitrary number of cointegration relationships. We show that the cointegration component allows capturing well-known features of commodity prices, i.e., upward sloping (contango) and downward sloping (backwardation) term-structures, smaller volatilities for longer maturities and an upward sloping correlation term structure. The model is calibrated to futures price data of ten commodities. The results provide compelling evidence of cointegration in the data.Implications for the prices of futures and options written on common commodity spreads (e.g., spark spread and crack spread) are thoroughly investigated. Meeting held in Amsterdam for helpful comments and suggestions. We are also grateful to AXPO AG for sharing data.

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