Pricing of commodity derivatives on processes with memory

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Spot option prices, forwards and options on forwards relevant for the commodity markets are computed when the underlying process S is modelled as an exponential of a process ξ with memory as e.g. a Lévy semi-stationary process. Moreover a risk premium ho representing storage costs, illiquidity, convenience yield or insurance costs is explicitly modelled as an Ornstein-Uhlenbeck type of dynamics with a mean level that depends on the same memory term as the commodity. Also the interest rate is assumed to be stochastic. To show the existence of an equivalent pricing measure Q for S we relate the stochastic differential equation for ξ to the generalised Langevin equation. When the interest rate is deterministic the process (ξ; ho) has an affine structure under the pricing measure Q and an explicit expression for the option price is derived in terms of the Fourier transform of the payoff function.