Pricing of Commodity Derivatives on Processes with Memory

Abstract

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 Volterra equation driven by a Lévy process. Moreover, the interest rate and a risk premium ρ representing storage costs, illiquidity, convenience yield or insurance costs, are assumed to be stochastic. When the interest rate is deterministic and the risk premium is explicitly modelled as an Ornstein-Uhlenbeck type of dynamics with a mean level that depends on the same memory term as the commodity, the process ( ξ ; ρ ) 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.