Pricing of commodity derivatives on processes with memory

TL;DR

This paper develops a model for pricing commodity derivatives where the underlying has memory, incorporating stochastic interest rates and explicit risk premiums, leading to analytical pricing formulas.

Contribution

It introduces a novel framework modeling the underlying as a process with memory and explicitly incorporates risk premiums and stochastic interest rates for commodity derivatives.

Findings

Explicit option pricing formulas derived using Fourier transforms.

Model accommodates memory effects in the underlying process.

Establishment of an equivalent pricing measure Q for the process.

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 {\xi} with memory as e.g. a L\'evy semi-stationary process. Moreover a risk premium \r{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 {\xi} to the generalised Langevin equation. When the interest rate is deterministic the process ({\xi}; \r{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.