A major advancement in the study of quantitative finance occurred when a number of disparate areas of modeling were consolidated into the affine jump-diffusion transform formalism (Duffie et al., 2000). The term *affine* simply denotes a constant term plus a linear term. Affine asset pricing generally involves defining state prices with the drift, diffusion, interest rate and jump process coefficients, which are affine functions. This structure ensures that the derivation of the model arrives at two unique Ricatti ordinary differential equations (ODEs). Often, a solution to the Ricatti equations is available, which gives the valuation equation in a closed form. This chapter will derive several affine models—of increasing complexity—that are relevant in commodity pricing.

In the affine mean reversion model, the logarithm of the spot is described by only one factor, a short-term variation , as given by

The short-term variation is described under a real-world measure as

This Ornstein–Uhlenbeck ...

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