Price Modelling in Carbon Emission and Electricity Markets

We present a model to explain the joint dynamics of the prices of electricity and carbon emission allowance certificates as a function of exogenously given fuel prices and power demand. The model for the electricity price consists of an explicit construction of the electricity supply curve; the model for the allowance price takes the form of a coupled forward-backward stochastic differential equation (FBSDE) with random coefficients. Reflecting typical properties of emissions trading schemes the terminal condition of this FBSDE exhibits a gradient singularity. Appealing to compactness arguments we prove the existence of a unique solution to this equation. We illustrate the relevance of the model at the example of pricing clean spread options, contracts that are frequently used to value power plants in the spirit of real option theory.