Analytical Solutions

Abstract

In this chapter we introduce two stochastic processes that have been proposed to model the behavior of a number of commodities. We analyze their statistical properties, and also introduce their versions under risk neutrality. This paves the way to the valuation of derivative assets (like annuities, futures and options) when the underlying asset price evolves according to any of these processes over time. The first stochastic model is the Geometric Brownian Motion. This non-stationary process implies that stock price returns follow a normal distribution. This clearly imposes a precise model for asset price changes. The second model is the Inhomogenous Geometric Brownian Motion. This process displays mean reversion. The current price tends to a given level in the long term (which can be itself stochastic) at a certain speed, but is continuously impacted by random shocks. The GBM is nested in this model. Now, Ito’s Lemma allows bridge the gap between the changes in the price of the underlying asset and those in the value of the derivative asset. Thus, one can go from a model for carbon allowance price (resp., coal) to a model for the value of the option to invest in a carbon capture unit (resp., a coal-fired station). We derive formulas for the value of finite-lived annuities under both stochastic models, and similarly for futures contracts. Then we consider the perpetual option as the limiting case of a finite-lived option to invest. We fully develop a number of examples, both analytically and numerically.