Economic models for commodity prices

The main purpose of the project was to study modelling of commodity prices with special emphasis on the link to economically fundamental factors. In the original proposal it was assumed that the class of so-called affine models should be given special attention. However, recent research conducted in equity markets have found that non-affine model specifications seem to have significant advantages compared to affine models. There are no corresponding studies carried out for commodities. Therefore it was decided to perform an empirical study of crude oil prices that, among other issues, addresses the issue of model-specification.

One of the most important results form the project is with regard to the empirical results regarding model specification. The results show that non-affine models give a significantly better fit to data compared to purely affine models also for the crude oil market. Even though the price dynamics for oil are very different form equity indexes the overall ranking of models points in the same direction and clearly favours non-affine specifications. These are important results since models used in this part of the project are commonly used for valuation and risk management in the market for crude oil derivatives.

A second important result is with regard to risk premia in the crude oil market. The project also studies compensation for different types of risk (risk premia). Estimating and understanding risk premia is a classical area in financial economics and is of great importance in both practical and academic work. The results point to a significant volatility risk premium in the crude oil market. This result is in line with other studies. The project is the first to present results for a jump risk premium. The results indicate that compensation for jump risk is not significant in statistical or economical terms. Furthermore, the empirical results show that the volatility risk premium is robust to the inclusion of jumps in the model specification.

A third important result is the implications for commodity prices in an equilibrium model with the reserve base as the most important factor of production. The theoretical part of the project studies how commodity prices are determined in equilibrium where a representative producer chooses method of production optimally. The reserve of the commodity is the only productive factor in the model. The dynamics, or time evolution, of the reserve can be set in one of two possible regimes. In one regime the commodity is extracted a constant rate and the reserve is declining. The producer can, at a cost, switch to a second regime in which new reserves are added to the existing base. In this regime the reserve can therefore increase temporarily. The producers problem of choosing how to optimally switch between the regimes is a stochastic impulse control problem which can be solved explicitly. A first result is that the spot price of the commodity follows a non-affine process in equilibrium. The spotprice will have different properties depending on in which regime the producer operates. The paper also studies forward prices for different times to maturity. The forward curve consists of forward prices with different times to maturity. In practice one typically distinguishes between forward curves with positive slope or negative slope. This is a crude classification and the curve can have many different shapes. In the model the shape of the curve is highly dependent on how close the revenue process is to certain threshold values where it becomes optimal to switch regime. Implied volatilities calculated form option prices are also studied. It is demonstrated that just as the slope of the forward curve the implied volatility is strongly dependent on the revenue process and its current location in relation to threshold values. Since the revenue process is stochastic and time varying this property is inherited by the slope of the forward curve and the volatility. The theoretical implications for the forward curve and implied volatilities are the most important results of the project.

The project has generated several new research questions. The results concerning modelspecification show that non-affine models are worth studying further. Naturally one important question is if these results also apply to other commodity markets. The same comment can be made regarding the results on risk premia. There are very few studies of this kind and more empirical studies are needed. The theoretical results brings the question of how empirically relevant the results are for different time horizons. Another area for future research is to extend the theoretical analysis in different directions. One example would be to allow for storage.

The results of the project has been organised in two separate papers. The empirical results are presented in the paper "Model dynamics and risk premia in the short term market for crude oil". The paper is strongly motivated due to recent research in equity markets who show that non-affine models often gives a better description of the properties of prices and volatilities. Comparable studies are largely absent for commodity markets. The paper is focused on crude oil which is often recognized as the most important global commodity. The financial dmarkets are also the largest for financial assets such as futures and options linked to crude oil. A large dataset containing both exchange traded futures and option contracts is used to estimate and evaluate a number of advanced asset pricing models. The implications of model choice for risk premia and the valuation of a new type of derivative assets called variance swaps is given special attention. The choice to study variance swaps is motivated from the fact that these assets are used in the estimation method and that they are linked to volatility indexes that are published on many markets today (including the crude oil market).

The theoretical results of the project are presented in the paper "An equilibrium model for commodity prices with regime switching reserve dynamics". The paper takes the reserve base and its evolution over time as the most important factor that determines prices in equilibrium. Pricing of the most common derivative assets, forward and options, are given special attention.

Open access in the project will be secured either via publication in journals with an open access alternative, or via parallel publication according to the guidelines of RJ.

Publications:

Larsson, K., "Model dynamics and risk premia in the short term market for crude oil", working paper.

Larsson, K., "An equilibrium model for commodity prices with regime switching reserve dynamics", working paper.