Overview
Most quantitative models of commodities try to model the price of a commodity, whose characteristics often mirror the quantitative characteristics of equity models. As such, much of the math of commodity modeling is shared with their equity cousins.
The most common mathematical assumption for modeling a commodity is that it follows a geometric brownian motion. However, there are well known corrections to the GBM model.
Mean reversion
Many commodities are known to be mean reverting. That is, once the price deviates from a given mean value, there exists a pull back to that mean.
{% \Delta log S = a [\mu - log S] \Delta t + \sigma dW %}
(see Hull)Here {% \mu %} is the mean reversion level of the log price. Any deviations of the logarithm of the price from {% \mu %} will create a drift back to that level.
Convenience Yield and Failure of Arbitrage
The typical way to trade commodities is through the futures market. That is, traders can avoid the need to actually exchange a physical commodity and yet still gain exposure to the movements in price, by taking a position in the futures market.
As a general rule, it is thought that the futures price should track the theoretical forward price. (that is, take the spot price and then apply the interest rate for the term of the futures contract to arrive at a future price). In commodities, the futures price can often deviate significantly from the hypothetical forward.
This deviation from theory is due to what is termed, the "convenience yield". That is, it is believed that actually holding a commodity can yield benefits that do not accrue to the holder of a futures contract.
Cost of Carry
The cost of carrying a commodity is defined to be the cost of actually holding the commodity. This will include the current interest rate (i.e. the cost of financing the purchase of the commodity), and the storage costs.
Futures Price
The futures price of a commodity is often given by the formula:
{% F = S e^{c-y}t %}
- {% c %} is the cost of carry
- {% y %} is the convenience yield
see Hull chap 5
Models
- Gibson and Schwartz - two factor model
- Seasonal Models
- Stochastic Volatility Models - a common extension to the basic commodity model is the stochastic volatility model.