Global demand for

crude oil is about 65 million barrels daily, and it is one of the most important commodities in the world economy. The price of oil has so many ramifications on life that governments rise and fall by it. During the 1973 war in the

Middle East, crude prices tripled. The fall of the

Shah of Iran in the late 70's again increased prices, as well as unprecedented

volatility. During these times, world industry and

GDP took a huge hit; resulting in a global slowdown.

Politics aside, we are again entering a time of uncertainty in the Middle East, resulting in (as well as resulting from) volatility in oil prices.

However, this time we are equipped with a more sophisticated market and more advanced hedging instruments to offset the risk of war. Using swaps, forward contracts, futures, and options, anybody with significant exposure to the price of oil can be protected (for a price).

Very few of these contracts, especially the more sophisticated/complicated ones, are exchange traded (Major exchanges are the NYMEX in New York and the IPE in London). Therefore, it is very important to effectively model how the price of oil moves as the underlying asset to these contracts, in order to price these options correctly and achieve the most effective hedge.

Many people are familiar with the binomial lognormal diffusion model used to model price processes for securities like stocks, elegantly expressed in the Black-Scholes model. The binomial assumption is that in a sufficiently small timestep, the stock price can either move up or down. The up or down move on the next timestep is independant of whether the stock had moved up or down during the last timestep. Unlike stocks, oil has been observed to follow a mean-reverting process (i.e., prices are not entirely Markov). This means that oil prices tend to get pulled back to a central value.

Here, we can express the risk neutral process followed by the commodity price, S as:

dln(S) = (θ(t) - α ln(S))dt +σ dt

We arrive at this expression by tweaking the stochastic differential equation for the expected value, and differentiating (both sides, of course) with respect to time. We will leave this as an exercise for the reader.

α and σ are constant parameters, and the Θ(t) term captures seasonality and trends. This model is effectively the Vasicek model for short term interest rates, for which mean reversion is also a key factor. The constant parameters can be found by doing a linear regression on historical data. This price process can also be modeled using a trinomial tree (as opposed to a binomial lattice for stocks). A very similar binomial tree can also be used for modeling some sorts of real options (Not so coincedentally, the ones that depend on commodity prices).

For crude oil, the reversion rate parameter, α, turns out to be about .5, and the volatility, σ, turns out to be 20%.