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%.

A friend of mine asked me a few days ago whether the status of oil as a non-renewable resource means that it must rise in price over the long term. It's an interesting question. Here is a fairly classical economic answer. The point here is not to consider the possibility of short-term price shocks, which are quite a different sort of phenomenon, and one not related directly to the scarcity or abundance of oil in the ground. Rather, the point is to consider whether the fact that oil reserves develop at a rate that is negligible compared to the rate at which they are exploited means that oil as a commodity is destined to become ever-dearer.

I would contend that there are three prices of oil which we can consider, and two of them are highly relevant. The first cost is simply the nominal cost of any particular grade of oil at a point in time. That is to say, the cost in whatever unit of currency you care to use. These are not directly comparable, over time, because they do not factor in inflation. The second, more relevant, price is the nominal price adjusted for the rate of inflation. While there are difficulties in properly assessing the rate of inflation, a price that takes it into account as can best be managed does a better job of reflecting what the price of oil is, relative to other commodities. Economists call this the 'real' price of oil. The third price, which I will get into further below, is the relative price of oil as a factor of production, from the perspective of any firm and firms in aggregate.

Examining the price trend, firstly, there are considerations of supply and demand. These are fundamentally related to the rate of oil extraction, not to the total available reserves. That said, those watching the levels of reserves might anticipate future scarcity (doing things like choosing less oil intensive technology or making bets on higher oil prices in commodity markets). Sticking to flows for the moment, there does seem to be a considerable extent to which oil production can be increased in the medium term. Especially given today's high oil prices, fields that were previously not commercially viable have become so. Likewise, fields that were depleted to the point where the cost of extracting an extra barrel of oil was at or below the value of that oil have become viable again. This kind of incentive will emerge whenever the real price of oil rises. The potential to bring new oilfields onstream in the medium turn should act to mitigate - though not eliminate - price rises in oil.

The next big issue is substitution. We use oil for a great many purposes: from powering vehicles to making plastics and fertilizers to generating electrical power. In some of these applications, it can be more easily replaced with alternatives than it can in others. While you would be hard-pressed to make many plastics without oil, electricity can certainly be generated in other ways. At present, the global system for distributing natural gas is far less extensive than the one for distributing oil. As greater scarcity and higher oil prices are experienced and anticipated, states will shift their energy production strategies towards those based on other technology. The degree to which such shifts can take place is called the elasticity of demand: the easier is it to substitute, the smaller price rises will be, both in the short and long term. In almost all cases, elasticity of demand becomes greater with time, as firms and individuals have more scope to modify their consumption and production choices.

Relative factor prices provide one market mechanism by which production choices are made. That is to say, if the price of an input - say labour - rises, firms will modify their production strategy in the short, medium, and long term to reduce the usage of that factor to an efficient level. How big the changes they make have to do with their anticipation of future movements in factor prices. Through the existence or anticipation of higher oil prices, firms will be driven to make production decisions that reduce their usage of oil, while increasing their usage of other commodities.

In the long term, major technological change also promises to help us shift away from oil. Biotechnology and genetic engineering promise ways to produce fuels and polymers from plants. Electrical generation based on renewable sources can offset that from hydrocarbons. Organizational change can also play a role. Power sources and power usage can be brought physically closer together, reducing the cost of transport. Likewise, the amount of travel undertaken by individuals and firms can be reduced through planning that minimizes it.

A final inductive point is that, while people have predicted for hundreds of years that all manner of minerals are in danger of running out, this has not taken place for any. Indeed, the real prices of commodities like gold, silver, and copper have been falling in the long term. For a lengthy statistical treatment of this, see Bjorn Lomborg's "The Skeptical Environmentalist." In particular, examine part three.

The short answer, then, is that we have reasons to believe that the real price of oil does not need to increase as the commodity itself becomes scarcer, provided the above assumptions about the capacity for factor substitution and technological change are accurate. Even if not, the same considerations indicate that price rises will at least be moderated in the medium and long-term. It should also be remembered that the overall phenomenon of economic growth increases the buying power of individuals and firms. That is to say, they can each afford more goods and services than they could before. As such, the total proportion of an individual or firms spending power devoted to oil need not grow at the same rate as the price of oil.

This has also been posted on my blog, at:

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