Lakefield Corporation’s oil trading desk buys and sells oil products (crude oil and refined fuels), options, and futures in international markets. The trading desk is responsible for buying raw material for Lakefield’s refining and blending operations and for selling final products. In addition to trading for the company’s operations, the desk also takes speculative positions. In speculative trades, the desk attempts to profit from its knowledge and information about conditions in the global oil markets.

  

One of the traders, Lisa Davies, is responsible for transactions in the cash market (as opposed to the futures or options markets). Lisa has been trading for several years and has seen the prices of oil-related products fluctuate tremendously. Figure 4.46 shows the prices of heating oil #2 and unleaded gasoline from January 1986 through July 1992. Although excessive volatility of oil prices is undesirable for most businesses, Lakefield’s oil trading desk often makes substantial profits in periods of high volatility.

  

The prices of various oil products tend to move together over long periods of time. Because finished oil products are refined from crude oil, the prices of all finished products tend to rise if the price of crude increases. Because finished oil products are not perfect substitutes, the prices of individual products do not move in lockstep. In fact, over short time periods, the price movements of two products can have a low correlation. For example, in late 1989 and early 1990, there was a severe cold wave in the northeastern United States. The price of heating oil rose from $0.60 per gallon to over $1 per gallon. In the same time period, the price of gasoline rose just over $0.10 per gallon.

  

Davies believes that some mathematical analysis might be helpful to spot trading opportunities in the cash markets. The next section provides background about a few important characteristics of fuel oils, along with a discussion of the properties of blended fuels and some implications for pricing.

  

Characteristics of Hydorcarbon Fuels  

  

The many varieties of hydrocarbon fuels include heating oil, kerosene, gasoline, and diesel oil. Each type of fuel has many characteristics, for example, heat content, viscosity, freezing point, luminosity,

  

      

  

volatility (speed of vaporization), and so on. The relative importance of each characteristic depends on the intended use of the fuel. For example, octane rating is one of the most important characteristics of gasoline. Octane is a measure of resistance to ignition under pressure. An engine burning lowoctane fuel is susceptible to “engine knock,” which reduces its power output. Surprisingly, octane rating is more important than heat content for gasoline. In contrast, the most important characteristic of kerosene jet fuel is its heat content, but viscosity is also important. High-viscosity fuels do not flow as smoothly through fuel lines.

  

For the types of fuels Lisa Davies usually trades, the most important characteristics are density, viscosity, sulfur content, and flash point, which are described next.When trading and blending other fuels, characteristics besides these four are important to consider.

  

Density The density of a substance is its mass per unit volume (e.g., grams per cubic centimeter). The density of water is 1 g/cc. A related measure is American Petroleum Institute gravity (API), which is measured in degrees. API is related to density by

  

      

  

where D is density measured in g/cc. Water has an API of 10°. Note that density and API are inversely related.

  

The specifications for kerosene jet fuel are nearly identical for all civilian airlines worldwide. Kerosene jet fuel should have an API gravity between 37° and 51°. Diesel fuel and heating oil are required to have an API not less than 30°. API is important for controlling the flow of fuel in a combustion engine. It can also be used to limit the concentration of heavy hydrocarbon compounds in the fuel.

  

Viscosity Viscosity refers to the resistance of a liquid to flow. A highly viscous liquid, such as ketchup or molasses, does not pour easily. Viscosity is measured by the amount of time a specified volume of liquid takes to flow through a tube of a certain diameter. It is commonly measured in units of centistokes (hundredths of stokes). Most fuel specifications place upper limits on viscosity. Less viscous fuel flows easily through lines and atomizes easily for efficient combustion. More viscous fuels must be heated initially to reduce viscosity.

  

Sulfur Content The content of sulfur is measured in percentage of total sulfur by weight. For example, a fuel with 2% sulfur content has 2 grams of sulfur for every 100 grams of fuel. Sulfur causes corrosion and abrasion of metal surfaces. Low sulfur content is important for maintaining the proper operation of equipment.

  

Flash Point The flash point of a substance is the lowest temperature at which the substance ignites when exposed to a flame. The product description of kerosene jet fuel from the American Society for Testing and Materials specifies a flash point of at least 100°F. The New York Mercantile Exchange futures contract for heating oil #2 specifies a flash point of at least 130°F. Flash-point restrictions are often prescribed for safety reasons.

  

Table 4.20 gives a description of some fuels and their prices on a given day. In Table 4.20, the units of viscosity are centistokes, sulfur is given in percentage by weight, and flash point is in degrees Fahrenheit. For convenience, all prices in Table 4.20 are given in dollars per barrel. In practice, the prices of heating oil, gasoline, and kerosene jet fuel are typically quoted in cents per gallon. (There are 42 gallons in a barrel.)

  

    Blending Fuels  

  

Because hydrocarbon fuels are made of similar compounds and have similar characteristics, a certain degree of substitutability exists among fuels. Different fuels can also be blended to form a new fuel. Next we describe how the characteristics of the individual fuels combine in the blended fuel.

  

Sulfur combines linearly by weight. This means, for example, that mixing equal weights of a 1% sulfur oil with a 3% sulfur oil produces a 2% sulfur oil. To a close approximation, sulfur combines linearly by volume (because the densities of oils are not very different). That is, combining 0.5 barrel of 1% sulfur oil with 0.5 barrel of 3% sulfur oil gives 1 barrel of very nearly 2% sulfur oil.

  

      

  

In general, to say that a certain property of oil combines linearly (by volume) means the following: Suppose xj barrels of oil j (for j _ 1, 2, . . . , n) are blended together to form one barrel of oil; that is,  Also suppose that cj is the measure of the property of oil j. Then if the property combines linearly, the measure of the property for the blended oil is a linear combination of the cj’s; that is

  

API gravity does not combine linearly, but density does combine linearly. For example, consider blending 0.5 barrel of oil that has a density of 0.8 g/cc with 0.5 barrel of oil with a density of 1.2 g/cc. The resulting barrel of oil has a density of 1.0 (= 0.8[0.5] + 1.2[0.5]). The 0.8 g/cc density oil has an API of 45.38 , and the 1.2 g/cc density oil has an API of -13.58°. If API combined linearly, the blended barrel of oil would have an API of 15.90°(= 45.38 [0.5] -13.58 [0.5]). However, an API of 15.90 corresponds to a density of 0.96 g/cc, not 1.0 g/cc.¹¹ 

  

Viscosity, measured in centistokes, does not combine linearly. However, chemical engineers have determined that viscosity can be transformed to another measure, called linear viscosity, which (nearly) combines linearly.12 Similarly, flash points measured in degrees Fahrenheit do not combine linearly. But chemical engineers defined a new measure, termed linear flash point, which does combine linearly.¹³  Table 4.21 summarizes the properties of the 12 fuels measured in units that combine linearly.

  

    Implications ofr Pricing  

  

Sulfur in oil is a contaminant. Therefore, oil with a low sulfur content is more valuable than oil with a higher sulfur content, all other characteristics being equal. This relationship can be seen in Table 4.20 by comparing the prices of fuels 1, 2, and 3 and fuels 5,

  

      

  

6, and 7. Lower-density oils are generally preferred to higher-density oils, because energy per unit mass is higher for low-density fuels, which reduces the weight of the fuel. Lower-viscosity oils are preferred because they flow more easily through fuel lines than oils with higher viscosities. High flash points are preferred for safety reasons. However, because flash point and linear flash point are inversely related, this means that oils with lower linear flash point are preferred to oils with higher linear flash point.

  

That fuels can be blended cheaply to form new fuels affects price as well. For example, fuel 2 and fuel 3 from Table 4.20 can be blended to form a fuel with the same API, viscosity, sulfur, and flash point as fuel 1. In particular, 0.1304 barrel of fuel 2 and 0.8696 barrel of fuel 3 can be blended to form one barrel of a new fuel, which, in terms of the four main characteristics, is identical to fuel 1. Because the cost of blending is small, prices combine nearly linearly. The cost to create the blended fuel is $16.80 per barrel ($16.80  0.1304[13.25]  0.8696[17.33]). If the price of fuel 1 were greater than $16.80, say $17.10, Lisa Davies could create an arbitrage. She could buy fuels 2 and 3 in the appropriate proportions, Lakefield Corporation could blend them together, and Davies could sell the blend at the price of fuel 1. The profit would be $0.30 per barrel minus any blending and transaction costs. However, the actual price of fuel 1 is $16.08, so this plan does not represent an arbitrage opportunity.

  

The no-arbitrage pricing principle is simply a generalization of the previous example. No arbitrage means that the price of any fuel must be less than or equal to the cost of any blend of fuels of equal or better quality. As mentioned earlier, better means larger API, lower viscosity, lower sulfur content, and higher flash point. In terms of linear properties, better means lower density, lower linear viscosity, lower sulfur content, and lower linear flash point. Any number of fuels (not just two) can be blended together.

  

Davies would like to develop a system that automatically checks the no-arbitrage pricing condition for all of the fuels. If the condition is violated, she would like to know the appropriate amounts of the fuels to buy to create the arbitrage, the profit per barrel of the blended fuel, and the characteristics of the blended fuel.

  

Suppose that 0.3 barrel of fuel 2, 0.3 barrel of fuel 3, and 0.4 barrel of fuel 4 are blended together.What is the cost of the blended fuel? What are the (linear) properties of the blended fuel (i.e., density, linear viscosity, sulfur content, and linear flash point)?

120 Figure 4.46 Price of Heating Oil #2 and Unleaded 100 Gasoline 80 60 40 20 Jan-86 Jan-87 Jan-88 Jan-89 Jan-90 Jan-91 Jan-92 Date Heating Ol #2 --------- Unleaded Gasoline Price (cents/galon)