PURPOSE

The futures price for a given commodity represents the 'market's' best estimate of what the cash price of the commodity will be at the maturity date specified on the futures contract. This does not mean, however, that the futures price at any point in time prior to the contract's maturity will be identical to the commodity's cash price at maturity. Therefore, while using futures markets can help to reduce uncertainty when pricing commodities to be delivered at a future date, there is still some risk involved in using futures prices to forecast cash prices. The purpose of this web site is to provide individuals with a means by which they may gain some understanding of the uncertainty involved with using futures prices to forecast cash prices.
 

PROBABILISTIC PRICE FORECASTS

It is useful to view a commodity's cash price at the maturity of a futures contract as being random. It is random in the sense that until the maturity date of the futures contract arrives, we are uncertain as to what the corresponding cash price will actually be. Therefore, using commodity futures prices to forecast a commodity's cash price at contract maturity involves some uncertainty. To help measure this uncertainty we can use options written on futures contracts to assign probabilities to ranges of potential prices. In other words, we can provide estimates of the likelihood that the cash price at the option's expiration will take on a value in some specified range of prices.

While the futures price provides a point forecast of the cash price at contract maturity, it says nothing about the potential range of prices within which the cash price may fall. Options prices, on the other hand, reveal information regarding the volatility of the cash prices at contract maturity (Technically, options written on futures expire prior to the futures contract maturing, but we ignore this complication here). Together futures prices and options prices can be used to determine the underlying probability distribution associated with the commodity's cash price at the maturity of the futures contract. The graphs found at this web site were generated using the information contained in futures and options prices. They are referred to as probabilistic price forecasts. These graphs indicate the probability of a commodity's cash price at the option's expiration being within some specified range of prices.
 

USING THE PROBABILISTIC PRICE FORECASTS

Before explaining how to use the probabilistic price forecasts found on this web site, there are a few concepts that need to be explained. Take a look at the graph below.
 

This is an example of a probabilistic price forecast. On the horizontal axis are potential cash prices (at the futures contract's maturity) for the commodity contract specified at the top of the graph. For example, this graph refers to the wheat futures contract that reaches maturity in May 1999. On the left-hand side of the graph are the probabilities. The curve is called the cumulative distribution function and it matches various prices (or more correctly, ranges of prices) on the horizontal axis with the probabilities on the left-hand side.

Here we explain how to use the graphs. To do this we will provide the reader with a statistical 'rule' and then apply the rule in an example to show how it is used in practice. We will refer to the commodity's actual cash price at the maturity of the futures contract as x. This is not a known price, it simply represents some possible value the cash price could take on at contract maturity. Let the letters a and b represent two specific prices on the horizontal axis. The user of the graph, based on their particular needs, chooses these prices.
 

      RULE 1:

      For any given cash price a, the probability that the actual cash price at the maturity of the futures contract, x, is
      less than or equal to a is represented by F(a), where F(a) is the point on the curve corresponding to some
      price a on the horizontal axis and represents some probability on the left-hand side. In statistical notation this
      is represented by

      Pr(x < or = a) = F(a)

      where Pr stands for the probability of whatever is inside the parentheses. In this situation it is the probability
      that the cash price at the maturity of the futures contract, x, is less than or equal to the chosen price a.
 

We can use the above graph to provide an example of this concept. It is a good idea to print the graph to facilitate using it in the examples. To print the graph (and the entire text on how to use the graphs) simply use your mouse to left-click on the print button found on the toolbar at the top of your web browser.
 

      EXAMPLE 1:

      Suppose we want to find the probability that the actual cash price at the maturity of the futures contract will be
      less than or equal to $2.50 (this is a). Using RULE 1 we write

      Pr(x < or = a) = F(a).

      Now, plugging in the 2.50 for a we have

      Pr(x < or = 2.50) = F(2.50) = 0.34 or 34%.

      We see that the probability is 34% that the cash price at the maturity of the futures contract will be less than or
      equal to $2.50.
 

To see how this probability was found using the graph, draw a line from the $2.50 point on the horizontal axis up to the curve and make a dot on the curve at this point. Next, draw a line from the dot on the curve to the left-hand side of the graph.  The corresponding number on the left-hand side of the graph then tells us what the probability of the actual cash price, x, being less than or equal to $2.50 is.
 

      RULE 2:

      For any given cash price a the probability of the actual cash price at the maturity of the futures contract, x,
      being greater than a is represented by 1 - F(a). In statistical notation this is

      Pr(x > a) = 1 - F(a).
 

Again, the above graph provides a convenient example of how to use this rule.
 

      EXAMPLE 2:

      Suppose we want to find the probability that the actual cash price at the futures contract's maturity is greater
      than $2.50. Using RULE 2 we have

      Pr(x > a) = 1 - F(a).

      Plugging in 2.50 for a we get

      Pr(x > 2.50) = 1 - F(2.50) = 1 - 0.34 = 0.66 or 66%.

      We see that the probability is 66% that the cash price at the futures contract's maturity will exceed $2.50.
 

Here we drew a line from the 2.50 mark on the horizontal axis to the curve and then over to the left-hand side of the graph.  However, unlike EXAMPLE 1, since we are interested in finding the probability of the actual cash price at the futures contract's expiration, x, being greater than $2.50 we need to subtract the probability on the left-hand side from one.
 

      RULE 3:

      For any given cash prices a and b, such that a < b, the probability that the actual cash price at the futures
      contract's maturity, x, is between a and b is represented by F(b) - F(a). In statistical notation this is
      represented by

      Pr(a < x < or = b) = F(b) - F(a).

      Notice that the higher price, b in this example, comes first in the subtraction problem.
 

The way this would actually be carried out in practice is to find the probability that the actual cash price, x, is less than or equal to a [Pr(x < or = a) = F(a)] and subtract it from the probability that the actual cash price, x, is less than or equal to b [Pr(x < or = b) = F(b)]. Both of these probabilities are found exactly as shown in EXAMPLE 1. Once again, we can use the above graph to provide an example of this concept.
 

      EXAMPLE 3:

      Suppose we are interested in finding the probability that the cash price at the futures contract's maturity will
      be between $2.50 and $2.70 (these are a and b respectively). Using RULE 3 we have

      Pr(a < x < or = b) = F(b) - F(a).

      Next we use RULE 1 to find F(b) and F(a)

      Pr(x < or = b) - Pr(x < or = a) = F(b) - F(a).

      Plugging in 2.50 for a and 2.70 for b we get

      Pr(x < or = 2.70) - Pr(x < or = 2.50) = F(2.70) - F(2.50) = 0.85 - 0.34 = 0.51 or 51%.

      We see that the probability is 51% that the actual cash price at the futures contract's maturity will be between
      $2.50 and $2.70.
 

What we learned form this example is that if someone was interested in finding the probability that the cash price, x, was between $2.50 and $2.70 they would subtract the probability of the actual cash price, x, being less than or equal to $2.50 [F(2.50)] from the probability of the actual cash price being less than or equal to $2.70 [F(2.70)]. Again, these probabilities are found using RULE 1 as illustrated in EXAMPLE 1.

Now you have the tools necessary to use the probabilistic price forecasts found on this web site.

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