5. Forwards, Eurodollars, and Futures 171
5.1 Forward Contracts: A Second Look
Section 1.2 introduced forward contracts in the setting of a one-period model. The payoff, at maturity, of such a contract depends on the real- ization of the security or commodity on which the contract is written at the
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maturity time. Consider for example a forward contract written on a secu- rity XXY, whereby one party pays the other $110 and in return receives the security.
Recall that the $110 is obtained by multiplying the security price at the initiation time by (1+r), where r is the interest rate in the market from the initiation time to the maturity (delivery) time of the security. Hence if at the delivery time the price of the security is above $110, the party with the obligation to purchase it for $110 profits and the net cash flow to this party is the price of the security — $110. If at the delivery time the price of the security is below $110 this party loses. Thus the payoff (the cash flow) to the party with the obligation to purchase the security (referred to as the party with the long position in the contract) for $110 can be depicted in a graph as follows.
> plot( x-110,x=50..150,labels=[‘stock price‘,‘payoff‘]);
In order to prepare ourselves for the study of other types of contracts, e.g., Futures and Swaps, where the payoff from the contract is obtained not only at one point of time but at multiple points of time, we cast the forward contract in a multipeirod setting.
From a conceptual perspective, a forward contract in a multiperiod set- ting is not much different from a forward contract in a one-period setting.
We can think about the multiperiod as one long time period. We are only concerned with the spot price of the asset on which the contract is written and the price of that same asset on the maturity date of the contract. Thus, if we think about the time until the maturity of the contract as one (longer) time period we can still apply the results obtained in Section 1.2. Namely, the forward price is equal to the future value of the asset. The forward price can be written ase(r(t)t)S(0)or as S(0)
d(t), whereS(0)is the spot price of the asset,t is the time to maturity, andr(t)andd(t) are, respectively, the risk-free rate and the discount factor spanning the time interval [0,t].
We see that we perceive the forward price as the future value of the asset, based on the term structure and the spot price of the asset.
We can visualize the payoff from a forward contract in a multiperiod setting in the following way. Let us consider a forward contract in which one party is committed to deliver a certain good (or a financial asset) at some future time T for a price of FFagreed upon now and to be paid at time T. Assume that the current time is 0, the forward contract matures at time one, and the forward price is $7. We consider the payoff from the point of view of the party with a long position in the contract. The cash flow resultant from this contract occurs only at time T =1 and will be S(T)−FFwhereS(T)is the spot price at timeT.
At any other time, of course, the cash flow from this contract is zero.
Hence, if we visualize the cash flows from this forward contract in the time span zero to six, the result will be as in the following figure, where the black linear graph represents the payoff from the contract. The procedure PlotForPyf generates the payoff. The parameters forPlotForPyfare, in order, the maturity time of the contract, the forward price, and the right end point of the time span to be displayed in the graph.
> PlotForPyf(1,7,6);
The reader can experiment with different values of the parameters and drag the graph to look at it from different perspectives in the online version.
This should convey to the reader that considering the forward contract in a multiperiod setting really does not change any of its characteristics relative to a forward contract in a one-period model.
At any point in time prior to the maturity of the contract, the cash flow from the forward contract, as seen in the preceding figure, is zero. Of course, the contract still has some value. This value can be positive or neg- ative, depending on the spot price of the asset and on the term structure (or equivalently on the discount factor function) at that point in time. Exam- ining forward contracts in a multiperiod setting facilitates the investigation of their value prior to maturity, which is our next topic.