5. Forwards, Eurodollars, and Futures 171
5.2 Valuation of Forward Contracts Prior to Maturity
Consider a forward contract requiring the delivery of a certain asset at time T. We assume that the current time is zero; hence the contract matures in T units of time. At the time the contract is written its value is zero. Indeed this is how the forward price is calculated. We will refer to this contract as the “old” forward contract. Suppose that some time has passed, say t
units of time, and the time to maturity of the old contract is T −t. If a forward contract on the same asset maturing in T−t units of time were written now, its value would be zero. The new forward price may differ (and most likely would) from the price the holder (long) of the old contract pays in order to take delivery of the underlying asset. The current value of the “old” contract, however, is not necessarily zero. The “old” forward price specified on the “old” contract, the delivery price, does not reflect the circumstances of the market today. The distinction between a forward price and a delivery price was moot in the setting of a one-period model.
Each forward contract specifies the price to be paid on the delivery date for the asset in question, the underlying asset. The delivery price is equal to the forward price on the date the forward contract is initiated. The forward price is set up in such a way that the initial value of the contract is zero. As market conditions evolve, the spot price changes and so does the forward price. The forward price, which is the future value of the spot price of the asset, changes also, while the delivery price remains the same. Hence, the forward price and the delivery price, though identical on the initiation date of the contract, need not be identical at any other point in time over the life of the forward contract.
Let us now get back to the question of the value of the “old” contract.
This contract obligates the one who holds it short to deliver the underlying asset in T−t units of time. Thus it is equivalent to paying out the price of the assetS(T), inT−tunits of time, and receiving the delivery price, which we denote by F0. Hence, holding a short position in the forward contract is equivalent to the cash flow F0−S(T) inT−t units of time.
Therefore, following the same guidelines as in the one-period model, we have to discount this cash flow to the present time to obtain its price.
The value ofS(T)is stochastic (random), and to get its current value we cannot use the discount factor to time T. However, we know the current price, S(t), of the underlying asset and therefore, the current priceS(t) must be the present value ofS(T).That the present value ofS(T)is the spot price of the asset can also be explained in a slightly different way.
Replicating S(T)at timeT costsS(t)today (assuming the asset provides no income) since buying it now, guarantees having exactlyS(T)at timeT. Suppressing default risk, the value ofF0is simply its present value based on the risk-free discount factor. Denoting the current time byt, and the
discount factor function by d we arrive at equation (5.1) for the value of the forward contract.
S(t)−d(T−t)F0 (5.1)
Given a discount factor function and the delivery price we can visualize the value of the forward contract as a function of time and of the spot price in a three-dimensional graph. Assume a forward contract that was initiated in the past with a delivery price of $70 and which matures in three units of time. Let us generate a discount factor functiondisfby runningNarbitB for a certain market. We will suppress the graphing of the discount factor function by adding a fifth input parameter and setting it equal to zero.
> NarbitB([[110,0,0],[8,108,0],[6,6,106]],[90,80,75], 3,disf,0);
The no-arbitrage condition is satisfied.
The discount factor for time, 1,is given by, 9
The interest rate spanning the time interval,[0,1],is given by,11 0.2222
The discount factor for time,2,is given by, 202
The interest rate spanning the time interval,[0,2],is given by,297 0.4703
The discount factor for time,3, is given by, 6535 10494
The interest rate spanning the time interval,[0,3],is given by, 0.6058 The function Vdis ([c1,c2,..]), values the cashflow [c1,c2,..]
The continuous discount factor is given by the function, ‘disf ’, (.) If we were interested in checking the goodness-of-fit of our continu- ous approximation, we would just need to inquire about the value of the variableSumAbsDivthat is defined by the procedure.
> SumAbsDiv;
0
A value of zero means that the continuous approximation coincides with the value of the discount factors on the payment dates of the bonds.
The following figure, produced by the procedureForVal, demonstrates the value of the forward contract as a function of the time to maturity and
of the spot price. The parameters for this procedure are, in order, the dis- count factor function, the range of the time to maturity in the plot, the range of the spot price in the plot, and the delivery price. The horizontal plane defines theValue=0 plane. The other manifold is the value of the forward contract for different combinations of the spot price and the time to matu- rity. The emphasized line is the intersection of the plane and the manifold.
It emphasizes the locus of points(SpotPrice, TimeToMaturity)at which the value of the forward contract is zero. Note of course, that when the time to maturity is zero, the forward contract has a zero value if the spot price and the delivery price coincide (at $70). At some timetprior to maturity the forward contract will have a value of zero if 70disf(t) =SpotPrice. Thus the line, the intersection of the plane and the manifold, is the graph of the functionSpotPrice=70dif(t)in theSpot Price – Time To Maturityplane.
> ForValue(disf,0..3,30..80,70);
Each point on the emphasized line represents a combination of its coor- dinates —t, time to maturity, andS, spot price — at which the value of the forward contract is zero. Hence, it is the combination at which a forward contract, on the same asset, with a forward price of $70 would be issued if the time to maturity weret and the spot price were S. If the delivery price of the “old” forward contract happens to coincide with the forward price in the market, then the value of the “old” forward contract is zero.
The exercises at the end of the chapter ask the reader to demonstrate this algebraically. Note that when the time to maturity is zero, the value of the forward contract is zero if the delivery price equals the spot price. In our case, that spot price is $70.
In the following figure, we study the value of the forward contract for a given discount factor function as the time to maturity approaches the cur- rent time. Once the discount factor function has been assumed, the only source of uncertainty regarding the value of the contract is the spot price.
It would be more realistic to visualize the evolution of the value of the for- ward contract as time moves forward and the current time approaches the maturity time. In doing so, however, there is another source of uncertainty besides the spot price that has to be considered. This is the interest rate that will prevail in the market from the current time to the maturity of the contract. Therefore, the state of nature is summarized by two numbers: the spot price and the interest rate.
We assume, as before, a forward contract with a delivery price of $70 and maturity in three units of time. A three-dimensional animation can be utilized to visualize the evolution of the value of the forward contract as a function of the interest rate and the spot price, as time approaches to maturity time. A static version of one frame of the animation is depicted in figure below
> plots[animate3d]({70*exp(-r*t)-Spt,0},r=0.01..0.18, Spt=30..100,t=0..3,axes=normal,title=‘Value of Forwards That Mature in t Units of Time as a Function of Spot Price and Interest Rate.‘,labels=[‘rate‘,‘spot
price‘,Value],orientation=[-24,66], titlefont=[TIMES,BOLD,10],
style=PATCHCONTOUR,axes=framed);
Note that when the time to maturity is zero (the last frame in the an- imation), the value of the forward contract is independent of the rate of interest.
In equation (5.1) the value of an “old” forward contract at timet, ma- turing inT−tunits of time, was stipulated in terms of the spot price of the assetS(t), the delivery price, and the discount factor asS(t)−d(T−t)F0.
Let us denote the forward price of a forward contract written on the same asset today, at time t, and maturing in T−t units of time by F(t). We can substitute for the spot priceS(t)in terms of the discount factor and the forward price of a contract maturing int units of time. According to our derivation the forward priceF(t)is S(t)
d(T−t). Hence,S(t) =F(t)d(T−t) and we arrive at another expression, equation (5.2), for the value of an
“old” forward contract at timet.
d(T−t) (F(t)−F0). (5.2) Thus the value of the “old” contract today is the present value of the difference between the delivery price of the “new” and “old” contracts.
It could be positive or negative, depending on whether the forward price today is smaller or larger than the forward price in the past. We therefore see that the value of the contract is positive if the delivery price is smaller than the forward price, i.e.,F0<F(t)and vice versa.
As in most cases, there is more to the result in equation (5.2) than just a mechanical substitution. An exercise at the end of this chapter elaborates on this, and points out some insights and certain financial courses of action open to an investor who wants to get out of a forward contract. Suppose you have a long position in an “old” forward contract on a particular asset.
If you enter an opposite forward contract now (take a short position) on the same asset with the same maturity date, then you no longer need to deliver the asset. To understand this, note that you had a long position in the contract which meant you would be receiving the asset at the maturity date. If you hold a short position, then you would be obligated to deliver what you would have received. At maturity, you will be paying the old forward price and receiving the new forward price.
To enter a forward contract costs nothing. Hence, the cash flow which is a consequence of the transaction of taking an opposite position in another forward contract is the difference between the two forward prices. This cash flow, however, will be transacted at the maturity time of the opposing contracts. The value of this cash flow now is, thus, the discounted value of the difference between the two forward prices. Hence, the value of the
“old” forward contract is given by equation (5.2).
The investigation of a multiperiod scenario facilitates an analysis of pricing forward contracts on assets that pay known cash flows during the life of the contract. A very similar argument allows us to find the forward price of a stock that pays a known and fixed dividend yield during the life of the contract. These topics are examined next, starting with a forward contract on a coupon-paying bond, where we also use the opportunity to exemplify the value of a forward contract prior to maturity.