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ward contract commits investors to buy or sell a predetermined amount of
one currency for another currency at a predetermined exchange rate. Thus,
a forward is really nothing more than a mutual agreement to exchange one
commodity for another at a predetermined date and price.
Can investors who want to own Canadian Treasury bills use the for-
ward market to hedge the currency risk? Absolutely!
The Canadian Treasury bills will mature at par, so if the investors want
to buy $1 million Canadian face value of Treasury bills, they ought to sell
forward $1 million Canadian. Since the investment will be fully hedged, it
is possible to state with certainty that the three-month Canadian Treasury
bill will earn

1100>1.16002 197.90>1.15122 13602
197.90>1.15122 1872
5.670% .

Where did the forward exchange rates come from for this calculation?
From the currency section of a financial newspaper. These forward values
are available for each business day and are expressed in points that are then
combined with relevant spot rates. Table 2.4 provides point values for the
Canadian dollar and the British pound.
The differential in Eurorates between the United States and Canada is
312.5 basis points (bps). With the following calculation, we can convert
U.S./Canadian exchange rates and forward rates into bps.

11.1600 1.15122 13602
316 basis points
1.1512 87
1.1600 the spot rate
1.1512 the spot rate adjusted for the proper amount of forward

We assume that the Canadian Treasury bill matures in 87 days. Although
316 bps is not precisely equal to the 312.5 bp differential if calculated from

TABLE 2.4 Forward Points May 1991
Country 3 Month 6 Month 12 Month

Canada 90 170 290
United Kingdom 230 415 700

Cash Flows

the yield table, consideration of transaction costs would make it difficult to
structure a worthwhile arbitrage around the 3.5 bp differential.
Finally, note that the return of 5.670 percent is 15 bps above the return
that could be earned on the three-month U.S. Treasury bill. Therefore, given
a choice between a three-month Canadian Treasury bill fully hedged into
U.S dollars earning 5.670 percent and a three-month U.S. Treasury bill earn-
ing 5.520 percent, the fully hedged Canadian Treasury bill appears to be the
better investment.
Rather than compare returns of the above strategy with U.S Treasury
bills, many investors will do the trade only if returns exceed the relevant
Eurodollar rate. In this instance, the fully hedged return would have had to
exceed the three-month Eurodollar rate. Why? Investors who purchase a
Canadian Treasury bill accept a sovereign credit risk, that is, the risk the gov-
ernment of Canada may default on its debt. However, when the three-month
Canadian Treasury bill is combined with a forward contract, another credit
risk appears. In particular, if investors learn in three months that the coun-
terparty to the forward contract will not honor the forward contract,
investors may or may not be concerned. If the Canadian dollar appreciates
over three months, then investors probably would welcome the fact that they
were not locked in at the forward rate. However, if the Canadian dollar depre-
ciates over the three months, then investors could well suffer a dramatic loss.
The counterparty risk of a forward contract is not a sovereign credit risk.
Forward contract risks generally are viewed as a counterparty credit risk. We
can accept this view since banks are the most active players in the currency
forwards marketplace. Though perhaps obvious, an intermediate step
between an unhedged position and a fully hedged strategy is a partially
hedged investment. With a partial hedge, investors are exposed to at least
some upside potential with a trade yet with some downside protection as well.

Most currency futures are rather straightforward in terms of their delivery
characteristics, where delivery often is made on a single day at the end of
the futures expiration. However, the fact that gaps may exist between the
trading hours of the futures contracts and the underlying spot securities can
give rise to some strategic value.

This section examined the similarities of forward and future cash flow
types across bonds, equities, and currencies, and discussed the nature of the


interrelationship between forwards and futures. Parenthetically, there is a
scenario where the marginal differences between a forward and future actu-
ally could allow for a material preference to be expressed for one over the
other. Namely, since futures necessitate a daily marking-to-market with a
margin account set aside expressly for this purpose, investors who short
bond futures contracts (or contracts that enjoy a strong correlation with
interest rates) versus bond forward contracts can benefit in an environment
of rising interest rates. In particular, as rates rise, the short futures posi-
tion will receive margin since the future™s price is decreasing, and this
greater margin can be reinvested at the higher levels of interest. And if rates
fall, the short futures position will have to post margin, but this financing
can be done at a lower relative cost due to lower levels of interest. Thus,
investors who go long bond futures contracts versus forward contracts are
similarly at a disadvantage.
There can be any number of incentives for doing a trade with a partic-
ular preference for doing it with a forward or future. Some reasons might

Investors™ desire to leapfrog over what may be perceived to be a near-
term period of market choppiness into a predetermined forward trade
date and price
Investors™ belief that current market prices generally look attractive now,
but they may have no immediate cash on hand (or perhaps may expect
cash to be on hand soon) to commit right away to a purchase
Investors™ hope to gain a few extra basis points of total return by actively
exploiting opportunities presented by the repo market via the lending
of particular securities. This is discussed further in Chapter 4.

Table 2.5 presents forward formulas for each of the big three.


We now move to the third leg of the cash flow triangle, options.
Continuing with the idea that each leg of the triangle builds on the other,
the options leg builds on the forward market (which, in turn, was built on

Cash Flows

TABLE 2.5 Forward Formulas for Each of the Big Three
Product Formula
No Cash Flows Cash Flows

Bonds S (1 RT) S (1 T (R Yc))
Equities S (1 RT) S (1 T (R Yc))
Currencies S (1 T(Rh Ro))

the spot market). Therefore, of the five variables generally used to price an
option, we already know three: spot (S), a financing rate (R), and time (T).
The two additional variables needed are strike price and volatility. Strike
price is the reference price of profitability for an option, and an option is
said to have intrinsic value when the difference between a strike price and
an actual market price is a favorable one. Volatility is a statistical measure
of a stock price™s dispersion.
Let us begin our explanation with an option that has just expired. If our
option has expired, several of the five variables cited simply fall away. For
example, time is no longer a relevant variable. Moreover, since there is no
time, there is nothing to be financed over time, so the finance rate variable
is also zero. And finally, there is no volatility to be concerned about because,
again, the game is over. Accordingly, the value of the option is now:

Call option value is equal to S K

S the spot value of the underlying security
K the option™s strike price

The call option value increases as S becomes larger relative to K. Thus,
investors purchase call options when they believe the value of the underly-
ing spot will increase. Accordingly, if the value of S happens to be 102 at
expiration with the strike price set at 100 at the time the option was pur-
chased, then the call™s value is 102 minus 100 2.
A put option value is equal to K S. Notice the reversal of positions
of S and K relative to a call option™s value. The put option value increases
as S becomes smaller relative to K. Thus, investors purchase put options
when they believe that the value of the underlying spot will decrease.
Now let us look at a scenario for a call™s value prior to expiration. In
this instance, all five variables cited come into play.
The first thing to do is make a substitution. Namely, we need to replace
the S in the equation with an F. T, time, now has value. And since T is rel-
evant, so too is the cost to finance S over a period of time; this is reflected


by R and is embedded along with T within F. And finally, a value for volatil-
ity, V, is also a vital consideration now. Thus, we might now write an equa-
tion for a call™s value to be

Call value F K V.

Just to be absolutely clear on this point, when we write V as in the last
equation, this variable is to be interpreted as the value of volatility in price
terms (not as a volatility measure expressed as an annualized standard devi-
ation).12 Since there is a number of option pricing formulas in existence
today, we need not define a price value of volatility in terms of each and
every one of those option valuation calculations. Quite simply, for our pur-
poses, it is sufficient to note that the variables required to calculate a price
value for volatility include R, T, and , where is the annualized standard
deviation of S.13
On an intuitive level, it would be logical to accept that the price value
of volatility is zero when T 0, because T being zero means that the option™s
life has come to an end; variability in price (via ) has no meaning. However,
if R is zero, it is still possible for volatility to have a price value. The fact that
there may be no value to borrowing or lending money does not automati-
cally translate into a spot having no volatility (unless, of course, the under-
lying spot happens to be R itself, where R may be the rate on a Treasury bill).14
Accordingly, a key difference between a forward and an option is the role of
R; R being zero immediately transforms a forward into spot, but an option
remains an option. Rather, the Achilles™ heel of an option is ; being zero
immediately transforms an option into a forward. With 0 there is no
volatility, hence there is no meaning to a price value of volatility.
Finally, saying that one cash flow type becomes another cash flow type
under various scenarios (i.e., T 0, or 0), does not mean that they some-
how magically transform instantaneously into a new product; it simply high-
lights how their new price behavior ought to be expected to reflect the cash
flow profile of the product that shares the same inputs.

It is common in some over-the-counter options markets actually to quote options
by their price as expressed in terms of volatility, for example, quoting a given
currency option with a standard three-month maturity at 12 percent.
The appendix of this chapter provides a full explanation of volatility definitions,
including volatility™s calculation as an annualized standard deviation of S.
Perhaps the most recent real-world example of R being close to (or even below)
zero would be Japan, where short-term rates traded to just under zero percent in
January 2003.

Cash Flows

Rewriting the above equation for a call option knowing that F S
SRT, we have

Call value = S + SRT K + V.

If only to help us reinforce the notions discussed thus far as they relate
to the interrelationships of the triangle, let us consider a couple of what-if?
scenarios. For example, what if volatility for whatever reason were to go to


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