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value. This is because the positive value of Od serves to minimize the nega-
tive value of carry. When Od has a value greater than zero (as is certainly
the case prior to expiration of the futures contract), the price of the futures
contract will be below the forward price of the CTD (since a forward does
not embody Od). For this reason many investors will refer to how futures
trade cheap to spot (trade at a price below spot owing to the delivery options
in the futures). While this is true by definition, it is not intended to refer to
relative value; the cheapness of futures to spot does not imply that the futures
investor is getting some kind of bargain, but rather that bond futures are
built differently from bond forwards and spot.
The following figures show potential scenarios for the value of Od over
time as well as the relationship of Od to carry in a total return context. Od
is a function of all the usual variables associated with an option: S, R, T, K,
and V. Figure 4.5 presents the scenario where S, R, and V are unchanged as
time goes to zero.
Figure 4.6 shows the total return relationship between Od and cost-of-
carry ( SRT). Since an investor is short both Od and carry, these contribute
to the total return in a positive way as time passes.
In sum, and as illustrated in Figure 4.7, prior to expiration a basis trade
includes elements of spot, futures, and options. The maximum profit of the
strategy if held to expiration will be the carry™s initial value, and it may be more


2
Recall that in Chapter 2 we stated that options are unique relative to spot and
forwards and futures since options embody the right (not the obligation) to do
something; to exercise or not to exercise. In the context of the delivery options
described here, the choices listed (what to deliver, when to deliver, and how to
deliver) all have some kind of value prior to expiration. The values may be derived
with traditional option pricing formulas or other methods. In sum, the term
“delivery options” is intended to be descriptive both as verb (as in “to choose
between delivering early or late in the delivery cycle”) and as noun (as in “the
calculated option price relevant for an expected CTD”).



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Financial Engineering



Value of
Od




O O
Time
Date of Date of
initial trade contract
expiration

FIGURE 4.5 Delivery option value over time.


Total return

Od contribution
Cost-of-carry plus Od




Cost-of-carry (“SRT)
contribution

O O
Time
Date of Date of
initial trade contract
expiration

FIGURE 4.6 Total return relationship between Od and cost-of-carry.

than that depending on the values of the various delivery options (and notably
if there were a beneficial change in CTD3). As shown, a relatively straight-
forward strategy like a basis trade can combine all three of the fundamental
cash flow elements. The triangle helps to show where key inter-relationships
begin and end.


3
A beneficial change in CTD via the quality option is simply this: If a new bond
should happen to become CTD over the life of a futures contract, it could be
profitable to change the S portion of the basis trade to a new underlying S.
Deciding whether this would be profitable requires performing what-if calculations
on the basket of bonds eligible to be switched with the spot that is currently used
in the given basis trade.




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122 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT



When T equals zero, as at the
Sd T(R Yc) Af Od
Bond Basis
expiration of the trade, then profit is
the full value of carry that was
originally shorted (assuming no
beneficial change in CTD and,
hence, no intrinsic value with Od ”
only time value, which is worthless
Futures at expiration).
Spot

When R equals zero, then the
F Sd Sd T(R Yc) Af value of carry is zero (noting that
S
Af may be zero or negative), and
Od remains alive until expiration of
the strategy. The profit of the
strategy depends on Od ™s value
when the trade was first initiated.
Od is a function of
S, T, R, K, and

Options

If V is zero, then the basis trade value becomes its
carry value. Zero volatility implies zero uncertainty
and, hence, no value in choosing something that is
already known, as with what to deliver or when to
deliver it; in short, all options within the delivery
options package are worthless.




FIGURE 4.7 Bond basis.

Securities lending (see Figure 4.8) consists of four steps, which are pre-
sented in the context of a gold transaction.

1. One investor (Investor A) pays the prevailing spot price for an ounce of gold.
2. Investor A immediately lends her gold for a prespecified amount of time
to Investor B in exchange for a loan of cash.
3. Investor A invests her loan of cash in a risk-free product (e.g., a
Treasury bill).
4. When a prespecified amount of time has passed (perhaps a month),
Investor A returns the loan of cash to Investor B, and Investor B returns
the loan of gold to Investor A.

In sum, Investor A is happy because she lent something (the gold) and
in exchange received a cash loan that she used to earn interest in a safe invest-
ment that otherwise would have just sat in her portfolio. Investor B, per-
haps a trading desk at an investment bank that specializes in these types of
transactions, is happy because of a satisfied need to borrow something
needed (gold) in exchange for a temporary loan (of cash). We can only pre-
sume that both Investor A and B were happy with the overall terms of the
loan transaction (namely the cash amounts paid and received); otherwise the
fundamental laws of economics suggest that the transaction would not have
been consummated in the first place.




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Financial Engineering




Spot Forward = Securities lending
Cash Gold

Borrow Loan



FIGURE 4.8 Use of spot and forward to create a securities lending strategy.


At this point readers may be asking what the real difference is between
a regular buy/sell transaction and the cash-and-carry trade just described.
After all, isn™t there one investor providing a security in exchange for cash
and another investor taking the security in exchange for cash? Yes. However,
a key difference is the mind-set of the two investors at the start of the trans-
action. Namely, both investors agree at the outset that the cash and securi-
ties involved are to be returned at some prespecified date in the future. There
also may be important differences in the tax treatment of a buy/sell versus
a lend/borrow strategy. This type of borrowing and lending of securities and
cash is commonplace, and is generally called securities lending. In the bond
market, it is often referred to as engaging in a repurchase agreement (or repo,
or reverse repo), as is discussed further in the next section.
Readers may have already surmised that a reverse repo (sometimes called
a cash-and-carry trade) is really a variation of a forward transaction; it is a
forward loan transaction where assets consisting of cash and securities guar-
antee the loan. Figure 4.9 illustrates this.
Why might investors be motivated to engage in a securities lending trans-
action as opposed to a simple forward transaction? From the perspective of
the investor lending the equity (or gold, or bond, or whatever), the differ-
ence between securities lending rate and the risk-free rate may be a favor-
able one. That is, the rate of return on the safe investment that is made with
the loan of money (in exchange for the loan of equity) could be advanta-
geous. And from the perspective of the investor borrowing the equity, the
ability to show the equity in a portfolio (if even for just a short period of
time) allows him or her to show a position in the security that suits a par-
ticular strategy or objective.
Earlier in this chapter it was said that a bond future™s CTD is determined
by the lowest total return (which, incidentally, happens to be the same cal-
culation for a total return for a basis trade). This total return value is some-
times called an implied repo rate (or implied securities lending rate), and it
is applicable for basis trades on bonds and equities or any other security type.
The reason is that the incentive for investors doing a basis trade rather than
a securities lending trade may be the simple difference between how they are
compensated for doing one trade over the other. Accordingly, an implied


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Investor A provides Investor B
Investor A agrees to accept a
with the forward price of the
security from Investor B in 3
The forward loan
security in exchange for the
months, and at the 3-month
security.
forward price agreed at trade
date.

O O
Trade date 3 months later

Investor B returns Investor
Investor A lends Investor B the
A™s security, and Investor A
security that is to be returned in
returns Investor B™s loan
3 months. In exchange,
plus interest. The dollar
Investor B agrees to lend Assets in support
amount of the interest is
Investor A cash over the 3- of the loan
equal to the difference
month period. The amount of
between the security™s spot
the cash lent is equal to the
and forward prices of 3
security™s spot price.
months earlier.

FIGURE 4.9 Reverse repo as a variation of a forward transaction.


securities lending rate might be more appropriately called a breakeven secu-
rities lending rate for the simple reason that if the true securities lending rate
were ever less than the breakeven securities lending rate, it would be desir-
able for investors to execute this arbitrage strategy:

Buy the spot security underlying the futures contract.
Go short an equal face amount of the futures contract.
Finance the spot security in the securities lending market.

Since the spot security can be financed at the lending rate for less than the
implied lending rate, the return earned on this strategy is an arbitraged profit,
and the profit is equal to the difference in the true and implied lending rates.
Since cost-of-carry can be positive, zero, or even negative, a product that
pays a dividend or a coupon will exhibit positive carry whenever the cur-
rent yield of the product is above its financing rate. With bonds, this is typ-

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