basket of securities to pick from.

2. There tends to be an economic incentive for delivering one or two spe-

cific bonds among the several that are eligible for delivery. In fact, the

most economical bond to deliver has a special name, and it is cheapest-

to-deliver (CTD).1 This ability to make a choice of which security to

deliver has an associated value, and it is one of three different delivery

options embedded in a CBOT bond futures contract. When a basis trade

is held to the expiration of the futures contract and there is no change

in CTD, we would expect the total return on the trade to be equivalent

to cost-of-carry adjusted for the delivery options. Specifically, with a

basis trade involving a coupon-bearing bond and a bond future, we have

Sd Fd CF,

where

Sd Pd (dirty price at time of trade)

Fd S(1 T(R Yc)) Af Od.

1

The formula to calculate which security is cheapest-to-deliver is nothing more than

a basis trade expressed as an annualized total return; that is, ((F S)/S) 360/T,

where F is calculated with the relevant conversion factor and T is time in days from

trade date to expiration of the futures contract. The bond that generates the lowest

rate of return is CTD.

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116 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

With CF 1, the basis trade is

Sd (S(1 T(R Yc)) Af Od),

SdT(R Yc) Af Od.

With our basis trade now equal to SdT(R Yc) Af Od instead of

simply SRT, we have a more complex situation to evaluate. The overall

value of the basis trade greatly depends on the relative values of R and Yc ,

as shown in Table 4.1.

Even though the forward accrued interest term ( Af) and delivery

options term (Od) are unambiguous in terms of their respective values (where

Af is either negative or zero, and Od is either positive or zero), the overall

situation remains complex owing to the uncertainty of how all relevant vari-

ables ultimately interrelate with one another. For example, even if SdT(R

Yc) results in a negative value, its negative value combined with Af may

or may not be enough to outweigh the positive value of Od. However, hav-

ing said all this, we can make some observations regarding potential values

as they march toward expiration. Quite simply, if T 0, as at the expiration

of the basis trade, both Od and SdT(R Yc) are zero as well. Accordingly,

at expiration, a basis trade will always end up with a maximum possible

return of SdT(R Yc). This return will be modified (if by much at all) by

the value of Af.

Thus, if going long the bond basis results in a negative price value (as

is the result in the base case of no cash flows where carry is SRT), a strat-

egy of going long the basis results in a short position in carry. Being short

carry generates a positive return as carry goes to zero. Conversely, if going

long the basis results in a price value that is positive (as may be the case with

a bond basis strategy where cash flows are now generated), then going long

the basis results in a long position in carry. In this instance being long carry

will generate a positive return as long as carry grows larger. Table 4.2 sum-

marizes these different profiles.

As a guide to thinking about potential returns with a basis trade strat-

egy, consider the following. For the base case of a basis trade involving an

underlying spot without cash flows (as with gold), and where we are going

long the basis (long S and short F), we end up with SRT (negative carry).

TABLE 4.1 Cost-of-Carry Value for Different Assumptions of R Relative to Yc

R Yc R Yc R Yc

SdT(R Yc) 0 SdT(R Yc) 0 SdT(R Yc) 0

Negative value Positive value Zero value

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Financial Engineering

TABLE 4.2 Buying/Selling the Basis to Be Short Carry under Various Scenarios

SRT SdT(R Yc) Af Od 0 SdT(R Yc) Af Od 0

Buy the basis Buy the basis Sell the basis

to be short carry to be short carry to be short carry

Figure 4.2 presents three scenarios for the value of carry as time to expira-

tion approaches. As shown, if S and R are unchanged over the investment

horizon, then carry shrinks in a linear fashion as time slowly erodes. By con-

trast, if S and R decline over time, then negative carry becomes even more

negative, though is eventually forced to zero at expiration. And if S and R

increase over time, then negative carry becomes less negative, though once

again it inevitably goes to zero.

If we now expand the base case of a basis trade to involve a cash

flow“paying product type, such as a coupon-bearing bond, let us assume we

have a normal or upward“sloping yield curve and positive carry. Figure 4.3

presents three scenarios for the value of carry as expiration nears. Again,

carry is SdT(R Yc) Af.

Overall we have a curious situation where our basis investor is looking

for one part of the strategy to shrink in value (the carry that she is short)

while at the same time being long something within the same strategy (the

delivery options). However, as time passes both carry and the delivery

options will shrink to zero because both are a function of time”that is, unless

the delivery options take on intrinsic value.

If the intrinsic value of the delivery options is zero over the life of the

strategy, then the return of the basis trade will simply be equal to the full value

of the carry at the time the trade was originated. If intrinsic value is not zero,

then the exercise of the delivery options will depend on the relationship

Value of SRT

O SRT with increasing values for S and R

SRT with values unchanged for S and R

SRT with decreasing values for S and R

O O

Trade date Expiration date

FIGURE 4.2 Three scenarios for the value of carry.

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118 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

Value of “SRT

O “Sd and R unchanged, Yc increasing

“Sd, R, and Yc unchanged

“Sd and R unchanged, Yc decreasing

O O

Trade date Expiration date

FIGURE 4.3 Three scenarios for the value of carry (expanded case).

between intrinsic value and the accrued value of carry. In other words, if exer-

cising a delivery option means that the basis trade will cease to exist, then

any carry value remaining in the basis trade is forfeited.

Figure 4.4 presents the relationship between the value of carry and the

value of the delivery options as expiration approaches.

As long as S, R, Yc, and are virtually unchanged over the life of the

basis trade, then the value of carry will decline in a relatively linear fashion,

as depicted. By contrast, the time decay pattern of Od (as with options gen-

erally) is more curvilinear, as discussed in Chapter 5.

Of all the options said to be embedded in Treasury futures, the three most

commonly cited are the quality option, the wildcard option, and the timing

or cost-of-carry option. Regarding the quality option and the 10-year

Treasury futures contract, any Treasury maturing in not less than 61/2 years

or more than 10 years from the date of delivery may be delivered into a long

contract. Although only one deliverable bond is generally CTD at any one

time, the CTD may change several times between a given trade date and deliv-

ery date. Unique profit opportunities are associated with each change in CTD,

and investors are free to switch into more attractive cash/future combinations

over time. The transitory behavior of the CTD has value to the holder of a

short futures position, and the quality option quantifies this value.

As to the wildcard option, on each day between the first business day of

the delivery month and the seventh business day before the end of the delivery

month, the holder of a short bond futures position has until 9 P.M. Eastern

Standard Time (EST) to notify the exchange of an intention to deliver.

“Delivery” means that deliverable securities are provided in exchange for a cash

payment. The investor who is short the futures contract sells the deliverable

securities, and the investor who is long the futures contract buys those securi-

ties. To determine how much ought to be paid for the delivered securities, an

invoice price is set at 3 P.M. EST. The invoice price is calculated from the future™s

settlement price at 3 P.M. EST on the day that a delivery notice is given. The

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119

Financial Engineering

This line represents the total return profile for the carry component of

the basis trade as time approaches zero (date of contract expiration),

and the threshold return that Od must rise above in order to have a

motive to exercise Od prior to expiration of the basis trade

The value of carry and

total return profiles are

shown with opposite

slopes because as carry's

Value of Total return value declines, the return

carry Value of carry on the basis trade

increases. This is because

an investor is short carry

in a basis trade.

These profiles are shown

as being linear, consistent

with the assumption that

Sd, R, Yc, and are

unchanged over time.

0 O O

Time

Date of Date of

initial trade contract

expiration

If the delivery options do not take on intrinsic value over the life of the basis

trade, then the value of Od will trend steadily toward zero along with carry.

However, if the delivery options take on intrinsic value (as via the quality option),

then the option may be exercised prior to the expiration of the basis trade.

FIGURE 4.4 Values of carry ( SRT) and total return of carry as time approaches zero.

cash market does not close until 5 P.M. EST, so there is a two-hour window of

opportunity when an investor holding a short future may profit from a decline

in the cash market. In actuality, the market often does not really close at 5 P.M.,

remaining open for as long as there is a trader willing to make a market. Indeed,

even if one is hard pressed to find a market maker in the United States after

5 P.M., it may not be difficult to find a market maker in Tokyo where the

trading day is just getting under way. The wildcard option thus values the

opportunity to profit from different trading hours for cash and futures.

Finally, the timing or cost-of-carry option attempts to quantify the opti-

mal time to make delivery. If there is a positive cost-of-carry, then there is an

incentive to put off delivery until the last possible delivery date. “Cost-of-carry”

means the difference between the return earned on a cash security and the cost

to finance that cash security in the repo market. If that difference is positive,

then there is a positive cost-of-carry. Cost-of-carry is usually positive when the

yield curve has a normal or positive shape. Conversely, if there is a negative

cost-of-carry, then there is an incentive to make delivery on the first possible

delivery date. Negative cost-of-carry exists if there is a negative difference

between the return earned on a cash security and the cost to finance that cash

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120 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

security in the repo market. Cost-of-carry is usually negative when the yield

curve has a negative or inverted shape. In sum, the cost-of-carry option may

be viewed as an option on the slope of the yield curve. The timing option has

its greatest value when the yield curve has a normal shape and the option is

priced to the latest possible delivery date during the delivery month.2

The various delivery options generally, including the yield shift option

or a new-auction option, can prove elusive to value and manage as some are

mutually exclusive and others are interdependent. Other texts go into

exhaustive detail; here it is sufficient to note that a short position in a futures

contract avails an investor with multiple choices that have value.

Again, the value for the basis prior to expiration is less than what it

would be at expiration since the delivery options would have no intrinsic