Choudhry. Fixed Income Securities Derivatives Handbook

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92 Introduction to Bonds

same article also notes two disadvantages: these curves are less ﬂ exible than

spline-based ones, and they may not ﬁ t the observed data as accurately.

James and Webber (2000), pages 444–445, suggests that because of the

limited number of parameters involved, Nelson and Siegel curves lack

ﬂ exibility and states further that they are accurate only for yield curves

with one hump, not for those with both a hump and trough.

Comparing Curves

The choice of curve depends on the user’s requirements and intended

application. It usually represents a trade-off between ease of computa-

tion and accuracy. The user must determine how well any curve meets

the following criteria, which are all met to a greater or lesser extent by the

methodologies discussed:

❑ Accuracy. Does the curve ﬁ t reasonably well?

❑ Flexibility. Is it ﬂ exible enough to accommodate a variety of yield

curve shapes?

❑ Model consistency. Is the ﬁ tting method consistent with a model

such as Vasicek or Cox-Ingersoll-Ross?

❑ Simplicity. Is the curve tractable—that is, reasonably simple to

compute?

A good summary of the advantages and disadvantages of popular mod-

eling methods can be found in James and Webber (2000), chapter 15.

Selected Cash and

Derivative Instruments

PART TWO

Part Two discusses selected instruments traded in the debt capi-

tal markets. The products—hybrid securities, mortgage-backed

bonds, and callable bonds—have been chosen to give the reader

an idea of the variety available in the market. Also described are

index-linked bonds and a structured product known as a col-

lateralized debt obligation (CDO). Some of the techniques for

analyzing these more complex products are explained.

This part also considers the primary ﬁ xed-income derivative

instruments. These are not securities in the cash markets and are

ﬁ xed-income derivatives (or interest rate derivatives) in the syn-

thetic markets.

The products discussed include interest rate swaps, options, and

credit derivatives. There is also a chapter on the theory behind for-

ward and futures pricing, with a case study featuring the price history

and implied repo rate for the CBOT long bond future.

This page is intentionally blank

CHAPTER 6

Forwards and Futures Valuation

nterest rate futures will be described in chapter 13. This chapter devel-

ops basic valuation concepts. The discussion is adapted, with permis-

sion, from section 2.2 of Rubinstein (1999).

Forwards and Futures

A forward is a contract between two parties in which one agrees to pur-

chase from the other a speciﬁ ed asset at a speciﬁ ed price for delivery at a

speciﬁ ed future date. The following discussion refers to these variables:

P = the current price of the underlying asset, also known as the spot

price

= the spot price of the underlying asset at the time of delivery

X = the delivery price speciﬁ ed in the forward contract

T = the term to maturity of the contract, in years, also referred to as

the time to delivery

r = the risk-free T-bill interest rate

R = the return of the payout, or its yield

F = the current forward price, that is, the current market prediction of

the underlying asset’s price on the delivery date

The payoff of a forward contract is given by expression (6.1).

Payoff = P

– X (6.1)

96 Selected Cash and Derivative Instruments

The payout yield, R, is the percentage of the spot price that is paid out

at contract expiry, i.e., R = (P

– X ) / P

. The forward contract terms are

set so that the present value of the payout (P

– X ) is zero. This means

that the forward price, F, on day one of the contract equals X. (Note that

the forward price is not the same as the value of the contract, which at this

point is zero.) From the initiation of the contract until its expiration, the

value of X remains ﬁ xed. The forward price, F, however, ﬂ uctuates, gener-

ally rising and falling with the spot price of the underlying asset.

Like forwards, futures contracts also represent agreements to purchase/

sell a speciﬁ ed asset at a speciﬁ ed price for delivery on a speciﬁ ed date.

While forwards, however, are custom instruments designed to meet indi-

vidual requirements, futures are standardized contracts that are traded on

recognized futures exchanges.

Commodity futures are settled by the physical delivery of the underly-

ing asset; many ﬁ nancial futures are settled in cash. A bond future—which

is written on a notional bond that can represent any of a set of bonds ﬁ t-

ting the contract terms, known as the contract’s delivery basket—is settled

by delivering to the long counterparty one of these bonds. Only a very

small percentage of either commodity or ﬁ nancial futures contracts are

delivered into—that is, involve the actual transfer of the underlying asset

to the long counterparty. This is because the majority of futures trading is

done to hedge or to speculate. Accordingly, most futures positions are net-

ted out to zero before contract expiry, although, if the position is held into

the delivery month, depending on the contract’s terms and conditions, the

long future may be delivered into.

Cash Flow Differences

Aside from how they are constructed and traded, the most signiﬁ cant

difference between forwards and futures, and the feature that inﬂ uences

differences between their prices, concerns their cash ﬂ ows. The proﬁ ts or

losses from futures trading are realized at the end of each day. Because of

this daily settlement, at expiration all that needs to be dealt with is the

change in the contract value from the previous day. With forwards, in

contrast, the entire payout occurs at contract expiry. (In practice, the situ-

ation is somewhat more complex, because the counterparties have usually

traded a large number of contracts with each other, across a number of

maturity periods and, perhaps, instruments, and as these contracts expire

they exchange only the net loss or gain on the contract.)

FIGURE 6.1 shows the daily cash ﬂ ows for a forward and a futures con-

tract having identical terms. The futures contract generates intermediate

cash ﬂ ows; the forward doesn’t. As with the forward contract, the delivery

price speciﬁ ed in the futures contract is set so that at initiation, the pres-

Forwards and Futures Valuation 97

ent value of the futures contract is zero. At the end of each day, the future

is marked to market at the close price. This will result in a proﬁ t or a

gain

—or neither, if the closing price is unchanged from the previous day’s

closing price, which technical traders call a doji—which is handed over to

the appropriate counterparty. Through this daily settlement, the nominal

delivery price is reset each day so that the present value of the contract is

always zero. This means that the future and nominal delivery prices of a

futures contract are the same at the end of each trading day.

As illustrated in ﬁ gure 6.1, the process works as follows: After day one,

the future price is reset from F to F

. The amount (F

– F ), if positive, is

handed over by the short counterparty to the long counterparty. If the

amount is negative, it is paid by the long counterparty to the short. On the

expiry day, T, of the contract, the long counterparty receives a settlement

amount equal to

−

−1

, which is the difference between the future

price and the price of the underlying asset. The daily cash ﬂ ows cancel

each other out, so that at expiry the value of the contract is identical to

that for a forward, that is (P

– F ).

All market participants in exchange-traded contracts trade with a

central counterparty, the exchange’s clearing house. This eliminates coun-

terparty risk. The clearing house is able to guarantee each deal, because

all participants are required to contribute to its clearing fund through

margining: each participant deposits an initial margin and then every day,

as proﬁ ts and losses are recorded, deposits a further variation margin as

FIGURE 6.1

Cash Flows for Forwards and Futures Contracts

Profit/Loss

Underlying asset price

–15

–10

–5

10 20 30 40 50 52.5 60 70 80 90 100

98 Selected Cash and Derivative Instruments

needed. Marking the futures contract to market is essential to this margin

process.

Daily settlement has both advantages and disadvantages. If a position

is proﬁ table, receiving part of this proﬁ t daily, as happens with a futures

contract, is advantageous because the funds can be reinvested while the

position is still maintained. On the other hand, a losing position saddles

the holder of a futures contract with daily losses not suffered by the holder

of a loss-making forward position.

Relationship Between Forward and Futures Prices

Under certain speciﬁ ed conditions, the prices of futures and forwards with

identical terms must be the same. Consider two trading strategies with

identical terms to maturity and written on the same underlying asset, one

using forward contracts and the other futures. Neither strategy requires an

initial investment, and both are self-ﬁ nancing—that is, all costs and future

funding are paid out of proceeds. Assume the following conditions:

❑ no risk-free arbitrage opportunities

❑ an economist’s perfect market

❑ certainty of returns

For the strategy employing forwards, r

contracts are bought, where r

is the daily return (or instantaneous money market rate) and T the matu-

rity term in days. The start forward price is F = X, and the payoff on expiry

rP F

−

()

The futures strategy is more involved, because of the margin cash ﬂ ows

that are received or paid daily during the term of the trade. On day one,

r contracts are bought, each priced at F. After the close that day, F

– F is

received. The position is closed out, and the cash received is invested at the

daily rate, r, up to the expiry date. The return on this investment is

T −1

Thus, on expiry the counterparty will receive

rF Fr

−

()

−

The next day, r

futures contracts are bought at a price of F

. At the

close, the cash ﬂ ow of

−

is received and invested at r

T–2

, generating

a return on expiry of

rF Fr

−

()

−

. This process is repeated until the

expiry date, which is assumed to be the delivery date. The return from

following this strategy is expression (6.2).

rF F rF F rF F

rP F rP F

TT T

12132

−

()

+−

()

+−

()

++ −

()

−

()

−

..... =

(6.2)

Forwards and Futures Valuation 99

This is also the payoff from the forward contract strategy. The key

point is that if equation (6.3) below holds, then so must (6.4).

rP F rP F

forward

future

−

()

=−

()

(6.3)

forward future

(6.4)

Forward-Spot Parity

The forward strategy can be used to imply the forward price, provided that

the current price of the underlying and the money market interest rate are

known.

FIGURE 6.2 illustrates how this works, using the one-year forward

contract whose proﬁ t/loss proﬁ le is graphed in ﬁ gure 6.1 and assuming an

initial spot price, P, of $50, a risk-free rate, r, of 1.05 percent, and a payout

yield, R, of 1 percent.

FIGURE 6.3 shows that the payoff proﬁ le illustrated in ﬁ gure 6.1 can

be replicated by a portfolio composed of one unit of the underlying asset

FIGURE 6.2

Forward Contract Profit/Loss Profile

TIME FORWARD CONTRACT FUTURES CONTRACT

0 0 0

1 0 F

– F

2 0 F

– F

3 0 F

– F

4 0 F

– F

5 0 F

– F

… 0 …

T – 1 0 F

T–1

– F

T–2

T P

– F P

– F

T–1

Total P

– F P

– F

100 Selected Cash and Derivative Instruments

whose purchase is ﬁ nanced by borrowing at the risk rate, which equals the

discount rate, for the term of the forward contract. At contract expiry the

loan is settled by paying a sum equal to

F /. .105 105

()

, which reduces to

F. Since no arbitrage opportunity exists, the cost of creating the portfolio

must be identical to the cost of the forward. As noted, the cost of the

forward contract at initiation is set at zero. Equation (6.5) must therefore

hold.

−+ =50 1 05 0F /.

(6.5)

F =× =50 105 5250..

The general relationship between the forward price and the spot

prices is demonstrated in

FIGURE 6.4. As in ﬁ gure 6.3, the ﬁ rst step is to

replicate a forward’s payoff proﬁ le with a portfolio composed of R

-T

units

of the underlying asset, whose creation is funded by borrowing a sum

equal to the present value of the forward price. Again, as in ﬁ gure 6.3, the

loan is settled on the forward expiry date by paying an amount equal to

Fr r F

TT−

()

×=

By design, the portfolio has a payoff that is identical to the forward’s:

−

()

. The cost of setting up the portfolio must therefore equal the

current price of the forward: if one were cheaper than the other, a trader

could make a risk-free proﬁ t by buying the cheaper instrument and short-

FIGURE 6.3

Forward Strategy

CASH FLOWS

START DATE EXPIRY

Buy forward contract 0 P

– F

Buy one unit of the underlying asset –50 P

Borrow zero present-value of

forward price F / 1.05 F

Total –50 + F / 1.05 P

– F

Result

Set –50 + F / 1.05 equal to zero (no-arbitrage condition)

Therefore F = 52.5

Forwards and Futures Valuation 101

ing the more expensive one. The current cost of the forward—its present

value—is zero. The cost of constructing the duplicating portfolio must

therefore be zero as well. This is expressed in equation (6.6a), which may

be solved for F as (6.6b).

−+=

−−

PR Fr

(6.6a)

FPrR

()

(6.6b)

The price of the forward contract is thus a function of the current

underlying spot price, the risk-free or money market interest rate, the

payoff, and the maturity of the contract. It can be shown that neither

FPrR

()

/ nor FPrR

()

/ is possible unless arbitrage is admitted.

The only possibility is (6.6b), which describes the state in which the

futures price is at fair value.

The Basis and Implied Repo Rate

This section introduces some terms used in the futures markets. The ﬁ rst

is basis: the difference between the price of a futures contract and the

current underlying spot price. The size of the basis is a function of issues

such as cost of carry, the net cost of holding the underlying asset from

FIGURE 6.4

Proof of Forward-Spot Parity

START DATE EXPIRY

Buy forward contract 0 P

– F

Buy R

–T

units of the underlying asset –PR

–T

Borrow zero present-value of

forward price Fr

–T

–F

Total –PR

–T

+ Fr

-T

– F

Result

Set –PR

–T

+ Fr

–T

= 0

Therefore F = P (r / R )