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bond which pays the positive deterministic cash flows c₁, . . . , c_mat the dates 1 < . . . < m≤ T^∗ is given implicitly by means of the formula

Bc(i) = m j=i+1

cj

(1 + ˜Yc(i))^j−i, (7.13)

where B_c(i) stands for the price of a bond at the date i < m.

In a continuous-time framework, we have the following definition of the bond’s yield-to-maturity.

Definition 7.2.2 If a bond pays the positive cash flows c₁, . . . , c_mat the dates T₁< . . . < T_m≤ T^∗, then its continuously compounded yield-to-maturity Y_c(t) = Y_c(t; c₁, . . . , c_m, T₁, . . . , T_m) is uniquely determined by the following relationship

B_c(t) = 

Tj>t

c_je^{−Yc(t)(Tj−t)}, (7.14)

where Bc(t) denotes the bond price at time t < Tm.

Note that on the right-hand side of (7.13) ((7.14), respectively), the coupon payment at time i (at time t, respectively) is not taken into account. Consequently, the price Bc(i) (Bc(t), respectively) is the price of a bond after the coupon at time i (at time t, respectively) has been paid. We focus mainly on the continuously compounded yield-to-maturity Yc(t). It is common to interpret the yield-to-maturity Yc(t) as a proxy for the uncertain return on a bond; this means that it is implicitly assumed that all coupon payments occurring after the date t are reinvested at the rate Y_c(t). Since this cannot, of course, be guaranteed, the yield-to-maturity should be seen as a very rough approximation of the uncertain return on a coupon-bearing bond. On the other hand, the return on a discount bond is certain, therefore the yield-to-maturity determines exactly the return on a discount bond. It is worthwhile to note that for every t, anFt-adapted random variable Y_c(t) is uniquely determined for any given collection of positive cash flows c₁, . . . , c_mand dates T₁, . . . , T_m, provided that the bond price at time t is known.

Let us conclude this section by observing that the bond price moves inversely to the bond’s yield-to-maturity. Moreover, it can also be checked that the moves are asymmetric, so that a decrease in yields raises bond prices more than the same increase lowers bond prices. This specific feature of the bond price is referred to as convexity. Finally, it should be stressed that the uncertain return on a bond comes from both the interest paid and from the potential capital gains (or losses) caused by the future fluctuations of the bond price. Therefore, the term fixed-income security should not be taken literally, unless we consider a bond which is held to its maturity.

7.3 Interest Rate Futures

Until the introduction of financial futures, the futures market consisted only of contracts for delivery of commodities. In 1975, the Chicago Board of Trade (CBT) created the first financial futures contract, a futures contract for so-called mortgage-backed securities. Mortgage-backed securities are bonds collaterized with a pool of government-guaranteed home mortgages. Since these securities are issued by the Government National Mortgage Association (GNMA), the corresponding futures contract is commonly referred to as Ginnie Mae futures.

7.3.1 Treasury Bond Futures

Treasury bond futures contracts were introduced on the Chicago Board of Trade in 1977. Nominally, the underlying instrument of a T-bond futures contract is a 15-year T-bond with an 8% coupon.

T-bond futures contracts and T-bond futures options trade with up to one year to maturity. As usual, the futures contract specifies precisely the time and other relevant conditions of delivery.

Delivery is made on any business day of the delivery month, two days after the delivery notice (i.e., the declaration of intention to make delivery) is passed to the exchange. The invoice price received by the party with a short position equals the bond futures settlement price multiplied by the delivery factor for the bond to be delivered, plus the accrued interest. The delivery factor, calculated for each deliverable bond issue, is based on the coupon rate and the time to the bond’s expiry date.

Basically, it equals the price of a unit bond with the same coupon rate and maturity, assuming that the yield-to-maturity of the bond equals 8%. For instance, for a bond with m years to maturity and coupon rate rc, the conversion factor δ equals

δ = 2m j=1

rc/2

(1 + 0.04)^j + 1

(1 + 0.04)^2m, (7.15)

so that δ > 1 (δ < 1, respectively) whenever r_c > 0.0 8 (r_c < 0.08, respectively). Note that the adjustment factor δ makes the yields of each deliverable bond roughly equal for a party paying the invoice price. In particular, if the settlement futures price²is close to 100, this yield is approximately 8%. At any given time, there are about 30bonds that can be delivered in the T-bond futures contract (basically, any bond with at least 15 years to maturity). The cheapest-to-deliver bond is that deliverable issue for which the difference

Quoted bond price − Settlement futures price × Conversion factor

is least. Put another way, the cheapest-to-deliver bond is the one for which the basis bⁱ_t= B_cⁱ(t)−ftδⁱ is minimal, where Bⁱ_c(t) is the current price of the i^th deliverable bond, ft is the bond futures settlement price, and δⁱ is the conversion factor of the i^th bond. Usually, the market is able to forecast the cheapest-to-deliver bond for a given delivery month. A change in the shape of the yield curve or a change in the level of yields often means a switch in cheapest-to-deliver bond, however.

This is because, as yields change, a security with a slightly different coupon or maturity may become cheaper for market makers to deliver. Before the delivery month, the calculation of the cheapest-to-deliver issue also involves the cost of carry (net financing cost) of a given bond; the top delivery choice is the issue with the lowest after-carry basis. Due to the change of yield level (or new bond issue) as time passes, the top delivery choice also changes. The possibility of such an event, which may be seen as an additional source of risk, makes the valuation of futures contracts and their use for hedging purposes more involved.

7.3.2 Bond Options

Currently traded bond-related options split into two categories: OTC bond options and T-bond futures options. The market for the first class of bond options is made by primary dealers and some active trading firms. The long (i.e., 30-year) bond is the most popular underlying instrument of OTC bond options; however, options on shorter-term issues are also available to customers. Since a large number of different types of OTC bond options exists in the market, the market is rather illiquid.

Most options are written with one month or less to expiry. They usually trade at-the-money. This convention simplifies quotation of bond option prices. Options with exercise prices that are up to two points out-of-the-money are also common. Bond options are used by traders to immunize their positions from the direction of future price changes. For instance, if a dealer buys call options from a client, he usually sells cash bonds in the open market at the same time.

Like all typical exchange-traded options, T-bond futures options have fixed strike prices and expiry dates. Strike prices come in two-point increments. The options are written on the first four delivery months of a futures contract (note that the delivery of the T-bond futures contract occurs only every three months). In addition, a 1-month option is traded (unless the next month is the

2It is customary to quote both the bond price and the bond futures price for a $100 face value bond.

7.3. INTEREST RATE FUTURES 123 delivery month of the futures contract). The options stop trading a few days before the corresponding delivery month of the underlying futures contract. The T-bond futures option market is highly liquid.

An open interest in one option contract may amount to $5 billion in face value (this corresponds to 50,000 option contracts). For a detailed description of the T-bond market, we refer to Ray (1993).

7.3.3 Treasury Bill Futures

The Treasury bill (T-bill, for short) is a bill of exchange issued by the U.S. Treasury to raise money for temporary needs. It pays no coupons, and the investor receives the face value at maturity.

T-bills are issued on a regular schedule in 3-month, 6-month and 1-year maturities. In the T-bill futures contract, the underlying asset is a 90-day T-bill. The common market practice is to quote a discount bond, such as the T-bill, not in terms of the yield-to-maturity, but rather in terms of so-called discount rates. The discount rate represents the size of the price reduction for a 360-day period (for instance, a bill of face value 100which matures in 90days and is sold at a discount rate 10% is priced at 97.5). Formally, a discount rate R_b(t, T ) (known also as bankers’ discount yield) of a security which pays a deterministic cash flow X_T at the future date T, and has the price X_t at time t < T, equals

R_b(t, T ) = XT− Xt

X_T

360

T− t, (7.16)

where T− t is now expressed in days. In particular, for a discount bond this gives Rb(t, T ) = 100− P (t, T )

100

360

T − t = (1− B(t, T )) 360 T− t,

where P (t, T ) (B(t, T ), respectively) stands for the cash price of a bill with face value 100 (with the unit face value, respectively) and T− t days to maturity. Conversely, given a discount rate Rb(t, T ) of a bill, we find its cash price from the following formula

P (t, T ) = 100



1− Rb(t, T )T− t 360

.

For a just-issued 90-day T-bill, the above formulae can be further simplified. Indeed, we have R_b(0, 90) = 4(1− P (0, 90)/100), P (0, 90) = 100

1−¹₄R_b(0, 90) . The bill yield on a T-bill equals

Yb(t, T ) = 100− P (t, T ) P (t, T )

360

T − t =1− B(t, T ) B(t, T )

360

T− t = Rb(t, T ) B(t, T ),

so that it represents the annualized (with no compounding) interest rate earned by the bill owner.

In terms of a bill yield Yb(t, T ), its price B(t, T ) equals

B(t, T ) = 1

1 + Y_b(t, T )(T− t)/360.

Let us now examine market conventions associated with T-bill futures contracts. In the T-bill futures contract, the underlying asset is the 90-day Treasury bill. In contrast to T-bills, which are quoted in terms of the discount rate, T-bill futures are quoted in terms of the price for a 100 face value bill. In particular, the T-bill futures quoted price at maturity equals 100 minus the T-bill quote (in percentage terms). The marking to market procedure is based, however, on the corresponding cash price of a given futures contract, which is calculated from the formula

P_t^b= 10 0 −¹₄(100− ˜f_t^b) = 100−¹₄R˜^f_t,

where ˜f_t^b is the market quotation of T-bill futures at time t, and ˜R^f_t = ˜R^f_t(T, T + 90) represents the futures discount rate over the future time interval [T, T + 90] implied by T-bill futures contract (note

that ˜R^f_t is expressed here in percentage terms). For instance, if the quotation for T-bill futures is f˜_t^b = 95.02, the implied futures discount rate equals ˜R^f_t = 4.98% and thus the corresponding cash futures price, which is used in marking to the market, equals

P_t^b= 10 0 −¹₄(100− 95.02) = 100 −¹₄4.98 = 98.755

per $100face value bill, or equivalently, $987,550per futures contract (the nominal size of one T-bill futures contract which trades on the CME is $1 million).

7.3.4 Eurodollar Futures

Since conventions associated with market quotations of the LIBOR rate and Eurodollar futures con-tracts are close to those examined above, we shall describe them in a rather succinct way. Eurodollar futures and futures options have traded on the CME since 1981 and 1985, respectively.³ Eurodollar futures and related Eurodollar futures options trade with up to five years to maturity. A Eurodollar futures option is of American style; one option covers one futures contract and it expires at the settlement date of the underlying Eurodollar futures contract. Formally, the underlying instrument of a Eurodollar futures contract is the 3-month LIBOR rate. At the settlement date of a Eurodollar futures contract, the CME surveys 12 randomly selected London banks, which are asked to give their perception of the rate at which 3-month Eurodollar deposit funds are currently offered by the market to prime banks. A suitably rounded average of these quotes serves to calculate the Eurodollar fu-tures price at settlement (cf. Amin and Morton (1994)). Let us stress that the LIBOR is defined as the add-on yield ; that is, the actual interest payment on a 3-month Eurodollar time deposit equals

“ LIBOR× numbers of days for investment/360” per unit investment. In our framework, the (spot) LIBOR at time t on a Eurodollar deposit with a maturity of τ days is formally defined as

l(t, t + τ ) = 360 τ

 1

B(t, t + τ )− 1 , or equivalently

B(t, t + τ ) = 1

1 + l(t, t + τ )τ /360. In particular, a 3-month LIBOR equals

l(t, t + 90 ) = 4 (B⁻¹(t, t + 90 )− 1), ˜l(t, t + 90 ) = 40 0 (B⁻¹(t, t + 90 )− 1),

where ˜l(t, t + 90) represents a 3-month LIBOR expressed in percentage terms. Eurodollar futures contract is always settled in cash (no physical delivery is possible). A Eurodollar futures price f_T^l on the settlement day T is tied to the current level of a 3-month LIBOR at time T (and thus to the price of a zero-coupon bond) through the conventional formula

f_T^l =

1− l(T, T + 90 )

=

 1− 4

B⁻¹(T, T + 90 )− 1 .

More specifically, the quoted price ˜f_T^l is given for 100$ of a nominal value, so that at maturity date T we have

f˜_T^l = 10 0

1− l(T, T + 90 )

= 10 0 − ˜l(T, T + 90) = 100 1− 4

B⁻¹(T, T + 90 )− 1 . Therefore, a 3-month futures LIBOR ˜l_t^f = 10 0− ˜f_t^l implicit in a Eurodollar futures contract converges to a 3-month LIBOR (expressed in percentage terms) as the time argument t tends to the settlement date T. Let us now focus on the valuation of a Eurodollar futures contract. A market quotation of Eurodollar futures contracts is based on the same rule as the quotation of T-bill futures. Explicitly,

3Eurodollar futures trade also on the LIFFE (since 1982) and SIMEX (since 1984).

7.4. INTEREST RATE SWAPS 125