Interest Rates And Related Contracts.

Laszlo Neogrady (1896-1962) was a Hungarian painter.

Zero-Coupon Bonds.

Let $T^{*}$ be a fixed horizon date for all market activities. A zero-coupon bond (a discount bond) is a bond that has no coupon payments. We assume that bonds are default-free; that is, the possibility of default by the bond’s issuer is excluded. The price of a zero-coupon bond at time $t$ that pays a sure $D$ at time $T\geq t$ is denoted by $p(t,T)$, which follows a strictly positive and adapted process on a filtered probability space $(\Omega ,{\cal F},\mathbb{P})$. Usually $D$ is assumed to be $1$. This means that the bond’s principal (also known as face value or nominal value) is one dollar. Since there are no other payments to the holder, a discount bond sells for less than the principal before maturity, i.e., at a discount (hence its name). This is because one could carry cash at virtually no cost, and thus would have no incentive to invest in a discount bond costing more than its face value. Various different interest rates are defined in connection with zero-coupon bonds, but we will only consider continuously compounded interest rates.

Definition. A zero-coupon bond with maturity date $T$, also called a $T$-bond, is a contract which guarantees the holder a cash payment of one unit on the date $T$. The price at time $t$ of a bond with maturity date $T$ is denoted by $p(t,T)$. $\sharp$

Obviously, we have $p(t,t)=1$ for all $t$. We shall assume that the price process $p(t,T)$ for $t\in [0,T]$ is adapted and strictly positive, and that, for every fixed $t$, $p(t,T)$ is continuously differentiable with respect to the variable $T$.

Term Structure of Interest Rates.

Let us consider a zero-coupon bond with a fixed maturity date $T\leq T^{*}$. The simple rate of return from holding the bond over the time interval $[t,T]$ equals
\[\frac{1-p(t,T)}{p(t,T)}=\frac{1}{p(t,T)}-1.\]
The equivalent rate of return with continuous compounding is commonly referred to as a (continuously compounded) yield-to-maturity on a bond. The term structure of interest rates is defined as the relationship between the yield-to-maturity on a zero-coupon bond and the bond’s maturity. Let $y(t,T)$ be the yield-to-maturity $T$. The term structure of interest rates, known also as the yield curve, is the function that relates the yield $y(t,T)$ to maturity $T$. It is obvious that, for arbitrary fixed maturity date $T$, there is a one-to-one correspondence between the bond price process $p(t,T)$ and its yield-to-maturity process $y(t,T)$. Given the yield-to-maturity process $y(t,T)$, the corresponding bond price process $p(t,T)$ is uniquely determined by the formula
\[p(t,T)=e^{-y(t,T)\cdot (T-t)}\mbox{ for all }t\in [0,T].\]
Then we have the following definition.

Definition. An adapted process $y(t,T)$ defined by the formula
\[y(t,T)=-\frac{1}{T-t}\ln p(t,T)\mbox{ for all }t\in [0,T)\]
is called the yield-to-maturity on a zero coupon bond maturing at time $T$. $\sharp$

At the theoretical level, the initial term structure of interest rates may be represented either by the family of current bond prices $p(0,T)$, or by the initial yield curve, as $p(0,T)=e^{-y(0,T)\cdot T}$ for all $T\in [0,T^{*}]$. In practice, the term structure of interest rates is derived from the prices of several actively traded interest rate instruments such as Treasury bills, Treasury bonds, swaps and futures. Note that the yield curve at any given day is determined exclusively by market prices quoted on that day. The shape of an historically observed yield curve varies over time. The observed yield curve may be upward sloping, flat, descending, or humped. There is also strong empirical evidence that the movements of yields of different maturities are not perfectly correlated. Also, the short-term interest rates fluctuate more than long-term rates. This may be partially explained by the typical shape of the term structure of yield volatilities, which is downward sloping. These features mean that the construction of a reliable model for stochastic behavior of the term structure of interest rates is a task of considerable complexity.

Forward Interest Rates.

Let $f(t,T)$ be the forward interest rate at date $t\leq T$ for instantaneous risk-free borrowing or lending at date $T$. Intuitively, $f(t,T)$ should be interpreted as the interest rate over the infinitesimal time interval $[T,T+dt]$ as seen from time $t$. As such, $f(t,T)$ will be referred to as the instantaneous, continuously compounded forward rate, or shortly, instantaneous forward rate. In contrast to bond prices, the concept of an instantaneous forward rate is a mathematical idealization rather than a quantity observable in practice. A widely accepted approach to the bond price modelling, due to Heath, Jarrow and Morton, is actually based on the exogenous specification of a family $f(t,T)$, for $t\leq T\leq T^{*}$, of forward rates. Given such a family $f(t,T)$, the bond prices are then defined by
\[p(t,T)=\exp\left (-\int_{t}^{T}f(t,u)du\right )\mbox{ for all }t\in [0,T].\]
On the other hand, if the family of bond prices $p(t,T)$ is sufficiently smooth with respect to maturity $T$, the implied instantaneous forward interest rate $f(t,T)$ is given by the formula
\begin{equation}{\label{museq115}}\tag{1}
f(t,T)=-\frac{\partial\ln p(t,T)}{\partial T}
\end{equation}
which can be seen as the formal definition of the instantaneous forward rate $f(t,T)$.

Alternatively, the instantaneous forward rate can be seen as a limit case of a forward rate $f(t,T_{1},T_{2})$ which prevails at time $t$ for riskless borrowing or lending over the future time interval $[T_{1},T_{2}]$. The rate $f(t,T_{1},T_{2})$ is in turn tied to the zero-coupon bond price by means of the formula
\[p(t,T_{2})=e^{-f(t,T_{1},T_{2})\cdot (T_{2}-T_{1})}\cdot p(t,T_{1})\mbox{ for all }t\leq T_{1}\leq T_{2},\]
or equivalently
\begin{equation}{\label{museq116}}\tag{2}
f(t,T_{1},T_{2})=\frac{\ln p(t,T_{1})-\ln p(t,T_{2})}{T_{2}-T_{1}}.
\end{equation}
Observe that $y(t,T)=f(t,t,T)$, as expected, investing at time $t$ in $T$-maturity bonds is clearly equivalent to lending money over the time interval $[t,T]$. On the other hand, under suitable technical assumptions, the convergence $f(t,T)=\lim_{U\downarrow T}f(t,T,U)$ holds for every $t\leq T$. For convenience, we focus on interest rates that are subject to continuous compounding. In practice, interest rates are quoted on an actuarial basis, rather than as continuously compounded rates. For instance, the actuarial rate (or effective rate) $a(t,T)$ at time $t$ for maturity $T$, i.e., over the time interval $[t,T]$, is given by the following relationship
\[(1+a(t,T))^{T-t}=e^{f(t,t,T)\cdot (T-t)}=e^{y(t,T)\cdot (T-t)}\mbox{ for all }t\leq T.\]
This means that the bond price $p(t,T)$ equals
\[p(t,T)=\frac{1}{(1+a(t,T))^{T-t}}\mbox{ for all }t\leq T.\]
Similarly, the actuarial forward rate (or effective forward rate) $a(t,T_{1},T_{2})$ prevailing at time $t$ over the future time period $[T_{1},T_{2}]$ is set to satisfy
\[p(t,T_{1})=p(t,T_{2})\cdot (1+a(t,T_{1},T_{2}))^{T_{2}-T_{1}}.\]
We formalize these terms as follows.

The forward rate for the period $[T_{1},T_{2}]$ concentrated at $t$ is defined by
\[f(t;T_{1},T_{2})=\frac{\ln p(t,T_{1})-\ln p(t,T_{2})}{T_{2}-T_{1}}.\]
The spot rate $r(T_{1},T_{2})$ for the period $[T_{1},T_{2}]$ is defined as
\[r(T_{1},T_{2})=f(T_{1};T_{1},T_{2})=-\frac{\ln p(T_{1},T_{2})}{T_{2}-T_{1}}.\]
The instantaneous forward rate with maturity $T$, concentrated at $t$, is defined by
\[f(t,T)=-\frac{\partial\ln p(t,T)}{\partial T}.\]
The instantaneous short rate at time $t$ is defined by
\[r_{t}=f(t,t).\]

Short-term Interest Rate.

Most traditional stochastic interest rate models are based on the exogenous specification of a short-term rate of interest. We write $r_{t}$ to denote the instantaneous interest rate (also referred to as short-term interest rate, or spot interest rate) for risk-free borrowing or lending prevailing at time $t$ over tth infinitesimal time interval $[t,t+dt]$. In a stochastic setup, the short-term interest rate is modeled as an adapted process, say $r_{t}$, defined on a filtered probability space $(\Omega, {\cal F},\mathbb{P})$ for some $T^{*}>0$. We assume tthat $r$ is a stochastic process with almost all sample paths integrable on $[0,T^{*}]$ with respect to the Lebesgue measure. We may then introduce an adapted money account process $B$ of finite variation and with continuous sample paths, given by the formula
\[B_{t}=\exp\left (\int_{0}^{t} r_{u}du\right )\mbox{ for all }t\in [0,T^{*}].\]
Equivalently, for almost all $\omega\in\Omega$, the function $B_{t}=B_{t}(\omega )$ solves the differential equation $dB_{t}=r_{t}B_{t}dt$. In financial interpretation, $B$ represents the price processes of a risk-free security which continuously compounds in value at the rate $r$. The process $B$ is referred to as an accumulation factor or a savings account in what follows. Intuitively, $B_{t}$ represents the amount of cash accumulated up to time $t$ from time $0$, and continually rolling over a bond with infinitesimal time to maturity.

Coupon-Bearing Bonds.

A coupon-bearing bond is a financial security which pays to its holder the amounts $c_{1},\cdots ,c_{n}$ at date $t_{1},\cdots ,t_{n}$ and the face value paid at time $t_{n}$ is $F$. Obviously, the bond price, say $p_{c}(t)$, at time $t$ can be expressed as a sum of the cash flows $c_{1},\cdots ,c_{n}$ discounted at time $t$
\[p_{c}(t)=\sum_{j=1}^{n} c_{j}\cdot p(t,t_{j})+F\cdot p(t,t_{n}).\]
Hence we see that a coupon-bearing bond is equivalent to a portfolio of zero-coupon bonds.

The main difficulty in dealing with bond portfolios is due to the fact that most bonds involved are coupon-bearing bonds, rather than zero-coupon bonds. Although the coupon payments and the relevant dates are pre-assigned in a bond contract, the future cah flows from holding a bond are reinvested at rates that are not known in advance. Therefore, the total return on a coupon-bearing bond which is held to maturity appears to be uncertain. As a result, bonds with different coupons and cash flow dates may not be easy to compare. The standard way to circumvent this difficulty is to extend the notion of a yield-to-maturity to coupon-bearing bond. We give below two versions of the definition of a yield-to-maturity. The first corresponds to the case of discrete compounding (such an assumption reflects more accurately the market practice). The second one assumes continuous compounding.

Yield-to-maturity.

Consider a bond which pays $n$ yearly coupons $c_{i}$ at the dates $1,\cdots ,n$, and the principal $F$ after $n$ years. Its yield-to-maturity at time $0$, denoted by $\tilde{y}_{c}(0)$, may be found from the following relationship
\begin{equation}{\label{museqa394}}\tag{3}
p_{c}(0)=\sum_{i=1}^{n}\frac{c_{i}}{(1+\tilde{y}_{c}(0))^{i}}+\frac{F}{(1+\tilde{y}_{c}(0))^{n}}.
\end{equation}
Suppose now that $c_{i}=c$ for all $i=1,\cdots ,n$. Since the coupon payments are usually determined by a preassigned interest rate $r_{c}>0$ (known as a coupon rate), expression (\ref{museqa394}) may be rewritten as
\[p_{c}(0)=\sum_{i=1}^{n}\frac{r_{c}F}{(1+\tilde{y}_{c}(0))^{i}}+\frac{F}{(1+\tilde{y}_{c}(0))^{n}}
=F\cdot\left (\sum_{i=1}^{n}\frac{r_{c}}{(1+\tilde{y}_{c}(0))^{i}}+\frac{1}{(1+\tilde{y}_{c}(0))^{n}}\right ).\]
It is clear that in this case, the yield does not depend on the face value of the bond. Notice that the price $p_{c}(0)$ equals the bond’s face value $F$ whenever $r_{c}=\tilde{y}_{c}$; in this case a bond is said to be priced at {\bf par}. Similarly, we say that a bond is priced below par (i.e. at a discount) when its current price is lower than its face value, $p_{c}(0)<F$, or equivalently, when its yield-to-maturity exceeds the coupon rate, $\tilde{y}_{c}(0)>r_{c}$. Finally, a bond is priced above par (i.e. at a premium) when $p_{c}(0)>F$, that is, $\tilde{y}_{c}(0)<r_{c}$.

In the case of continuous compounding, the corresponding yield-to-maturity $y_{c}(0)$ is defined by the relation
\[p_{c}(0)=\sum_{i=1}^{n}c_{i}\cdot e^{-y_{c}(0)\cdot t_{i}}+F\cdot e^{-y_{c}(0)\cdot t_{n}},\]
where $p_{c}(0)$ stands for the current market price of the bond. Let us now focus on zero-coupon bonds (i.e., $c=0$ and $F=1$). The initial price $p(0,n)$ of a zero-coupon bond can be easily found provided its yield-to-maturity $\bar{y}(0,n)$ is known. Indeed, we have
\[p(0,n)=\frac{1}{1+\bar{y}(0,n))^{n}}.\]
Similarly, in a continuous-time framework, we have $p(0,T)=e^{-y(0,T)\cdot T}$, where $p(0,T)$ is the initial price of a unit zero-coupon bond of maturity $T$, and $y(0,T)$ standards for its yield-to-maturity. We adopt the following definitions of the yield-to-maturity of a coupon-bearing bond in a discrete-time and in a continuous-time setting.

The discretely compounded yield-to-maturity $\tilde{y}_{c}(i)$ at time $i$ on a coupon-bearing bond which pays the positive deterministic cash flows $c_{1},\cdots ,c_{n}$ at the dates $1<\cdots <n\leq T^{*}$ is given implicitly by means of the formula
\begin{equation}{\label{museq119}}\tag{4}
p_{c}(i)=\sum_{j=i+1}^{n}\frac{c_{j}}{(1+\tilde{y}_{c}(i))^{j-i}},
\end{equation}
where $p_{c}(i)$ stands for the price of a bond at the date $i<n$.
In a continuous-time framework, if a bond pays the positive cash flows $c_{1},\cdots ,c_{n}$ at the dates $t_{1}<\cdots <t_{n}\leq T^{*}$, then its continuously compounded yield-to-maturity $y_{c}(t)$ is uniquely determined by the following relationship
\begin{equation}{\label{museq1110}}\tag{5}
p_{c}(t)=\sum_{t_{j}>t}c_{j}\cdot e^{-y_{c}(t)\cdot (t_{j}-t)},
\end{equation}
where $p_{c}(t)$ denotes the bond price at time $t<t_{n}$.

Note that on the right-hand side of (\ref{museq119}) (resp. (\ref{museq1110})), the coupon payment at time $i$ (resp. at time $t$) is not taken into account. Consequently, the price $p_{c}(i)$ (resp. $p_{c}(t)$) is the price of a bond after the coupon at time $i$ (resp. st time $t$) has been paid. We focus mainly on the continuously compounded yield-to-maturity $y_{c}(t)$. It is common to interpret the yield-to-maturity $y_{c}(t)$ 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 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 of a zero-coupon bond is certain, therefore the yield-to-maturity determines exactly the return on a zero-coupon bond. It is worthwhile to note that for every $t$, an ${\cal F}_{t}$-adapted random variable $y_{c}(t)$ is uniquely determined for any given collection of positive cash flows $c_{1},\cdots ,c_{n}$ and dates $t_{1},\cdots ,t_{n}$, provided that the bond price at time $t$ is known.

Market Conventions.

The market conventions related to U.S. government debt securities differ slightly from the generic definition adopted above. Debt securities issued by the U.S. Treasury are divided into three classes: bonds, notes and bills. The Treasury bill (T-bill, for short) is a zero-coupon bond (it pays no coupons), and the investor receives the face value at maturity. Maturity of a T-bill is no longer than one year. Treasury note and bonds, T-notes and T-bonds for short, are coupon securities. T-bonds have more than $10$ years to maturity when issued, T-notes have shorter times to maturity. Bonds and notes are otherwise identical. The U.S. Treasury pays bond-holders total annual interest equal to the coupon rate. However, a $n$-year government bond pays coupons semi-annually in equal installments, say at times $t_{j}=j\cdot\delta$ where $\delta =1/2$ and $j=1,\cdots ,2n$. The quoted “yield-to-maturity” $\hat{y}_{c}(0)$ on a government bond, more correctly called a bond equivalent yield, is based on thefollowing relationship
\begin{equation}{\label{museq1111}}\tag{6}
p_{c}(0)=\sum_{j=1}^{2n-1}\frac{r_{c}F/2}{(1+\hat{y}_{c}(0)/2)^{j}}+\frac{(1+r_{c}/2)F}{(1+\hat{y}_{c}(0)/2)^{2n}},
\end{equation}
where $r_{c}$ is the coupon rate of a bond and $F$ stands for its face value. By simple algebra, one finds that formula (\ref{museq1111}) may be rewritten as follows
\[p_{c}(0)=\frac{r_{c}F}{\hat{y}_{c}(0)}+\frac{F(1-r_{c}/\hat{y}_{c}(0))}{(1+\hat{y}_{c}(0)/2)^{2n}}.\]
The yield at time $i$ is implicitly defined by means of the relationship
\[p_{c}(i)=\sum_{j=i+1}^{2n-1}\frac{r_{c}F/2}{(1+\hat{y}_{c}(i)/2)^{j}}+\frac{(1+r_{c}/2)F}{(1+\hat{y}_{c}(i)/2)^{2n}},\]
where $\hat{y}_{c}(i)$ is the yield-to-maturity on a bond at time $i$ after the $i$th interest payment. Note that the interest rate $\hat{y}_{c}(0)$ is annualized with no compounding. The compounded annualized yield $\hat{y}_{c}^{e}(0)$, which equals
\[\hat{y}_{c}^{e}(0)=(1+\hat{y}_{c}(0)/2)^{2}-1,\]
is commonly referred to as the effective annual yield.

Interest Rate Futures.

The LIBOR (the London Inter-Bank Offer Rate) is the rate of interest offered by banks on deposits from other banks in Eurocurrency markets. LIBOR represents the interest rate at which banks lend money to each other; it is also the floating rate widely used in interest rate swap agreements in international financial markets. LIBOR rates are determined by trading between banks, and change continuously as economic conditions change.

The most heavily traded interest rate futures contracts are those related either to Treasury bonds, notes and bills, or to the LIBOR rate. Typical contracts from the first category are: Treasury bond futures, Treasury notes futures, Treasury bill futures, $5$-year Treasury note furutes, and $2$-year Treasury bond futures. The Eurodollar futures and $1$-month LIBOR futures, which have as the underlying instrument the $3$-month and $1$-month LIBOR rates, respectively, are examples of futures contracts from the second category.

The most important interest rate options are: $T$-bond futures options, $T$-note futures options, $5$-year $T$-note futures options, Eurodollar futures options, and $1$-month LIBOR futures options. The nominal size of the option contract usually agrees with the size of the underlying futures contract; for instance, it amounts to \$100,000 for both $T$-bond futures and $T$-bond futures options, and to \$1 millon for Eurodollar futures and the corresponding options.

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 nest month is the 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 onterest in one option contract may amount to \$5 billion in facr value (this corresponds to 50,000 option contracts).

Interest Rates and Related Contracts.