Fixed Income Notation

Tutorial Contents

1. Fixed Income Notation
2. Fixed Income Probability Measures
3. Multi-Currency Markets
4. The HJM Framework
5. Basic Fixed Income Instruments
6. Caps, Floors and Swaptions
7. CMS Swaps, Caps and Floors
8. Bermudan Swaptions
9. Structured Notes and Exotic Swaps
10. Callable Libor Exotics and TARNs
11. Volatility Derivatives
12. Yield Curves and Curve Construction
13. Managing Yield Curve Risk
14. Vanilla Models With Local Volatility
15. Vanilla Models With Stochastic Volatility
16. Introduction to Term Structure Models
17. One-Factor Short Rate Models I
18. One-Factor Short Rate Models II
19. Multi-Factor Short Rate Models
20. The Libor Market Model I
21. The Libor Market Model II
22. Risk Management and Sensitivity Computations
23. P&L Attribution
24. Payoff Smoothing
25. Vegas in the Libor Market Model

The archetypal fixed income security is the bond. A bond is an instrument of indebtedness of the issuer to the holder. The holder pays an upfront fee to the issuer in exchange for a series of pre-determined future
cash-flows. Hence, the issuer borrows money now in order to finance their needs, and pays the holder back across the lifetime of the bond along with interest. Governments and corporations both borrow money by issuing bonds, utilising the upfront fees buyers pay for the instrument in order to finance their activities.

The fundamental object we consider is that of a zero-coupon bond (ZCB). A ZCB makes a single payment, at the so-called maturity date of the bond. In contrast to a standard coupon bond, where interest payments are made throughout the lifetime of the bond, the interest paid on a ZCB is entirely baked-into the market price of the instrument. At its maturity, the holder receives the par (or face) value of the bond.

While there are large and liquid markets for zero-coupon bonds, for example US Treasury bills (T-bills) for short-term bonds, we shall see that in a liquid and well-functioning market, the prices of longer-term term ZCBs can be extracted from the prices of more liquid, coupon bearing bonds.

Hence, our analysis of ZCB prices is primarily theoretical, and in the following we will assume the existence of a continuous set of ZCB prices for a large range of maturities.

Bonds and Forward Rates

Zero-coupon bonds make a single payment at their maturity date $T$. For ease of calculation, we normalise this payment to 1 unit of currency, and refer to a ZCB with maturity $T$ as a $T$-bond. Since I'm a proud Brit, unless otherwise specified, we will assume a base currency of Pound sterling, £.

Definition (T-Bonds)

A ZCB of maturity $T$ (or $T$-bond) is a contract paying 1 unit of currency on the maturity date $T$. We will denote its value at the time $t \in [0,T]$ by $P(t, T).$

The next fundamental quantity that we are interested in is that of a forward rate. To get to this, we introduce the concept of a forward bond. The best way to motivate this concept is through an example.

Suppose we are interested in purchasing, at some future time $T$, a ZCB maturing at $T + \tau$, $\tau > 0$. Today, at time $t < T$, the price of such a bond can be locked in as follows:

1. Today, purchase one $(T+\tau)$-bond for $P(t, T+\tau)$.
2. To finance the purchase of the $(T+\tau)$-bond, sell short $\tfrac{P(t,T+\tau)}{P(t,T)}$ $T$-bonds. This strategy has no upfront cost, since
   $P(t, T+\tau) + \left( \frac{P(t,T+\tau)}{P(t,T)} \right) P(t,T) = 0.$

Despite costing nothing to setup, a flow of $-\tfrac{P(t,T+\tau)}{P(t,T)}$ will take place at time $T$ due to the maturity of our $T$-bond position. Likewise, a flow of $1$ will take place at time $T+\tau$ due to our position in the $(T+\tau)$-bond. In other words, our strategy fixes the time $T$ price of the $(T+\tau)$ bond at

a quantity known as the time $t$ forward price for the ZCB spanning $[T, T+\tau]$.

Definition (Forward bond)

The time $t$ price for a forward bond with expiry date $T$ into a bond with maturity date $T+\tau$, $\tau > 0$ is given by $P(t,T, T+\tau)$.

Often, it is convenient to characterize a forward bond price by a discount rate. There are numerous ways to do this, but one is the continuously compounded forward yield $y(t,T,T+\tau)$, defined implicitly by

Definition (Continuously compounded forward yield)

The time $t$ value for the continuously compounded forward yield corresponding to the forward bond price for the interval $[T,T+\tau]$ is given by $y(t,T,T+\tau)$.

The time between the expiry of the forward contract and the maturity of the underlying bond, $\tau$, is often called the tenor of the forward bond or forward yield.

The above yield is motivated by the idea that the issuer of the ZCB invests their proceeds into a continuously compounding account. The time scaling $\tau$ term in the exponential above ensures that yields are comparable across tenors, making yields a useful metric for comparing different ZCBs.

Most actual market quotes, however, are based on discrete-time compounding of proceeds. Accordingly, we define a simple forward rate $L(t,T,T+\tau)$ as

Definition (Simple forward rate)

The time $t$ value for the simple forward rate corresponding to the forward bond price for the interval $[T,T+\tau]$ is given by $L(t,T,T+\tau)$.

Again, $\tau$ is the tenor of the forward rate. For an arbitrary set of dates $T = T_0 < T_1 <\cdots < T_n$, define the tenor structure $\left\{ \tau_1,\ldots,\tau_n\right\}$ by $\tau_i := T_i - T_{i-1}$. Notice that the forward bond prices can be recovered from the forward rates by simple compounding:

We will always assume that the spot rates $L(t,T) := L(t, t, T)$, $t \le T$ are the Libor rates quoted in the interbank market. Libor rates are quoted on values of $\tau = T - t$ ranging from one week ($\tau = \tfrac{1}{52}$) all the way through to 12 months ($\tau = 1$). Spot Libor rates are incredibly important in the interest-rate derivatives market. They form the benchmark for a number of instruments, such as interest rate swaps and Eurodollar futures.

Note that for a number of reasons, the market is currently in the process of transitioning away from Libor to the so called risk free rates (RFRs): Sterling Overnight Interbank Average (SONIA) and Secured Overnight Financing Rate (SOFR) in the UK and the US respectively.

Another important quantity is known as the instantaneous forward rate at time $T$, denoted by $f(t,T)$. This quantity can be seen as the forward rate at time $t$ for the infinitesimal interval $[T, T+ \text{d}T]$. Hence, it is given by

Definition (Instantaneous forward rate)

The time $t$ value for the instantaneous forward rate for the infinitesimal interval $[T,T+\text{d}T]$ is given by $f(t,T)$.

Using the definition of $L(t,T,T+\tau)$ and simple calculus we see that

Integrating both sides and taking exponentials shows that

By comparison with the continuously compounded yield formula we also have

or, upon using the mean value theorem,

Finally, by multiplying both sides of (1.2) by $\tau$ and differentiating, we see have

The quantity $r(t) := f(t,t)$ is another important quantity in the interest rate modeling world. It is known as the short rate and is an $\mathcal{F}(t)$-measurable random variable. Loosely speaking, $r(t)$ can be thought of as the overnight rate at time $t$.

Definition (Short rate)

The instantaneous forward rate at time $t$ for the infinitesimal interval $[t,t+\text{d}t]$ is known as the short rate and is given by

Futures Rates

A quantity closely related to the forward rate is the futures rate. Through the Eurodollar futures market, investors can enter into a security paying

at time $T$. At inception, i.e. time 0, a Eurodollar futures contract can be entered into for no upfront cost, but with the implicit obligation of the holder paying

per unit of notional at time $T$, in return for the payout (1.3). Here, the quantity $F(t,T,T+\tau)$ is known as the simple futures rate at time $t$ for the period $[T,T+\tau]$.

Definition (Simple futures rate)

The time $t$ value for the simple futures rate for the interval $[T,T+\tau]$ is given by $L(t,T,T+\tau)$.

The important difference between a forward and a futures contract is that the futures rate is marked to market each day, meaning that the day's change in the futures rate is immediately credited (or debited) from the contract holder's account with the futures exchange. This difference is in place to mitigate the credit risk associated with either party in the contract defaulting on their payments and leads to differences in the modeling of futures relative to forwards. Specifically, after holding the contract for a period of $\Delta = 1$ day, the futures contract holder would experience a cash flow of

Continuing this process to maturity, denoting day $i$ by $\Delta_i$, the futures holder receives a cumulative cash-flow of

where we have used the fact that $F(T,T,T+\tau)$ must equal $L(T,T,T+\tau)$ to avoid delivery arbitrage (if they were not, we could enter the futures contract at maturity and immediately make a profit (or loss)) by selling the underlying.

As with forward rates, we can define the instantaneous futures rate $q(t,T)$ via:

Definition (Instantaneous futures rate)

The time $t$ value for the instantaneous futures rate for the infinitesimal interval $[T,T+\text{d}T]$ is given by $q(t,T)$.

Annuity Factors and Par Rates

In contrast to ZCBs, most fixed income securities involve multiple cash-flows taking place on a predetermined schedule of dates, referred to as a tenor structure, $0 \le T_0 \cdots < T_N$. Given such a tenor structure, for any two integers $k$ and $m$ satisfying $0 \le k < N$, $m > 0$ and $k + m \le N$, we define the annuity factor $A_{k,m}$ by

Definition (Annuity factor)

The time $t$ annuity factor for the tenor structure $0 \le T_0 \cdots < T_N$, with $k$ and $m$ satisfying $0 \le k < N$, $m > 0$ and $k + m \le N$ is given by $A_{k,m}(t)$ as defined above.

Annuity factors supply us with a compact way of expressing the price of coupon-bearing bonds. For instance, if a coupon bond makes $m$ interest payments, each of a value $c \tau_n$ and each paid at the time $T_{n+1}$ for $n = k,\ldots,k+m-1$, we can consider each of these payments as a separate ZCB scaled by notional $c \tau_n$, so the security has time $t$ value

If the security also involves a final payment of principal, as is the case for standard coupon-bearing bonds, the coupon bearing bond has time $t$ price

where the bond has been normalized to pay unit notional. The time $t$ forward price for expiry $T_k$ is then given by

The value of the coupon payment $c$ for which this expression is at par, i.e. 1 is known as the forward par rate, or, when used in the context of swap pricing, as the forward swap rate. Denoting this quantity by $S_{k,m}(t)$ and rearranging gives

Definition (Forward par/swap rate)

The time $t$ forward swap rate for the tenor structure $0 \le T_0 \cdots < T_N$, with $k$ and $m$ satisfying $0 \le k < N$, $m > 0$ and $k + m \le N$ is given by $S_{k,m}(t)$ as defined above.

From the definition of the simple forward rate $L(t,T_n,T_{n+1})$ we then see that the above can be rewritten as

where $L_n(t) := L(t,T_n, T_{n+1})$.

Hence, one interpretation of the forward swap rate is that it is simply a weighted average of simple forward rates on the tenor structure. The time $T_k$ is often referred to as the fixing date, or expiry of the swap rate $S_{k,m}$, while the length of the corresponding swap $T_{k+m} - T_k$ is often called the tenor of the swap rate.