Basic Fixed Income Instruments

Interest Rate Modeling

[Quantitative Finance] · August 25, 2020 · 16 mins to read

Tutorial Contents

In this section, we will introduce the various fixed income markets along with an overview of the most basic products that trade. The simpler and more liquid of these products typically serve as targets to calibrate the parameters or our models to. The more complex and more illiquid ones, where accurate pricing is not readily available, are the contracts that our models are ultimately designed to price and hedge.

The vast majority of securities we analyse are tied to the so-called Libor rates - the London Interbank Offered Rates - forward looking interest rates set by a group of banks as an estimate of the rate they would be charged were they to borrow from other banks. Traditionally, Libor rates form a benchmark for the majority of exotic securities along with more basic instruments such as swaps, FRAs, caps and swaptions. The industry is currently in the process of shifting away from Libor towards so called risk free rates. In the UK, the rate of choice is known as SONIA - Sterling Overnight Index Average. For the time being, Libor is still in full force and will form the basis of our analysis.

Fixed Income Markets and Participants

Fundamentally, an interest rate represents the cost of borrowing and lending - to the borrower it is the price they must pay for borrowing - to the lender it represents the compensation they demand for tying up their capital and sacrificing the ability to earn a return of their own.

In general, the interest rate demanded for borrowing/lending for different lengths of time will not be equal. In usual conditions, a risk premium is demanded by the market for lending for longer periods of time. A participant tying up their money for a long period of time faces greater risk of an adverse move in economic events that they will be unable to respond to timely, than a particpant lending overnight. This effect is known as the liquidity spread. The dependence of interest rates versus time is described as the term structure of interest rates.

When comparing interest rates across time to maturity, it is important that each rate belongs to an instrument with the same credit quality. For example, it is valid to assume that a 2 year US Treasury Note has a similar credit risk (or risk of default) to a 10 year US Treasury Bond, etc. On the other hand, a 10 year T-Bond has a very different credit risk to a 10 year Bond issued by Greece. The difference in the yields or rates offered by these instruments is primarily driven by the different probabilities of the two countries honouring their obligation to repay these bonds. While this effect is important, when we analyse a particular term structure, we are trying to isolate the effects entirely due to the varying lengths of time for which we borrow and eliminate any credit risk.

The vast majority of fixed income instruments are traded OTC - over-the-counter. This market runs parallel to the exchange markets. The OTC market can be visualised as a network of banks and institutions trading among each other under terms governed by standardised agreements spelled out by ISDA - International Swaps and Derivatives Association. Central to OTC markets are interest rate dealers, trading desks within banks specialised in fixed income trading. These desks provide provide liquidity in various securities across the rates spectrum.

Institutions include mortgage companies, pension funds, insurance companies and hedge funds. These companies either seek to profit by engaging in trading activities (primarily hedge funds), or to hedge their exposures to interest rate risks, and achieve superior returns on their investments (pension funds and insurance companies). The exotic end of the fixed income markets is driven by issuers, companies that sell structured notes, instruments delivering appealing returns to investors via embedded optionality retained by the issuers.

Corporates form the remaining participants in the fixed income markets. These are the remainder (and majority) of companies, whose primary activities are not directly linked to fixed income markets. These participants raise capital by borrowing from banks or by issuing bonds - in doing so they expose themselves to interest rates and other market risks, which they may hedge through swaps and other fixed income derivatives.

Certificates of Deposit and Libor Rates

The most basic fixed income security is the CD - certificate of deposit. A CD is simply a contract returning a notional amount $N$ after a prespecified term and at a prespecified interest rate. Terms range from a week up to a year or more, with 3 and 6 months being the most popular terms. If 1 unit is invested at a date $T$ for a period of $\tau$ years, then $N = 1$ and the holder receives

1 + \tau L

at the time $T + \tau$ , where $L$ is the interest rate quoted on the CD. The rate is quoted as a simple rate, i.e. with a compounding frequency equal to $\tau$ . By definition of Libor itself, the average value of $L$ will be equal to the Libor rate for the tenor $\tau$ . Spot Libor rates for various tenors are calculated daily and quoted on Bloomberg.

If $P(T,T+\tau)$ is the discount factor to date $T + \tau$ on the date $T$ , then the discounted value of the CD should be equal to 1 at $T$ and is given by

1 = P(T, T + \tau)(1 + \tau L).

In particular, if $P(T, T + \tau)$ is the Libor based discount factor, then, as we might expect, we find that

L = L(T,T,T+\tau) = \frac{1}{\tau}\left(\frac{1 - P(T,T,T+\tau)}{P(T, T, T+\tau)}\right).

Forward Rate Agreements (FRAs)

FRAs - Forward Rate Agreements are similar to CDs in that they allow investors to lock in an interest rate for a given period of time. However, with a FRA, the interest rate is locked in for a period of time that starts in the future. Contracts providing such a guarantee in general are called forward contracts. A FRA for the period $[T, T+\tau]$ is a contract to exchange a fixed rate payment (agreed on the initiation date of the contract) against a payment based on the time $T$ spot Libor for tenor $\tau$ . While all payments on a FRA are exchanged on date $T$ , the contract is structured so that payments are made in $T+\tau$ currency units.

Formally, suppose today is date $t$ , $t \le T$ , and we enter a FRA on unit notional with a rate $K$ . From the perspective of the fixed rate payer, the net payment at $T$ is given by

V(T) = \tau \left( \frac{L(T, T, T+\tau) - K}{1 + \tau L(T,T,T+\tau)}\right) = \tau P(T,T+\tau) (L(T,T,T+\tau) - K)

where the discount factor $\tfrac{1}{1 + \tau L(T,T,T+\tau)}$ is applied to roll the payment into $T+\tau$ currency units. To discount this payment back to time $t$ , it is most convenient price under the $T$ measure. Doing so, we find that

\begin{aligned} V(t) & = \tau P(t,T) \mathbb{E}_t^{T}\left[P(T,T+\tau) (L(T,T,T+\tau) - K) \right] \\ & = P(t,T) \mathbb{E}_t^{T}\left[P(T,T+\tau) \left(\frac{1}{P(T, T+\tau)} - 1 - \tau K \right) \right] \\ & = P(t,T) \mathbb{E}_t^{T}\left[ 1 - P(T,T+\tau)( 1 + \tau K ) \right] \\ & = P(t, T) - P(t, T+\tau)(1 + \tau K). \end{aligned}

A FRA is constructed so that there is zero upfront cost to either party. Hence, the fixed rate $K$ is set to satisfy $V(t) = 0$ . Rearranging for $K$ we see that

K = L(t, T, T+\tau) = \frac{1}{\tau}\left(\frac{1}{P(t, T, T+\tau)} - 1 \right).

From this we can see that the forward Libor rate $L(t, T, T+\tau)$ can be interpreted as the break-even rate on a FRA in the interbank market.

Eurodollar Futures

We saw above that a FRA allows a market participant to lock in favourable rates for future periods (or to speculate on the future direction of rates). However, FRAs trade only on the OTC market and are thus only open to institutions participating in this market. An alternative way to achieve a similar goal is to purchase a Eurodollars futures contract. These contracts trade on the CME - Chicago Mercantile Exchange. The ED Futures - Eurodollars futures contract - settles at maturity $T$ at

100\cdot (1 - L(T,T,T+\tau)).

The futures rate $F(t, T, T+\tau)$ at time $t$ is defined to be the rate such that the quoted futures price at time $t$ is equal to

100\cdot (1 - F(t,T,T+\tau)).

As with all futures contracts, ED futures are marked to market daily. Confusingly, the actual amount settled between counterparties is determined by the daily change in the actual futures price defined by

N_{\text{ED}} \cdot \left(1 - \frac{1}{4}F(t,T,T+\tau) \right),

where $N_{\text{ED}}$ is the notional principal of the contract. For ED futures on the CME, this value is set to $\$1,000,000$ . In particular, for a 1bp move in the rate $F(t, T, T+\tau)$ , the CME holder exchanges $\$25$ .

ED futures are standardised. Available contracts expire on four dates - one each in March, June, September and December over the next 10 years. Due to this standardisation, ED futures are liquid and trade regurlarly.

Fixed for Floating Swaps

Swaps are OTC derivatives in which two counterparties exchange one stream of cash flows on a preset schedule against another on a (potentially different) preset schedule. These two streams are called the legs of the swap. A vanilla IR swap is a swap in which one leg is a stream of fixed payments and another a stream of floating rate payments, often Libor based. The legs are denominated in the same currency, have the same notional and expire on the same date. Payment streams are made on a pre-defined schedule of contiguous time intervals, called periods. Typically, the floating rate is fixed at the start of each period, and both legs are paid out at the end of the period.

Swaps of varying maturities are traded between banks and financial institutions both to hedge interest rates exposures, and to make bets on future rates. Corporates also use swaps in conjunction with bond issuance to transform their bonds into synthetic floating rate bond positions.

To define a fixed-floating swap, one specifies a tenor structure (an increasing sequence of maturities), typically spaced equidistantly:

0 \le T_0 < T_1 \cdots < T_N, \quad \tau_n = T_{n+1} - T_n.

In a vanilla swap with fixed rate $K$ , one party pays simple interest based on the rate $K$ in return for simple interest payments based on the Libor rate on date $T_n$ for the period $[T_n, T_{n+1}]$ , $n = 0,\ldots,N-1$ . The payments are exchanged at the end of each period, $T_{n+1}$ . In practice, payments are netted and only their difference changed hands. From the fixed rate payers perspective, the net cash flow of the swap at $T_{n+1}$ is thus

\tau_n (L_n(T_n) - K), \quad L_n(t) := L(t, T_n, T_{n+1}), n = 0,\ldots N-1.

Dates at which Libor rates are observed are called fixing dates and dates where payments occur are called payment dates. The time $t$ , $0 \le t \le T_0$ value of the swap to the fixed rate payer is given by

\begin{aligned} V(t) & = \sum_{n=0}^{N-1}\tau_n \beta(t) \mathbb{E}_t^{\mathbb{Q}}\left[\beta(T_{n+1})^{-1} (L_n(T_n) - K) \right] \\ & = \sum_{n=0}^{N-1}\tau_n P(t,T_{n+1}) \mathbb{E}_t^{T_{n+1}}\left[ L_n(T_n) - K \right] \\ & = \sum_{n=0}^{N-1} \tau_n P(t, T_{n+1}) \left(L_n(t) - K \right) \\ & = \left(\sum_{n=0}^{N-1} \tau_n P(t,T_{n+1}) \right) \left(\frac{\sum_{n=0}^{N-1}\tau_n P(t,T_{n+1})L_n(t)}{\sum_{n=0}^{N-1} \tau_n P(t,T_{n+1})} - K \right) \\ & = A(t)(S(t) - K), \end{aligned}

where the first equality follows from changing measure from the risk-neutral measure to the $T_{n+1}$ forward measure, and $A(t)$ and $S(t)$ are the annuity and swap rates respectively, i.e.

\begin{aligned} A(t) & := \sum_{n=0}^{N-1} \tau_n P(t,T_{n+1}), \\ S(t) & := \frac{\sum_{n=0}^{N-1}\tau_n P(t,T_{n+1})L_n(t)}{\sum_{n=0}^{N-1} \tau_n P(t,T_{n+1})}. \end{aligned}

General Swaps

In general, interest rate swaps are not required to pay a constant fixed rate, or for the notional to remain constant throughout the lifetime of the swap. A non-standard swap with notional schedule $\{q_n\}_{n=0}^{N-1}$ and fixed rate schedule $\{k_n\}_{n=0}^{N-1}$ has the value

\begin{aligned} V(t) & = \beta(t)\sum_{n=0}^{N-1} \tau_n q_n \mathbb{E}_t^{\mathbb{Q}} \left[\beta(T_{n+1})^{-1} (L_n(T_n) - k_n) \right] \\ & = \sum_{n=0}^{N-1} \tau_n q_n P(t, T_{n+1})(L_n(t) - k_n). \end{aligned}

Increasingly, swaps linked to overnight rates (Fed funds/SONIA) are traded. One example is the OIS - overnight index swap. The most liquid of all overnight rate based swaps, the OIS pays a compounded overnight rate versus fixed rate payments. Assuming we have the same tenor structure as before, and denoting $\{t_{n,i}\}_{i=1}^{K_n}$ the collection of business days in the period $[T_n,T_{n+1})$ , so that $T_n = t_{n,1} < \cdots < t_{n, K_n} < T_{n+1}$ , the net payment of the OIS with fixed rate $K$ at time $T_{n+1}$ is given by

\tau_n (\hat{L}_n - K),

where the floating rate $\hat{L}_n$ for the $n$ th period of the swap is given by

\hat{L}_n = \frac{1}{\tau_n} \left( \prod_{i=1}^{K_n - 1} (1+(t_{n,i+1} - t_{n,i})L(t_{n,i}, t_{n,i}, t_{n,i+1})) - 1 \right).

Here, $L(t,t,s)$ represents the overnight rate from the day $t$ to the next business day $s$ .

Libor-in-Arrears Swaps

For a Libor-in-arrears swap, Libor rates are observed (fixed) at the end of each period rather than at the start. Thus, the value of a Libor-in-arrears fixed-for-floating swap is equal to

V(t) = \beta(t)\sum_{n=0}^{N-1} \tau_n \mathbb{E}_t^{\mathbb{Q}}\left[\beta(T_{n+1})^{-1}(L_{n+1}(T_{n+1}) - K) \right].

Whereas with the vanilla fixed-for-floating swap, discounting from $T_{n+1}$ to $T_n$ led to cancellation of discount factors in the Libor rate and a simple expression for the value of the swap, here we cannot perform the same trick and the value of the Libor-in-arrears swap is model dependent.

Libor-in-arrears swaps are popular in upward-sloping rates environments. In such an environment, the break-even fixed rate on the Libor-in-arrears swap looks more attractive than that of the equivalent vanilla swap, thus increasing the desirability of the swap to those wishing to receive fixed.

Averaging Swaps

Swaps are not restricted to observing Libor rates on either the start or end day of a pay period. Another popular IR swap is the averaging swap, i.e. a swap where the floating rate is determined as an average of Libor rates observed at regular intervals between each period. To see this precisely, let $\{t_{n,i}^f, t_{n,i}^s, t_{n,i}^e \}_{i=1}^{K_n}$ be a collection of date triplets defining rates to be used in calculating the payment in period $n$ - the fixing, start and end dates. Defining a set of weights $w_{n,i}$ summing to 1, $i=1,\ldots,K_n$ , the floating rate $\hat{L}_n$ for the period $[T_n, T_{n+1}]$ is defined by

\hat{L}_n := \sum_{i=1}^{K_n} w_{n,i} L(t_{n,i}^f, t_{n,i}^s, t_{n,i}^e).

For the fixed rate payer, the time $t$ averaging swap value is thus given by

V(t) = \beta(t)\sum_{n=0}^{N-1} \tau_n \mathbb{E}_t \left[\beta(T_{n+1})^{-1}(\hat{L}_n - K) \right]. \tag{5.1}

Calculating (5.1) is again a model dependent computation.

Swaps linked to the average Fed fund rate are also common. Particularly noteworthy are Fed funds/Libor basis swaps which pay the average of the Fed funds rate versus a Libor based payment for that period. This instrument is an example of a floating for floating single currency basis swap. Another closely related contract is the Fed funds futures traded on the CBOT - Chicago Board of Trade. These use the 30 day running average of the Fed funds rate for settlement.