IRS, Caps, Floors and Swaptions

Fundamentals

I review here a few basic definitions relevant to the interest-rate world.

Definition 1 (Zero-coupon bond.) A $T$-maturity zero-coupon bond (pure discount bond) is a contract that guarantees its holder the payment of $1\$$ at time $T$, with no intermediate payments. The contract value at time $t < T$ is denoted by $P(t,T)$. Clearly, $P(T,T) = 1$ $\forall T\in[0,\infty)$.

Definition 2 (Continuously-compounded spot interest rate.) The continuously-compounded spot interest rate prevailing at time $t$ for the maturity $T$ is denoted by $R(t,T)$ and is the constant rate at which an investment of $P(t,T)$ units of currency at time $t$ accrues continuously to yield a unit amount of currency at maturity $T$.

\[ \begin{align*} R(t,T) := - \frac{\ln P(t,T)}{\tau(t,T)} \end{align*} \tag{1}\]

The continuously-compounded interest rate is therefore a constant rate that is consistent with the zero-coupon-bond prices such that:

\[ \begin{align*} e^{R(t,T)\tau(t,T)}P(t,T) = 1 \end{align*} \tag{2}\]

from which we can express the bond price in terms of the continuously compounded rate $R$:

\[ \begin{align*} P(t,T) = e^{-R(t,T)\tau(t,T)} \end{align*} \tag{3}\]

Definition 3 (Simply-compounded spot interest rate.) The simply-compounded spot interest rate prevailing at time $t$ for the maturity $T$ is denoted $L(t,T)$ and is the constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from $P(t,T)$ units of currency at time $t$, when accruing occurs proportionally to the investment time.

\[ \begin{align*} P(t,T)(1 + L(t,T)\tau(t,T)) = 1 \end{align*} \tag{4}\]

So, the bond price can be expressed in terms of $L$ as:

\[ \begin{align*} P(t,T) = \frac{1}{1 + L(t,T)\tau(t,T)} \end{align*} \tag{5}\]

Definition 4 (Annually-compounded spot interest rate.) The annually-compounded spot interest rate prevailing at time $t$ for the maturity $T$ is denoted by $Y(t,T)$ and is the constant (annualized) rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from $P(t,T)$ units of currency at time $t$, reinvesting the obtained amounts once a year. We have:

\[ P(t,T)(1+Y(t,T))^{\tau(t,T)} = 1 \tag{6}\]

Equivalently,

\[ Y(t,T) = \left[\frac{1}{P(t,T)}\right]^{\frac{1}{\tau(t,T)}} - 1 \tag{7}\]

Definition 5 (Zero-coupon curve.) The zero-coupon curve(sometimes also referred to as the yield curve) at time $t$ is the graph of the function

\[ T \mapsto \begin{cases} L(t,T) & t < T \leq t + 1 \text{ years }\\ Y(t,T) & T \geq t + 1\text{ years } \end{cases} \tag{8}\]

Definition 6 (Discounting Curve.) The discounting curve at time $t$ is the plot of the function:

\[ T \mapsto P(t,T), \quad T > t \tag{9}\]

Such a curve is also referred to as the term structure of discount factors.

Definition 7 (Simply-compounded forward interest rate.) The simply compounded forward interest rate prevailing at time $t$ for the expiry $T > t$, maturity $S > T$ and is defined by:

\[ \begin{align*} F(t;T,S) := \frac{1}{\tau(T,S)}\left(\frac{P(t,T)}{P(t,S)} - 1\right) \end{align*} \tag{10}\]

Definition 8 (Instantaneous forward rate.) The instantaneous forward interest rate prevailing at time $t$ for the maturity $T > t$ is denoted by $f(t,T)$ and is defined by:

\[ \begin{align*} f(t,T) &= \lim_{S \to T^+} F(t;T,S) \\ &= \lim_{S \to T^+} \frac{1}{\tau(T,S)}\frac{P(t,T) - P(t,S)}{P(t,T)} \\ &= -\frac{1}{P(t,T)}\lim_{S \to T^+} \frac{P(t,S) - P(t,T)}{\tau(T,S)}\\ &= -\frac{1}{P(t,T)}\lim_{h\to 0} \frac{P(t,T+h) - P(t,T)}{h}\\ &= -\frac{1}{P(t,T)} \frac{\partial}{T}(P(t,T))\\ &= - \frac{\partial}{\partial T}(\ln P(t,T)) \end{align*} \tag{11}\]

so we also have:

\[ P(t,T) = \exp\left(-\int_{t}^T f(t,u)du\right) \tag{12}\]

Classical LIBOR Rate Model

Let’s start with the classical LIBOR rate model. Suppose that bank B enters into a contract at time $t$ with bank A, to borrow 1 EUR at time $T_0$ and return 1 EUR plus the interest cost at time $T_1$. What’s the fair interest rate, that bank A and bank B can agree on? The MTM value to bank A is:

\[ \begin{align*} V(t) &= P(t,T_0) \mathbb{E}^{T_0}[-1|\mathcal{F}_t] + P(t,T_1)\mathbb{E}^{T_1}[1+\tau K|\mathcal{F}_t]\\ 0 &= -P(t,T_0) + P(t,T_1)(1+\tau K) \end{align*} \]

where $\tau=\tau(T_0,T_1)$ is the day-count fraction between $[T_0,T_1]$

Spot LIBOR Rate

The fair rate for an interbank lending deal with trade date $t$, starting date $T_0$ (typically 0d or 2d after $T$) and maturity date $T_1$ is:

\[ \begin{align*} L(t;T_0,T_1) = \frac{1}{\tau}\left[\frac{P(t,T_0)}{P(t,T_1) - 1}\right] \end{align*} \]

Panel banks submit daily estimates for interbank lending rates to the calculation agent. The relevant periods $[T_0,T_1]$ considered are $1m$, $3m$, $6m$ and $12m$. LIBOR rate fixings used to be the most important reference rates for interest rate derivatives. Nowadays, overnight rates have become the key reference rates.

References

Chapter 1, Interest Rate Models - Theory and Practice, Damiano Brigo and Fabio Mercurio.

--- title: "IRS, Caps, Floors and Swaptions" author: "Quasar" date: "2025-01-24" categories: [Rates Modelling] image: "image.jpg" toc: true toc-depth: 3 format: html: code-tools: true code-block-border-left: true code-annotations: below highlight-style: pygments --- # IRS, Caps, Floors and Swaptions ## Fundamentals I review here a few basic definitions relevant to the interest-rate world. :::{#def-zero-coupon-bond} ### Zero-coupon bond. A $T$-maturity zero-coupon bond (pure discount bond) is a contract that guarantees its holder the payment of $1\$$ at time $T$, with no intermediate payments. The contract value at time $t < T$ is denoted by $P(t,T)$. Clearly, $P(T,T) = 1$ $\forall T\in[0,\infty)$. ::: :::{#def-continuously-compounded-spot-interest-rate} ### Continuously-compounded spot interest rate. The continuously-compounded spot interest rate prevailing at time $t$ for the maturity $T$ is denoted by $R(t,T)$ and is the constant rate at which an investment of $P(t,T)$ units of currency at time $t$ accrues continuously to yield a unit amount of currency at maturity $T$. $$ \begin{align*} R(t,T) := - \frac{\ln P(t,T)}{\tau(t,T)} \end{align*} $${#eq-continuously-compounded-spot-interest-rate} The continuously-compounded interest rate is therefore a constant rate that is consistent with the zero-coupon-bond prices such that: $$ \begin{align*} e^{R(t,T)\tau(t,T)}P(t,T) = 1 \end{align*} $${#eq-zero-coupon-bond-price-equation-1} from which we can express the bond price in terms of the continuously compounded rate $R$: $$ \begin{align*} P(t,T) = e^{-R(t,T)\tau(t,T)} \end{align*} $${#eq-zero-coupon-bond-price-equation-2} ::: :::{#def-simply-compounded-spot-interest-rate} ### Simply-compounded spot interest rate. The simply-compounded spot interest rate prevailing at time $t$ for the maturity $T$ is denoted $L(t,T)$ and is the constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from $P(t,T)$ units of currency at time $t$, when accruing occurs proportionally to the investment time. $$ \begin{align*} P(t,T)(1 + L(t,T)\tau(t,T)) = 1 \end{align*} $${#eq-simply-compounded-spot-interest-rate} So, the bond price can be expressed in terms of $L$ as: $$ \begin{align*} P(t,T) = \frac{1}{1 + L(t,T)\tau(t,T)} \end{align*} $${#eq-bond-price-in-terms-of-a-simply-compounded-spot-rate} ::: :::{#def-annually-compounded-spot-interest-rate} ### Annually-compounded spot interest rate. The annually-compounded spot interest rate prevailing at time $t$ for the maturity $T$ is denoted by $Y(t,T)$ and is the constant (annualized) rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from $P(t,T)$ units of currency at time $t$, reinvesting the obtained amounts once a year. We have: $$ P(t,T)(1+Y(t,T))^{\tau(t,T)} = 1 $${#eq-annually-compounded-spot-interest-rate} ::: Equivalently, $$ Y(t,T) = \left[\frac{1}{P(t,T)}\right]^{\frac{1}{\tau(t,T)}} - 1 $${#eq-annually-compounded-spot-interest-rate-as-func-of-bond-price} :::{#def-zero-coupon-curve} ### Zero-coupon curve. The zero-coupon curve(sometimes also referred to as the *yield curve*) at time $t$ is the graph of the function $$ T \mapsto \begin{cases} L(t,T) & t < T \leq t + 1 \text{ years }\\ Y(t,T) & T \geq t + 1\text{ years } \end{cases} $${#eq-zero-coupon-curve} ::: :::{#def-discounting-curve} ### Discounting Curve. The discounting curve at time $t$ is the plot of the function: $$ T \mapsto P(t,T), \quad T > t $${#eq-discounting-curve} Such a curve is also referred to as the term structure of discount factors. ::: :::{#def-simply-compounded-forward-rate} ### Simply-compounded forward interest rate. The simply compounded forward interest rate prevailing at time $t$ for the expiry $T > t$, maturity $S > T$ and is defined by: $$ \begin{align*} F(t;T,S) := \frac{1}{\tau(T,S)}\left(\frac{P(t,T)}{P(t,S)} - 1\right) \end{align*} $${#eq-simply-compounded-forward-interest-rate} ::: :::{#def-instantaneous-forward-rate} ### Instantaneous forward rate. The instantaneous forward interest rate prevailing at time $t$ for the maturity $T > t$ is denoted by $f(t,T)$ and is defined by: $$ \begin{align*} f(t,T) &= \lim_{S \to T^+} F(t;T,S) \\ &= \lim_{S \to T^+} \frac{1}{\tau(T,S)}\frac{P(t,T) - P(t,S)}{P(t,T)} \\ &= -\frac{1}{P(t,T)}\lim_{S \to T^+} \frac{P(t,S) - P(t,T)}{\tau(T,S)}\\ &= -\frac{1}{P(t,T)}\lim_{h\to 0} \frac{P(t,T+h) - P(t,T)}{h}\\ &= -\frac{1}{P(t,T)} \frac{\partial}{T}(P(t,T))\\ &= - \frac{\partial}{\partial T}(\ln P(t,T)) \end{align*} $${#eq-instantaneous-forward-rate} so we also have: $$ P(t,T) = \exp\left(-\int_{t}^T f(t,u)du\right) $${#eq-discount-curve-and-instantaneous-forward-rate} ::: ## Classical LIBOR Rate Model Let's start with the classical LIBOR rate model. Suppose that bank B enters into a contract at time $t$ with bank A, to borrow 1 EUR at time $T_0$ and return 1 EUR plus the interest cost at time $T_1$. What's the fair interest rate, that bank A and bank B can agree on? The MTM value to bank A is: $$ \begin{align*} V(t) &= P(t,T_0) \mathbb{E}^{T_0}[-1|\mathcal{F}_t] + P(t,T_1)\mathbb{E}^{T_1}[1+\tau K|\mathcal{F}_t]\\ 0 &= -P(t,T_0) + P(t,T_1)(1+\tau K) \end{align*} $$ where $\tau=\tau(T_0,T_1)$ is the day-count fraction between $[T_0,T_1]$ ## Spot LIBOR Rate The fair rate for an interbank lending deal with trade date $t$, starting date $T_0$ (typically 0d or 2d after $T$) and maturity date $T_1$ is: $$ \begin{align*} L(t;T_0,T_1) = \frac{1}{\tau}\left[\frac{P(t,T_0)}{P(t,T_1) - 1}\right] \end{align*} $$ Panel banks submit daily estimates for interbank lending rates to the calculation agent. The relevant periods $[T_0,T_1]$ considered are $1m$, $3m$, $6m$ and $12m$. LIBOR rate fixings used to be the most important reference rates for interest rate derivatives. Nowadays, overnight rates have become the key reference rates. # References {.appendix} - *Chapter 1, [Interest Rate Models - Theory and Practice](https://www.amazon.co.uk/Interest-Rate-Models-Practice-Inflation/dp/3540221492), Damiano Brigo and Fabio Mercurio.*