# Credit Derivatives

These are some notes on credit derivatives from a mathematical point of view.

It is WIP and based mostly off Modelling single-name and multi-name Credit Derivatives (O'Kane, 2009).

Contents
Fundamentals
Notation Table
Discount Curves
Short Rates
Discount processes
Constant Rate Case
Annuities
Credit Events
Risky Annuties
Hazard Rates
Event-Triggered Payments

# Fundamentals

## Notation Table

We adopt the following notation throughout this document:

Notation Meaning
$s$ Time parameter
$t$ Time parameter
$\xi$ Time parameter
$T$ Maturity date of contract
$\tau$ Credit event
$Z(s,t)$ discount curve
$r(t)$ short rate process
$D(t)$ discount process
$\Gamma(t)$ annuity process
$A(t)$ annuity valuation process
$Q(t,T)$ survival curve
$\lambda(t)$ hazard rate process
$\hat Z (t,T)$ risky discount curve
$\hat A (t)$ risky annuity valuation process
$I$ Indicator variable
$\mathcal F$ filtration

Assume that all stochastic processes, unless otherwise specified, are adapted to some common filtration $\mathcal F$. We us $\mathbb{E}_s [X]$ as shorthand for the conditional expectation of the random variable $X$ with respect to $\mathcal F_{s}$ for some time parameter $s$:

$\mathbb{E}_s [X] = \mathbb{E}\left[X | \mathcal F_s\right].$

Time is represented with a real number $s$, with $s=0$ usually indicating the time at which some contract was created.

## Discount Curves

Consider a hypothetical bond maturing at time $t$ (the maturity or tenor) having the following properties:

Unit-Notional
At time $t$ the bond holder will be paid one unit of currency.
Zero-Coupon
No payments to the bond holder will be made before time $t$.
Risk Free
The notional is guaranteed to be paid in full and on time.

Denote the fair value of this bond ascertained at time $s$ ("today") as $Z(s,t)$. Here, $Z$ is known as the discount curve. This can be thought of as a bitemporal function. But more precisely, $Z$ is a parametrized class of stochastic processes. That is, for a fixed $t$, $s\mapsto Z(s,t)$ is a stochastic proccess.

We make the following assumptions on $Z$:

• $Z(s,t) > 0$
• $Z(s,t+h)=Z(s,t)\mathbb{E}_s[Z(t,t+h)]$
• For fixed $s$, $t\mapsto Z(s,t)$ is Riemann integrable.
• (Optional) For fixed $s$, $t \mapsto Z(s,t)$ is smooth.

The fourth assumption is optional. Most of the theory we develop won't require it. But it does aid in the definition of the short rate and discount process.

Exercise. From above assumptions, show that $Z(s,s)=1$.

## Short Rates

Assume that $t \mapsto Z(s,t)$ is smooth.

Define $Z'$ as the derivative of $Z$ with respect to its second argument. That is, $Z'(s,t)=\frac{\partial Z(s,t)}{\partial t}$. Then

\begin{aligned} Z'(s,t) &= \lim_{h\to 0} \frac{Z(s,t+h)-Z(s,t)}{h} \\ &= \lim_{h\to 0} \frac{Z(s,t)\mathbb{E}_s[Z(t,t+h)]-Z(s,t)}{h} \\ &= Z(s,t) \mathbb{E}_s\left[\lim_{h\to 0} \frac{Z(t,t+h)-1}{h}\right] \\ &= Z(s,t)\mathbb{E}_s\left[Z'(t,t)\right] \end{aligned}

Consequently,

\begin{aligned} Z(s,t) &= \mathbb{E}_s\left[ \exp\left(\int_s^t Z'(\xi,\xi)d\xi \right)\right] \\ &= \mathbb{E}_s\left[ \exp\left(-\int_s^t r(\xi)d\xi \right)\right]. \end{aligned}

Here, $r(t)=-Z'(t,t)$ is the short rate process. It can be thought of as the "instanteous" interest rate at time $t$.

## Discount processes

The discount process corresponding to the short rate process $r(t)$ is defined as

$D(t) = \exp\left(- \int_0^t r(\tau)d\tau\right)$

With this notation, the discount curve may be written as

$Z(s,t)=\frac{\mathbb E_s [ D(t) ]}{D(s)}.$

## Constant Rate Case

In the case that $r$ is a constant-value stochastic process, e.g. $r(t)=r_o$, then the discount process is just an exponential function $D(t)=e^{-r_ot}$. And the discount cuve is $Z(s,t)=e^{r_o(s-t)}$.

## Annuities

An annuity is a schedule of future payments. Denote $\Gamma (t)$ as the total payments of an annuity up to and including time $t$.

Assuming zero risk, the fair value of the remaining annuity at time $t$ is defined as the following Riemann-Stieltjes integral:

$A(t) = \int_t^\infty Z(t,\xi) d\Gamma (\xi).$

Discrete Case
Consider an annuity consisting of discrete payments $\gamma_1,\ldots,\gamma_N$ at times $\xi_1,\ldots,\xi_N$. Then
$A(t) = \sum_{n=1}^{N} \gamma_n Z(t,\xi_n)I(t>\xi_n),$
where $I$ is the indicator variable. Here $t_N$ is the maturity of the annuity.
Smooth Case
Suppose that $\Gamma$ is smooth. And define $\gamma(t) = \Gamma' (t)$ as the annuity rate. Then
$A(t) = \int_t^\infty Z(t,\xi)\gamma (\xi) ds.$
Smooth Case With Maturity
Suppose that $\Gamma' (t) = \gamma(t)I(t < T)$. Then the annuity value process can be written as
$A(t)=\int_t^T Z(t,\xi)\gamma(\xi)d\xi.$
Here, $T$ is the maturity of the annuity.

## Credit Events

A credit event is a contractually-obligated event wherein a debt security (e.g. a bond) has been determined to not be fully honored.

Mathematically, a credit event $\tau$ is a stopping time. That is, a random variable representing some point in time.

It's distribution can be described by the survival curve $Q$:

\begin{aligned} Q(s,t) & = \text{Pr}(\tau > t|\mathcal F_s) \\ & = \mathbb{E}_s [I(\tau > t)] \end{aligned}

This is the probability ascertained at time $t$ that a credit event will not occur before or during time $T$.

## Risky Annuties

Consider an annuity $\Lambda (t)$ that is risky. That is, scheduled payment occuring at or after a credit event $\tau$ are unrealized. The forward-looking fair value of this annuity is given by the risky value process:

$\hat A(t) = \int_t^\infty Z(t,\xi) Q(t,\xi) d\Gamma(\xi)$

## Hazard Rates

The hazard rate $\lambda (t)$ is the instanteous likelihood that a credit event will occur at time $t$. Mathematically, it is a stochastic process $\lambda$ defined by:

$\lambda(t) = \left. -\frac{\partial Q(t,s)}{\partial s} \right\vert_{s=t}.$

Intuitively, $\lambda(t) dt$ is the likelihood that a credit event will occur between times $t$ and $t+dt$.

Equivalently, the survival curve may be derived from an a priori defined hazard rate:

$Q(t,T)= \mathbb E \left[ \exp \left( -\int_t^T \lambda(s)ds \right) \middle\vert \mathcal F_t \right].$

The relationship between $\lambda$ and $Q$ is analogous to that of $r$ and $Z$.

## Event-Triggered Payments

Consider a payment of one unit of currency paid out at time $\tau$ if $\tau < T$ for some tenor $T$. This is a form of insurance. And it's value at time $t$ is given by

$\int_{s=t}^{s=T} Z(t,s) dQ(t,s) = \int_t^T Z(t,s) \lambda(s) ds.$

$\Gamma^{\text{ins}}(t) = \gamma_o^{\text{ins}}t + I(\tau < t)(\gamma_o^{\text{ins}}(t-\tau))$
Note that this is necessarily a stochastic process since it depends on the stopping time $\tau$.