# Credit Derivatives

These are some WIP notes on credit derivatives.

Contents
Fundamentals
Notation Table
Discount Curves
Short Rates
Discount processes
Annuities
Credit Events
Risky Annuties
Hazard Rates
Insurance Payments

# Fundamentals

## Notation Table

Notation Meaning
$\mathcal F$ filtration
$Z(t,T)$ discount curve
$r(t)$ short rate process
$D(t)$ discount process
$\Gamma(t)$ annuity process
$A(t)$ annuity valuation process
$\tau$ credit event
$Q(t,T)$ survival curve
$\lambda(t)$ hazard rate process
$\hat Z (t,T)$ risky discount curve
$\hat A (t)$ risky annuity valuation process

Assume that all stochastic processes, unless otherwise specified, are adapted to some common filtration $\mathcal F$.

## 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. This payment is the notional.
Zero-Coupon
No payments to the bond holder will be made before time $T$ (otherwise, such payments are called coupons).
Risk Free
The notional is guaranteed to be paid in full and on time.

Denote the fair value of this bond ascertained at time $s$ as $Z(s,t)$. The bitemporal function $Z$ is called the (risk-free) discount curve. It represents the market demand for money.

Some assumed propeties for all real values $t\geq s$ and $h$:

• For fixed $s$, $s\mapsto Z(s,t)$ is a stochastic process with respect to the filtration $\mathcal{F}_s$.
• $Z(s,t) > 0$
• $Z(s,t+h)=Z(s,t)\mathbb{E}[Z(t,t+h)|\mathcal F_s]$

We can deduce from the above properties that $Z(t,t)=1$. This is because, applying the second property: $Z(t,t)=Z(t,t+0)=Z(t,t)Z(t,t+0)=Z(t,t)^2$.

## Short Rates

Let us additionally assume that $t \mapsto Z(s,t)$ is smooth.

Define $Z'$ as the derivative of $Z$ with respect to its second argument. That is, $Z'(smt)=\frac{\partial Z(s,t)}{\partial t}$. 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}[Z(t,t+h)|\mathcal{F}_s]-Z(s,t)}{h} \\ &= Z(s,t) \mathbb{E}\left[\lim_{h\to 0} \frac{Z(t,t+h)-1}{h}\middle\vert\mathcal{F}_s\right] \\ &= Z(s,t)Z'(t,t) \end{aligned}

Consequently,

\begin{aligned} Z(s,t) &= \mathbb{E}\left[ \exp\left(\int_s^t Z'(\tau,\tau)d\tau \right) \middle\vert \mathcal{F}_s \right] \\ &= \mathbb{E}\left[ \exp\left(-\int_s^t r(\tau)d\tau \right) \middle\vert \mathcal{F}_s \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 $t$.

That is, the interest rate of a zero coupon bond during the period $(s,t)$ is as follows:

$\frac{1}{Z(s,t)}\frac{1-Z(s,t)}{t-s}.$

Taking the limit $s\to t$ would then yield $r(t)$.

## Discount processes

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

$D(t) = \exp \int_0^t r(\tau)d\tau.$

With this notation, the discount curve may be written as

$Z(s,t)=\frac{\mathbb E [ D(t) | \mathcal F_s]}{D(s)}.$

## Annuities

An annuity $\Gamma$ is a schedule of future payments. Denote $\Gamma (t)$ as the total payments up to time $t$. $\Gamma$ may be modeled as either a deterministic function or a stochastic process.

Assuming zero risk, the (forward-looking) fair value of this annuity at time $t$ is defined as the following Riemann-Stieltjes integral:

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

Discrete Case
Consider an annuity consisting of discrete payments $\gamma_1,\ldots,\gamma_N$ at times $t_1,\ldots,t_N$. Then
$A(t) = \sum_{n=1}^{N} \gamma_n Z(t,t_n)I(t>t_n),$
where $I$ is the indicator variable.
Smooth Case
Suppose that $\Gamma$ is smooth. And define $\gamma(t) = \Gamma' (t)$ as the annuity rate. Then
$A(t) = \int_t^\infty Z(t,s)\gamma (s) ds.$

## 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 the the survival curve:

$Q(t,T) = \text{Pr}(\tau > T|\mathcal F_t).$

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:

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

## 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 $Z$.

## Insurance 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

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

The Riemann-Stieltjes integral on the left side of the above equation is with respect to the parameter $s$.