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 $s$.