# (λ Credit Derivatives)

These are some WIP notes on credit derivatives.

# 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 $t$ as $Z(t,T)$. The bitemporal function $Z$ is called the (risk-free) discount curve. It represents the market demand for money.

Some assumed propeties:

• $Z(t,t)=1$
• For fixed $T$, $t\mapsto Z(t,T)$ is a stochastic process.
• For fixed $t$, $T\mapsto Z(t,T)$ is smooth.

## Interest Rates

The (risk-free) interest rate from time $t$ to time $s$ is given by

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

For $s\to t$, this approaches the short rate $r(t)$:

\begin{aligned} r(t) &= \lim_{s\to t} \frac{1}{Z(t,s)}\frac{1-Z(t,s)}{s-t} \\ &= -\left. \frac{\partial Z(t,s)}{\partial s} \right\vert_{s=t}. \end{aligned}

The short rate is a stochastic process.

Alternatively, it is possible to calculate the discount curve from the short rate process:

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

## Discount processes

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

$D(t) = \int_0^t r(s)ds.$

With this notation, the discount curve may be written as

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

## 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 $s$.