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:
 UnitNotional
 At time $T$ the bond holder will be paid one unit of currency. This payment is the notional.
 ZeroCoupon
 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 (riskfree) 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 (riskfree) interest rate from time $t$ to time $s$ is given by
For $s\to t$, this approaches the short rate $r(t)$:
The short rate is a stochastic process.
Alternatively, it is possible to calculate the discount curve from the short rate process:
Discount processes
The discount process corresponding to the short rate process $r(t)$ is defined as
With this notation, the discount curve may be written as
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 (forwardlooking) fair value of this annuity at time $t$ is defined as the following RiemannStieltjes integral:
 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 contractuallyobligated 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:
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 forwardlooking fair value of this annuity is given by the risky value process:
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:
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:
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
The RiemannStieltjes integral on the left side of the above equation is with respect to the parameter $s$.