These are some WIP notes on credit derivatives.
|short rate process|
|annuity valuation process|
|hazard rate process|
|risky discount curve|
|risky annuity valuation process|
Assume that all stochastic processes, unless otherwise specified, are adapted to some common filtration .
Consider a hypothetical bond maturing at time (the maturity or tenor) having the following properties:
- At time the bond holder will be paid one unit of currency. This payment is the notional.
- No payments to the bond holder will be made before time (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 as . The bitemporal function is called the (risk-free) discount curve. It represents the market demand for money.
Some assumed propeties for all real values and :
- For fixed , is a stochastic process with respect to the filtration .
We can deduce from the above properties that . This is because, applying the second property: .
Let us additionally assume that is smooth.
Define as the derivative of with respect to its second argument. That is, . Then
Here, is the short rate process. It can be thought of as the "instanteous" interest rate at time .
That is, the interest rate of a zero coupon bond during the period is as follows:
Taking the limit would then yield .
The discount process corresponding to the short rate process is defined as
With this notation, the discount curve may be written as
An annuity is a schedule of future payments. Denote as the total payments up to time . 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 is defined as the following Riemann-Stieltjes integral:
- Discrete Case
Consider an annuity consisting of discrete payments
at times . Then
where is the indicator variable.
- Smooth Case
Suppose that is smooth. And define
as the annuity rate. Then
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 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 that a credit event will not occur before or during time .
Consider an annuity that is risky. That is, scheduled payment occuring at or after a credit event are unrealized. The forward-looking fair value of this annuity is given by the risky value process:
The hazard rate is the instanteous likelihood that a credit event will occur at time . Mathematically, it is a stochastic process defined by:
Intuitively, is the likelihood that a credit event will occur between times and .
Equivalently, the survival curve may be derived from an a priori defined hazard rate:
The relationship between and is analogous to that of and .
Consider a payment of one unit of currency paid out at time if for some tenor . This is a form of insurance. And it's value at time is given by
The Riemann-Stieltjes integral on the left side of the above equation is with respect to the parameter .