(λ Credit Derivatives)

These are some WIP notes on credit derivatives.

Fundamentals

Notation Table

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

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

Discount Curves

Consider a hypothetical bond maturing at time T T (the maturity or tenor) having the following properties:

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

Some assumed propeties:

Interest Rates

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

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

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

r ( t ) = lim s t 1 Z ( t , s ) 1 Z ( t , s ) s t = Z ( t , s ) s s = 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 ) = E [ exp ( t T r ( s ) d s ) | F t ] . 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 ) r(t) is defined as

D ( t ) = 0 t r ( s ) d s . D(t) = \int_0^t r(s)ds.

With this notation, the discount curve may be written as

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

Annuities

An annuity Γ \Gamma is a schedule of future payments. Denote Γ ( t ) \Gamma (t) as the total payments up to time t t . Γ \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 t is defined as the following Riemann-Stieltjes integral:

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

Discrete Case
Consider an annuity consisting of discrete payments γ 1 , , γ N \gamma_1,\ldots,\gamma_N at times t 1 , , t N t_1,\ldots,t_N . Then
A ( t ) = n = 1 N γ n Z ( t , t n ) I ( t > t n ) , A(t) = \sum_{n=1}^{N} \gamma_n Z(t,t_n)I(t>t_n),
where I I is the indicator variable.
Smooth Case
Suppose that Γ \Gamma is smooth. And define γ ( t ) = Γ ( t ) \gamma(t) = \Gamma' (t) as the annuity rate. Then
A ( t ) = t Z ( t , s ) γ ( s ) d s . 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 τ \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 ) = Pr ( τ > T F t ) . Q(t,T) = \text{Pr}(\tau > T|\mathcal F_t).

This is the probability ascertained at time t t that a credit event will not occur before or during time T T .

Risky Annuties

Consider an annuity Λ ( t ) \Lambda (t) that is risky. That is, scheduled payment occuring at or after a credit event τ \tau are unrealized. The forward-looking fair value of this annuity is given by the risky value process:

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

Hazard Rates

The hazard rate λ ( t ) \lambda (t) is the instanteous likelihood that a credit event will occur at time t t . Mathematically, it is a stochastic process λ \lambda defined by:

λ ( t ) = Q ( t , s ) s s = t . \lambda(t) = \left. -\frac{\partial Q(t,s)}{\partial s} \right\vert_{s=t}.

Intuitively, λ ( t ) d t \lambda(t) dt is the likelihood that a credit event will occur between times t t and t + d t t+dt .

Equivalently, the survival curve may be derived from an a priori defined hazard rate:

Q ( t , T ) = E [ exp ( t T λ ( s ) d s ) | F t ] . Q(t,T)= \mathbb E \left[ \exp \left( -\int_t^T \lambda(s)ds \right) \middle\vert \mathcal F_t \right].

The relationship between λ \lambda and Q Q is analogous to that of r r and Z Z .

Insurance Payments

Consider a payment of one unit of currency paid out at time τ \tau if τ < T \tau < T for some tenor T T . This is a form of insurance. And it's value at time t t is given by

s = t s = T Z ( t , s ) d Q ( t , s ) = t T Z ( t , s ) λ ( s ) d s . \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 s .