Deriving the CDS Protection Leg from Discounting and Survival Probabilities
Summary
The document works through a derivation for the protection leg of a credit default swap. The payment at default is the notional amount multiplied by one minus the recovery rate, discounted from the valuation date to the default time, and counted only if default occurs before maturity.
It rewrites the discount factor to default as the ratio of discount factors at default and valuation dates. The expected discounted payment is then expressed as an integral over default times, using the density of default implied by the decline in survival probability. This yields a negative integral of the discount factor times the change in survival probability over the contract horizon. The derivation clarifies the role of the default-time distribution; it does not address calibration, dependence between rates and default, or conventions beyond the stated setup.
Key ideas
- The protection payment scales with notional and the loss fraction, one minus recovery.
- Discounting to the default time can be written as a ratio of discount factors.
- The probability of default in a time interval is represented by the decline in survival probability.
- The expected protection payment becomes an integral over the contract horizon weighted by discount factors and changes in survival.
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Full text
# CDS protection/contingent leg pricing, taking expectation of interest and hazard rates
# CDS protection/contingent leg pricing, taking expectation of interest and hazard rates
The Pricing and Risk Management of Credit Default Swaps, with a Focus on the ISDA Model Screenshot: Pricing protection leg of a CDS, by OpenGamma
In the screenshot above, I am having trouble understanding the maths between equation 13 and equation 14.
Notation:
- $N$ = notational payment, e.g., £100
- $RR$ = recovery rate, the percentage of the $N$ recovered upon default, e.g., you get back 40%
- $\tau$ = time of default
- $t_v$ = valuation date
- $T$ = maturity date
- $\mathbb{I}_A$ = indicator function for event $A$
- $r(s)$ = instantaneous short rate at time $s$
- $P(t)$ = discount factor from time $t > 0 = $start date
- $Q(t)$ = survival probability at time $t$
What I have tried:
$$\mathbb{E}\left[e^{-\int_{t_v}^{T}r(s)ds}\mathbb{I}_{\tau<T}\right] = \int_{-\infty}^{\infty} \tau e^{-\int_{t_v}^{T}r(s)ds}\mathbb{I}_{\tau<T} d\tau = \int_{0}^{T} \tau \frac{P(\tau)}{P(t_v)}d\tau$$
From here, I cannot see how equation 14 is derived.
## Answer by Vivek Patel (score 1, accepted)
https://quant.stackexchange.com/a/35650
After some more trying, I think I have it. \begin{equation} \label{eq1} \begin{split} N(1-RR)\ \mathbb{E}\left[ e^{-\int_{t_v}^{\tau}r(s)ds} \mathbb{I}_{\tau<T} \right] & = N(1-RR)\ \mathbb{E}\left[ \frac{P(\tau)}{P(t_v)} \mathbb{I}_{\tau<T} \right] \\ & = \frac{N(1-RR)}{P(t_v)}\ \mathbb{E}\left[ P(\tau) \mathbb{I}_{\tau<T} \right] \\ & = \frac{N(1-RR)}{P(t_v)}\ \int_{-\infty}^{\infty} -\frac{dQ(s)}{ds}P(s) \mathbb{I}_{s<T} ds \\ & = -\frac{N(1-RR)}{P(t_v)}\ \int_{0}^{T} P(s) \frac{dQ(s)}{ds} ds \\ & = -\frac{N(1-RR)}{P(t_v)}\ \int_{0}^{T} P(s)\ dQ(s) \end{split} \end{equation}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.