Credit Default Swaps: Valuation

CDS valuation is done by calculating the ‘survival probability curve’.
The ‘survival probability curve’ is the very fundamental tool which gives market implied probabilities of reference entity that it doesn’t suffer a credit event prior to a give time horizon.

Lets do the maths

After every spread being paid by the protection buyer two things might happen:
1. Survival – which gives you a probability value of 1-q i.e. the reference entity survives for one more period.
2. Default – which gives you a probability value of q i.e.credit event occurred with deliverable obligation trading at the recovery value of R.

Probabilities build up stage

Situation            Payment               Probability          Probability of survival    Probability of Event
t=1
Survival            -s  (spread)                        1-q(1)                  p(1)*{1-q(1)}                  1-p(1)
Credit event     (1-R)-s                   q(1)

t=2
Survival            -s  (spread)                        1-q(2)                   p(2)*{1-q(2)}                 p(1)-p(2)
Credit event     (1-R)-s                   q(2)

Hence, when you calculate the NPV of a CDS then formula goes like this:

NPV= Summation {(1-R)*( probability of credit event )* risk free discount} – Summation { cash flow during survival) * probability of survival * risk free discount}
or,
NPV= Summation [ (1-R)* { p( i-1 )-p( i ) } * d ( i ) – Summation [ s *p ( i -1 )* d( i ) ]

Here, (1-R) is assumed loss in default and d ( i ) is LIBOR

consider at t=1, NVP = 0 then solving the equation will give you  q (1) =   s
                                                                                                                        (1-R)

This equation tells the conditional default probability on the spread.

Lets assume 1 year  CDS at 50 bps spread having recovery (R) at 50%, what would be the probability of survival or default ?

q(1)= 0.005/(1-0.5)= 0.01

thus p(1)= 1-q(1) = 1-0.01
= 99% is the probability of survival.

So, we got to know how to calculate NPV on CDS, probability of survival on which we can make series of probabilities thus the ‘survival probability curve’ as well.

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