Credit Default Swap - III

In last two post on CDS I have written about CDS basics and replication of CDS using the money and bond market instruments. Now we are prepared to understand the CDS pricing.
Premium of new CDS referencing a non distressed entity determine by the arbitrage relationship with cash spreads of that entity.In couple of last years CDS market has evolved as a very high liquid market and it has become mainstream market place to price credit risk. Even banks have started using the CDS spreads to decide the loan interest rates for their borrowers. This has helped banks to reduce monitoring of their borrowers, the-transformation-of-banking-tying-loan-interest-rates-to-borrowers-credit-default-swap-spreads.

CDS premiums should be determined by the following factors:

• The default probability of the reference entity;
• The expected recovery rate of the deliverable obligations;
• The maturity of the swap;
• The default probability of the protection seller, as well as the default correlation between the protection seller and the reference entity.
• discount curve

Like Interest rate swap, a CDS should have 0 present value at inception of the swap. Value of both the legs ,premium paying leg and contingent payment leg, of CDS should be equal. The premium is set at a level

that equates the PVs of the two leg. After inception, however, as expectations change, the PVs of the two legs will be different.If the on-market CDS quotes for a reference entity have tightened, protection sellers
will have a positive mark-to-market (“MTM”), and vice versa when premiums widen. In either case, the market price of this off-market (seasoned) CDS is different from its contract premium.

Quantitative model do not get used for on the run CDS pricing as premium depends on the arbitrage relationship with cash spread and demand supply factors. Quantitative models normally used to price off the run CDS, taking inputs from market quotes. 

Default Probability -

Both the legs have uncertain cash flows: premium are paid until the default of reference entity or maturity , payment of (1-recovery rate)*notional happens only in event of default. So to price the CDS we require the default probabilities. These default probabilities are calibrated from market quotes of premiums.

Credit Triangle - 

Probability of default , recovery rate and premium make a credit triangle. We can calculate third variable with given values of  two variables using below equation. CDS premium includes hazard rate and loss given default (1-recovery rate) so to calculate the hazard rate we need to figure out what is the value of recovery rate. Hazard rate in this equation is default probability of reference entity in that time period so it means that reference entity has survived all previous period and get defaulted in this period only.

                                  Premium =  Hazard rate*(1- recovery rate)

Normally recovery rate for the investment grade security (senior unsecured) assumed as 40%. This has now become market standard. Recovery swap market is thin, in future it is possible that recovery rate can be observed from quotes of this market.

In the credit triangle, For a given premium, higher recovery rate assumptions mean higher default

probabilities. On the other hand, given the same recovery rate assumption, higher premiums obviously mean higher default probabilities.

With recovery rate assumption hazard rate can be calculated from the cds market quote. Such hazard rate will be required for each period, like term structure of hazard rate. PV of premium leg will be calculated using hazard rate term structure and discount curve. 

A term structure of non-conditional default probabilities, which is effectively the difference in survival probabilities between two adjacent periods is required to calculate the PV of contingent default payment leg.

With Hazard rate ,term structure of default probabilities and discount curve we can price the off the run CDS using the same quantitative model.

Assumptions in the model

1) Recovery is a fixed percentage of par (40% for IG issues), independent of the model and constant over time

2) The interest rate (i.e., the discount rate) and default processes are independent of each other

3) Hazard rates are constant in one particular period.