| Математика
Key words: CDS, credit curves, reduced-form models, credit triangle, hazard rates and unconditional default probability density
Credit spreads

There are two important curves when dealing with interest rate derivatives.
These are spot (discount) and forward curves.
In order to deal with credit derivatives, such as CDS, we need both interest rate and credit curves.
Credit curves help to quantify credit risk.
Credit curves show credit spreads demanded for buying or selling protection plotted against different maturities.

Credit curve
There are two approaches to modeling credit risk: structural models and reduced form models.
Structural models attempt to model the path of an entity’s assets and liabilities while reduced form models derive default probabilities from liquid fixed-income securities.

There are two degrees of freedom associated with reduced-form models: the probability of default and the recovery rate R.
In the corporate market, recovery rates are generally somewhere in the neighborhood of 40%.

Taking into account these two parameters as inputs, we can derive credit spreads from fixed-income securities. Let’s see the following example.
Corporate bonds are considered to be risky investments and thus P, the price of the bond, can be shown as:

Binomial model

This binomial model shows that:

Binomial model pricing

Let’s assume that one-year probability of default p = 0.8% and R = 40%. The one year risk-free rate is 2% yields:

Binomial model pricing formula

If we value this bond without taking into account the probability of default, we clearly see the difference in pricing:

Binomial pricing formula
Credit spread causes the difference between these two valuations and thus it is easy to derive the spread:

Binomial credit pricing formula
It is possible to show that one can accurately approximate the credit spread using the credit triangle formula:

Credit triangle
Using this equation we recalculate an approximate value of the spread again:

Credit triangle spread

The logic behind this formula is very simple.
If we have a credit curve and thus can calculate credit spread for a particular security, we can derive probability of default.
Having probability of default, we can calculate survival probability as (1-p) and these both parameters can be used as inputs to CDS pricing.
This standard credit pricing approach has been widely adapted to price and value a variety of less-liquid risky instruments.

In the example we were dealing with unconditional probability of default, i.e. the rate of default at a future time assuming only that the reference entity is alive now.

In practice, there are two ways of quantifying probability of default.
These are hazard rates and unconditional default probability density.
Both are related with each other and can be used to price CDS contracts.

Hazard rate is known as a conditional probability of default in a period.
This is the probability of a company to default over the period given that it has not defaulted up to the start of the period.