The value of the CDS to a buyer is going to be the risk-neutral value of the protection they receive when the bond defaults - the risk-neutral values of the premiums they pay. We are going to assume that the default event is evenly distributed on the premium interval delta.
Now we will break apart these two factors that make up the value of the CDS.
The risk-neutral value of a single premium payment is:
Remember that I(t(k)) shows that there is no default and B(t(k)) is the discounting factor (or the cash account or the value of a zcb). Once again, as in the last lesson, we can break this down into simpler terms.
The risk-neutral value of of all the premium payments is just the sum of the above statement:
The risk-neutral value of the accrued interest if there is a default event between tau = (t(k-1), t(k)] is:
Since the default event is uniformly distributed on delta, the expected value of the uniform distribution of delta is simply (delta/2) which is how we get the above statement.
Therefore combining these values, the risk-neutral value of the premium and the accrued interest is:
We next moved on to the value of the protection.
The risk-neutral value of the protection is:
In this pricing, we assume that R is known, but in reality R is known only on default. So we are assuming that these CDSs have been around for a while so we know what R will be.
The par spread, or S(par), is that makes the value of the spread that makes the contract equal to zero. S(par) is:
This is simply found by taking the value of the protection minus the value of the premium and accrued interest, setting this equal to zero, and solving for S.
We then suppose that q(t(k)) = (1-h)*q(t(k) - 1). We can then approximate the par spread to be:
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