0:18

In this module, we are going to start with a very simple CDS pricing formula.

Â The value of the CDS to a buyer is going to be the risk neutral value of the

Â protection minus the risk neutral value of the premiums.

Â We're going to assume that the default event is uniformly distributed over the

Â premium interval delta. So let's walk through each of the pieces

Â that go into constructing the value for the CDS.

Â The risk-neutral value for single-premium payment on date t k, is going to be delta

Â times S. Delta is the period, times S, which is

Â the spread, times N, which is a notional amount, times the expected value under

Â the risk-neutral measure at time zero, of I t k divided by B t k.

Â Recall. I t k is the prop, is the indicative

Â function that the entity does not default, is not in default.

Â And B t k is a cash account at time t k. So this going to be here simply is a

Â discounted value of the event that there is no default up to time t k, discounted

Â back to time 0. Using the cash account B t k.

Â Again we're going to assume as we have done in the modules corresponding to

Â defaultable bonds that I t k and B t k are independently distributed, and

Â therefore I can take the expectation separately.

Â If I take the expectation separately I'll end up getting Q t k which is the time

Â zero probability. That the bond doesn't default up to time

Â t k. This quantity which is Z t k 0 is simply

Â the expectation or under the risk neutral measure of 1 over B t k, which is nothing

Â but the price of a zero-coupon bond which pays $1 at time t k.

Â This can further be simplified as the discount rate.

Â 2:20

Up to time t k. So what's the risk neutral value of all

Â the premium payments? Just sum this overall times k, that's k

Â going from one through n, delta times S time N, which is the coupon payment, this

Â is the probability that this coupon has to be paid, this is the discounted value

Â of that quantity. So the expected discounted value of the

Â coupons is what the coupon payments are going to be.

Â What about the accrued interest. We've got to assume, that if t k is here,

Â and this dot, some default time tau happens, it's uniformly distributed over

Â this. So the value of coupon that you're going

Â to be paying, would be approximately half.

Â So that's what this quantity here is. About, this is about, approximately half

Â the coupon. And this comes from the fact that we are

Â assuming that the default is uniformly distributed over the interval t k minus 1

Â to t k. Now, what is the probability that the

Â default occurs at time t k? It's I t k minus 1 minus I t k, divided

Â by B t k to discount things over, again. Just taking the expectation and using the

Â fact that the default is going to be independent of the interest rate

Â dynamics. We end up getting that this quantity is

Â going to be delta S N divided by 2 q t k minus 1 minus q t k.

Â This is the probabilities of the two indicators times Z t k 0 which is the

Â price of a zero-coupon bond. Which pays $1 at time t k.

Â We can dis, we can simplify this further and write this quantity as a discount

Â phase. If you add up the two quantities, this

Â tell you the risk-neutral value of the premium and the accut, accrued interests

Â can be approximated and the approximation here comes from the fact that we're

Â approximating this uniform distribution. Is also another subtle approximation

Â which I don't want to go in too much detail and that comes from the fact.

Â That if you look at the, hazard rates, those are going to be going monotonically

Â down, so even if you assume that the period, that it's going to be in the

Â interval is going to be uniform, the probabilities are going to be slightly

Â different. We're going to approximate all of that

Â and assume that it's sort of a flat probability, in that interval.

Â Okay, to sum that up, you get an expression, and we're going to be using

Â this expression in the next page to try to figure out what the par value is going

Â to be. What is the risk-neutral present value of

Â the protection or the contingent payment? It's 1 minus R times N.

Â This is going to be the amount of protection that you have to be paid.

Â 4:49

It's another subtle amount here. We're trying to price the CDS.

Â We're going, but in the pricing we're assuming that the R is known.

Â But really R gets known only on default. So in some sense we're making an implicit

Â assumption that these CDSs have been around, and so we have a good idea what

Â the expected recovery is going to be on particular CDS that we are going to be

Â pricing. What is I t k minus 1 minus I t k, this

Â is the event that a default happens at time t k, and, times B t k which is a

Â discount that I have to do in order to bring the quantities back to time zero.

Â Again I've gone through two steps. We can first write it as the price of

Â zero-coupon bond or you can directly go to the discount and I'm directly going to

Â the discount here. That is the quantity that is going to be

Â the contingent payment or the protection payment.

Â So S par which is the par spread is defined to be the spread that makes the

Â value of the contract exactly equal to 0, you compare this term with that term.

Â This term involves S and solve for S and you end up getting to the solve for, the

Â value, the notional amount goes away. It's 1 minus R.

Â This is protection. Down here it's premium plus accrued

Â interest. If you assume that the default

Â probabilities remain sort of flat over the entire premium interval so q t k is

Â going to be 1 minus some hazard rate h times q t k minus 1.

Â Then you can approximate that power spread to be 1 minus R times h divided by

Â 1 minus h over 2. And, if you recall, back in the module,

Â the first module in CDS we had said that this is approximately equal to 1 minus R

Â times h, and this is typically because h is assumed to be pretty close to zero.

Â When h starts becoming pretty close to 1, the approximation is not valid and one

Â has to use a better approximation. As you would intuitively see this is

Â increasing in the hazard rate h and decreasing in the recovery rate R.

Â In the rest of this module, I'm going to show you this pricing using a

Â spreadsheet. So in this spreadsheet we're going to

Â price the hypothetical two year CDS that we had introduces in the module.

Â The principal amount for the hypothetical CDS was $1 million.

Â The recovery was set as 45%. So 1 minus R which is the amount that we

Â are going to have to pay is 55% Arbitrarily I'm just setting the interest

Â rate here to be 1% per annum. So using that interest rate, I can

Â computer out the discount value, which is right here.

Â So it's just going to be, the quarterly discount is going to be R divided by 4

Â times the month count divided by 3. So that, this interest rate directly

Â gives me the discount rate. The hazard rate.

Â I took it from the calibration worksheet that we worked with for the bonds.

Â So, there, we computed the hazard rate for a six month default probability.

Â Here, we are talking about three month default probability.

Â So I just took that value, and divided it by 2.

Â 8:08

How did I compute the survival probabilities?

Â I took the same formula. It's going to be the survival probability

Â 1 times step before times 1 minus the hazard rate.

Â The only difference is before I was using survival probabilities in absolute

Â numbers, here I'm looking at them at percentages.

Â What about the fixed payments, which are the fixed coupon payments that.

Â The buyer has to make. This is simply going to be the quantity

Â of the spread divided by 4, and I've left off the notional amount, and I'm going to

Â look at the notional amount in a moment. So that's the fixed payment that's going

Â to happen. Now in this column, row F, what we have

Â done is we have taken the fixed payments and found an expected value, the expected

Â value which takes into account the probability of default.

Â So that's e7 times d7, divided by 100. And this 100 just comes because the

Â probability of survival is written as percentages.

Â So, the values in this column simply reflect the fact that fixed coupon

Â payment has to be paid only if the reference entity survives up to that

Â time. This is the present value.

Â We have taken the expected value, multiplied it by the notional amount, and

Â because, these spreads are in basis points, I multiplied it by 0.0001 to

Â covert into absolute numbers. And multiplied it by b7, which is the

Â discount value to get the present value. Of the coupon payments.

Â Done this for all time periods. That tells me what the total fixed coupon

Â payments are. But we still have to figure out what the

Â accrued interest is. And, in order to compute the accrued

Â interest, we need the default probabilities.

Â So here's the default probability. This default probability is just a

Â survival probability. Times the hazard rate, and I put that

Â along this column all the way through. What is the accrued interest?

Â If you click on that, it's we've assumed it to be half of what the coupon payment

Â is going to be, because we have assumed that this is going to be half way

Â between. Times the default probability again

Â divided by 100 to convert the default probability, which is in percentages,

Â into absolute numbers. What is the present value?

Â You take the notional amount, N, which is of $1,000,000 times the accrued interest,

Â times the discount times again 0.0001 to convert the basis points into.

Â absolute numbers. So this is going to be the present value

Â of the accrued interest, in different periods.

Â Finally, what about the protection. What is the expected value of the

Â protection, it's the default probability times 1 minus R divided by 100.

Â The 100 again to take the default probabilities and convert them into

Â absolute numbers. What is the present value of the

Â protection? You multiply by the notional amount,

Â multiply by the discount rate, that tells me what is the present value of the

Â protection for the different time periods.

Â So the premium leg now is going to be the sum of the present value of all the

Â premiums and sum of all the present value of the accrued interest.

Â And that tells you what the buyer has to pay.

Â What the buyer will receive is just the sum of all the expected present values of

Â the protection payments. This is the net value of the swap at time

Â zero. The difference between the premium leg

Â and the protection leg. So, since the value of the CDS is

Â positive. It, it suggests that this spread is set

Â too low. it's a good deal for the buyer.

Â The buyer is getting protection at too lower rate.

Â If you increase that spread, to say from 218 to 220 basis points, then you end up

Â getting that that spread is too high, you end up getting that the difference

Â between the premium leg and the protection leg is negative.

Â Which means that it's not been, it's not a good deal for the buyer.

Â This spread has been set too high. You can compute out what the correct

Â spread is going to be. So we're going to set the objective which

Â is the value and try to solve for the value equal to zero by changing the

Â variable cells S, it's just a reference cell.

Â And I'm assuming that it's going to be positives.

Â If you solve that and you wait for Solver you end up getting that the spread is

Â going to 218.89 which is slightly smaller then then spread that we had started

Â with. And this brings us to the end of this

Â module.

Â