金融工程是一个多学科交叉的领域涵盖从财经，数学，统计，工程和数值计算方法。FE&RM Part 1 这门课的重点会放在是用简单随机模型来给包括股票，固定收益工具，信用以及抵押支持证券在内的各种资产类别衍生品证券定价。我们也将思考其中证券在金融危机中的角色。这门课程一个引人注目的特别栏目是对Emanuel Derman （著名“宽客”，畅销书“宽客人生”的作者）的一段采访。我们希望完成这门课的学生将能够开始理解金融工程背后的高深原理。但更重要的是，我们希望你们也能理解这一理论在实践上的局限性以及明白为什么总是应该用批判的眼光看待金融模型。接下来的FE&RM Part 2将会进一步深入学习衍生品定价模型但是它也会关注资产配置和资产组合优化以及其他金融工程的应用实例，像实物期权，商品衍生品，能源衍生品和程序化交易。
Pricing Defaultable Bonds
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来自 Columbia University 的课程
金融工程与风险管理，第 1 部分
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从本节课中
Term Structure Models II and Introduction to Credit Derivatives
Calibration of term-structure models; the Black-Derman-Toy and Ho-Lee models. Limitations of term-structure models and derivatives pricing models in general. Introduction to credit-default swaps (CDS) and the pricing of CDS and defaultable bonds.
与讲师见面
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
This is the second module on pricing defaultable bonds.
In this module, we're going to extend, the idea that we had developed in the
previous module, to pricing coupon bearing bonds, and also show you how
these are done in practice using several bonds and calibration on an Excel
spreadsheet. In this module we're going to assume that
the hazard rates h i j are state independent.
This ensures that the default probability is going to be independent of the
interest rate dynamics. It'll be easier for us to keep track of
the events by defining a quantity called q t which at the risk mutual probability
that the bond survives until date t. A simple recursion defines what q t is
going to be. So look at q t plus one, which is a
probability that the bond survives up to date t plus 1.
This is going to be the probability that the bond survives up to date t, and the
conditional probability that the bond survives one extra period which is 1
minus h t. And therefore you can write q t plus 1 as
just a product of k going from 0 to t, 1 minus h k.
Let I t denote the indicator that the bone survives up to time t.
So I t is going to be 1, if the bond is not in default at time t, and its going
to be 0 otherwise. Then the indicative variable that the
default occurs exactly at time t is going to be the difference between I t minus 1
and I t. I t minus 1 equals 1 and I t equals 0,
tells me that exactly that the bond defaulted at time t.
From the definition of I t, it immediately follows that the expected
value, under the risk-neutral measure of I t is exactly equal to q t.
Once we have this indicator val, variable I t, we can define various events using
this indicator value, variable. And it's going to be easier for us to
keep track of various events that happen to the bond.
We're going to assume that the random recovery rate, R tilde, is going to be
independent of the interest rate dynamics under the risk neutral measure Q.
And we're going to let R, without the tilde, denote the expected value of R
tilde under the risk measure, risk neutral measure Q at time 0.
Recall that R tilde is a fraction of the face value F 8 on default.
Here are the details of the pricing. We are going to assume that the current
date is equal to 0. t 1 to t n are the future date at which
the coupons are going to be paid. The coupon on date t k is paid only if i
t k, the indicator vari, variable that indicates whether the bond is in default
or not, is equal to 1, meaning that the bond is not in default.
Therefore the random cash flow associated with the coupon payment on date t k is c
times I t k. The randomness comes from the fact that
although the coupon payment is deterministic, the fact whether the bond
is in, not in default or is in default, is going to be a random quantity.
The face value F, is going to be paid on date t n only if I t n is equal to 1.
Therefore, the random cash flow associated with the face value payment on
date t n is going to be a deterministic quantity F times the random quantity I t
n. The recovery itself is random, so
recovery at date t k is going to be R tilde t k.
The random recovery rate times the face value F.
But this random quantity is going to be paid only if the default occurs on date t
k, which have been indicator vari, variable I t k minus 1 minus I t k.
Therefore, the random cash-flow associated with the recovery on date t k
is going to be R tilde t k times F times I t k minus 1 minus I t k.
Now that we have all the random cash-flows associated with a bond, a
defaultable bond with some random recovery, we can now price this bond by
just discounting all of these random cash-flows with respect to the risk
neutral measure, and that's what we're going to do on the next slide.
Let B t denote the value of the cash account at time t, then the price at time
0 of a defaultable fixed coupon bond is given by, the expected value under the
risk neutral measure of the random cash-flows discounted by the cash
account. So C times I t k is the random cash flow
associated with the bond, with the bond coupon payment, it has to, and it occurs
at time t k, therefore it has to be discounted by the cash amount at time t
k. F times I t n is going to be the random
cash flow associated with the face value. It's going to have to be discounted, so
there's no b t, but it's just 1. And, at the rate b t n, R tilde t k times
F is going to be the random cash-flow associated with the recovery at time t k,
so it's discounted by B t k, but this extra variables has the default, of
course at time t k. Now, we have, we have assumed that the
default is independent, of the interest rate dynamics.
So the first expectation, I can split it up into two expectations.
This expectation is the expectation of the default, the second expectation is
just the expectation with respect to the risk neutral dynamics of the interest
rate, or the short rate. And this split, happens, only because
I've assumed that the default and the interest rate dynamics are independent.
Again, I'm going to split up the next term into two terms.
One that corresponds to the default and one the other one that corresponds to the
interest rate dynamics. And finally with the same, same thing
with the last term, this corresponds to default.
And this other one corresponds to the interest rate dynamics.
[SOUND]. Now we know from the definition of the
indicator function that I expectation under Q sub 0 of I t k is nothing but Q t
k. In the expectation under Q sub 0 of I t n
is nothing but Q t n. Similarly, this quantity expectation
under Q sub 0 of I t k minus 1 is Q t k minus 1, and E sub 0 super Q of I t k is
Q t k. So the, all of these are just coming from
the definition of the indicator function of the expectations of that indicator
function under the risk neutral measure. What happens to these quantities?
These are nothing but the prices of the zero coupon bond.
So that particular quantity is nothing but the price at time 0 of a 0 coupon
bond that pays $1 at time t k. This quantity over here, is the price of
the 0 coupon bond at ti, that pays $1 at time t n.
And similarly again, this is the same repeat, it's a 0 coupon bond paying $1 at
time t k. We can further simplify this exp,
expression. This quantity is nothing but the discount
rate up to time t k, discount rate up to time t n, and discount rate up to time t
k. Here's a formula that tells me what the
price is. The only thing that is going into this
formula are the discount rates, which are determined by the short rates, and the
probabilities of default that are going to be determined by hazard rates.
So in principle, if I had the prices of lots of defaultable bonds with fixed
coupon payments, I can infer from there what the hazard rates are going to be.
So my story is going to be, I'm going to assume that the interest rate is dynam,
is deterministic and known. We could have calibrated this and you
have done this in another module. To keep the story simple and focus on the
hazard rates, we are going to assume that the interest rate dynamics are going to
be deterministic. If that is deterministic, then I can
write, the model price of a defaultable bond as a function of the hazard rates.
I'm going to assume that I have some observed prices of the market bonds and I
can use those prices, and compare it with the model price, get an error.
So here's the market price for the Ith bond.
Here's the model price for the Ith bond. I compare the two, take the square.
That is going to be the error that I'm going to be making on the Ith bond.
And I take the sum over all possible bonds, that's the total error, and the
calibration problem that I'm going to be facing, is to minimize over H F H.
And I'm going to show you this, using a numerical example in the associated
spreadsheet. So here's a simple spreadsheet that I'm
going to be working with. In this sheet, I'm just, I'm going to
assume, that the discount rate is given to me.
So the discount rate is assumed to be 5% per annul, deterministic.
And from that interest rate, I can compute out what the discount rates are
going to be. For the six month discount rate, it's
just going to be half of this. So if you look at the formula, it's just
going to be 1 plus R F divided by 2 times the number of half year periods that have
elapsed. So this is simple, this is something that
we have done before. Now, in this particular worksheet, I'm
going to assume that the hazard rate, which is going to be the six month hazard
rate, is fixed at 0.02. What does that mean?
So right now, it's time 0. So the survival probability is 1 because
the bond exists. Now from this survival probability, I
want to compute the survival probability and the default probability in six months
from now. The hazard rate of time, 0 is 0.02, so
the probability that I default, in the next six months, is simply going to be
the probability, survival probability times the default probability.
The conditional probability, of default is the hazard rate.
The probability that I have survived right now is 1.
I take the product of that, that gives me the default probability.
What is the probability that I survive, it's the probability that I'm surviving
right now times 1 minus H, where H is the hazard rate.
That gives the survival probability in six months.
What is the survival probability in 12 months?
It's, again the same thing, which is 0.98, which is the survival probability
at six months times 1 minus the hazard rate.
What is the default probabilitiy? It's the surval probability times the
hazard rate. So all of this table has been computed
using, the survival probabilities and default probabilities form the hazard
rate. Now let's see what happens to a coupon.
So here is a bond, which is a one year bond so it expires in one year.
It me, it has two coupon payments and the face value payment.
I'm going to assume here that the face value is 100, and the coupon is 5%,
therefore the coupon is 5. The recovery rate is 10%.
So what happens? So the coupon and face value payments are
going to be 5 in six months and 105 in one year.
The recovery is going to be 10% of the face value, so it's going to be $10 if a
default occurs in six months. It's going to be another $10 if a default
occurs in 12 months or a year. How do I compute what is the expected
value of the payments? So if you look at this formula, what's
going to happen? If the bond survives in six months,
you're going to get the coupon payment. So it's going to be 5 times the survival
probability, which is 0.98, plus 10, which is the recovery times the default
probability 0.02, and this happens in six months, and therefore you have to
discount it back using the discount rate, which is 0.98.
Similarly, if you look at this one, it's the same formula again.
It's the coupon plus 105, which is going to be paid only if you are going to be
surviving at time 12 months or in 1 year. So it's H8 times C8.
And, in the case that you default, it's going to be I8 times D8, which is the
default probability times the recovery. It has to be discounted back.
So now you're going to use the discount rate of 0.5.
Sum all of that, you end up getting what the price of this particular bond is
going to be. Similarly, if you look at, here is
another example of a bond, it's a two year bond with 8% recovery, same story.
[SOUND]. This should be 2%, so that coupon payment
is 2%. So therefore, here are the coupon
payments. Here are the recovery rates.
If you look at the formula, it's exactly the same.
Coupon payment times the probability of survival, recovery times the probability
of default, discounted back to time 0. Sum it all up, and you end up getting
what the price of this bond is going to be.
Great. So we know how to price bonds, given the
calibrate, given the hazard rates. Now I'm going to show you the next
spreadsheet, what happens when you calibrate.
So here, what I've done, is I have created for you, same bonds as was there
on the last sheet. I took the true price of the bond, and to
it, I added a small random quantity. I took the true price that was there and
then I added about $0.10 of randomness. You can play with this and see what
happens when you add more randomness. Here, I'm assuming, just to keep things
simple, that the hazard rate, the six month hazard rate of default is going to
be constant for a year. So what I've done is that, in the first
year, the six month hazard rate is an unknown quantity, but in the next six
months it's exactly equal. So if you click on this, I've just made
it equal to A6. Similarly, over here I've made it equal
to A6, A6 and so on. Once I know the hazard rate, I can
compute the survival probability to the default probability, and I can compute
the model price. This is the model price that has been
computed using whatever these hazard rates are going to be.
Now, what I'm going to do, is I'm going to compare the model price with the true
price, compute the error. This is nothing but the model price minus
the true price squared. I have five different bonds.
I'm going to sum up all of those errors. And this is going to be the sum of the 5
errors. Add them up, and then I'm going to
minimize it. So, if I use solver, what I'm trying to
do is, before I go to solver, let me randomly create some instances here.
[SOUND]. So there's this a random instance of what
happens to the boat. Now I'm going to go to Solver.
And if you look at Solver, all I'm trying to do is, minimize the error of J21.
Which is this error quantity here. By changing the variable cells, A6, A8,
A10, A12, and A4, and the reason, I left off A16, is because this is the hazard
rate that's going to be in the future and it's not going to matter.
The A7, A9, A11, A13, and A15 have been left off because, I'm just assume that
this is going to be equal over the entire year.
And the reason I made that assumption is because I only have bonds that I, that
are expiring in the years, and not at six months.
And therefore I will not be able to calibrate the next six months of asset
rate. So we minimize that.
We hit solve. And, it found a solution, and the minimum
error it found was 0.01. And, the prices that you end up getting
is pretty to what you started of with. It's slightly different, it's 0.0201
instead of 0.002. Here it's just 0.19 and so on.
And that happens just because the error is small.
Some of the bonds, it's able to compute it correctly, some of the bonds it's it
has small errors. And so the overall error turns out to be
0.01.
