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 Mortgage-Backed Securities
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來自 Columbia University 的課程
金融工程与风险管理，第 1 部分
1485 個評分
從本節課中
Introduction to Mortgage Mathematics and Mortgage-Backed Securities
Basic mortgage mathematics; mechanics of mortgage-backed securities (MBS) including pass-throughs, principal-only and interest-only securities, and CMOs; pricing of MBS; MBS and the financial crisis.
與講師見面
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
In this module, the final module on mortgage backed securities, we're going
to discuss prepayment modeling and the pricing and deed of mortgage backed
securities. We'll see an example of a prepayment
model and we'll see some of the factors that go into a prepayment model, but we
will also mention that there are very few prepayment models that are available in
the public domain. When it comes to pricing mortgage-backed
securities, it is a very difficult exercise.
We need to use Monte Carlo simulation, typically, to price these securities.
They're very complicated, and they're very computationally intensive to price.
So, we will just be discussing the big picture here.
A little bit about prepayment modeling and a little bit about the pricing of
mortgage-backed securities. In many respects, the prepayment model is
the most important feature of any residential mortgage backed security
pricing engine. Term structure models, which are also
required to price mortgage backed securities are well understood in the
financial engineering community. But this is not true in prepayment
models. For example we can think of our binomial
model that we saw that we used on price fixed income derivatives.
That's an example of a term structure model and there are many term structure
models that are out there in the quantitative finals community.
They're well understood and people understand how to work with them.
A mortgage backed security is exposed both to interest rate risk, which is why
you need a term structure model but also it is exposed to prepayment risk.
And so you would also need a model of prepayments, a model of how home owners
in the underlying mortgage boosts of how they are going to prepay.
So a prepayment model is also very important when it comes to processing
mortgage backed securities. The problem is that there's relatively
little publicly available information concerning prepayment rates.
So it is very difficult to construct good prepayment models, accurate prepayment
models, and to calibrate these prepayment models as well, in other words, to figure
out what the correct parameter values are inside these prepayment models.
One very well known publicly available model is due to Richard and Roll from
1989. And they model the conditional prepayment
rate, the CPR, whose definition we recall.
The CPR is the rate at which a given mortgage pool prepays.
It is expressed as a percentage of the current outstanding principle level in
the underlying mortgage pool. It is also expressed, by the way, as an
annual rate. And we have seen that before as well.
So let's take a brief look at this model due to Rich and Roll.
I want to emphasize that, the vast majority of the prepayment models that
are out there are private, they are not published, nobody knows what they are.
They're just used by banks and other investment companies to invest in
mortgage backed securities. And they don't share their models with
other people. So it is very difficult to find good
prepayment models that are out there in the public domain.
So this is the prepayment model of Rich and Roll.
It looks very complicated, but I don't want anybody watching this video to think
that the need to understand everything inside here.
The idea is just to give us an, a sense of how one might go about constructing a
prepayment model. And in particular a model for the CPR.
So this model of Richard and Roll assumes the following; So they assumes that the
CPR time k is equal to the RI which is the Refinancing Incentive times the H
which is the Seasoning Multiplier. Times the MM which is the Monthly
Multiplier times BM which is the Burnout Multiplier, now what are all these
multipliers. Well the refinancing incentive is
something we've discussed before, lets not worry about the form of this here
really that's not relevant, the only point I want to emphasis here is this.
WAC if you, you recall is the weighted average Coupon in the underlying mortgage
pool. And RK10 here, we're assuming, is the ten
year spot interest rate. Now, why is this differnece relative?
Well, remember we said that homeowners have an option to prepay their mortgage.
So, for example, suppose the weighted average coupon was, let's take an example
suppose WACwas equal to 10%, and suppose RK, in other words the ten year interest
rate at time k, was equal to 3%. Well then this difference here is 7%.
And it represents how much money in, homeowners could make by prepaying their
mortgage. And refinancing at a lower interest rate.
Now they wouldn't be able to refinance at 3%.
This rate would be too low. But maybe they'd be able to refinance at
r plus 2%. Which would be 5%.
So therefore, they could prepay their mortgage where, on which they're
currently paying 10%, and refinance at 5%, that would be a large gain.
So this number here, WAC minus arc A10, represents the incentive to refinance,
and that's why we call it R I. And so the larger this number is, the
more prepayments you're going to get. So in fact this number here, although it
may be difficult to see it when you've got this inverse tan function.
But this RIk here will increase as WAC minus RK10 increases.
And therefore it will increase the CPR. Age is what's called the seasoning
multiplier, so again T is measured in months and it's a minimum of 1NT over 30.
This is just meant to capture the fact that early in the lifetime of a mortgage
a homeowner is very unlikely to prepay. After all if they've just taken out a
mortgage maybe it's because they just bought a new home.
It's very unlikely they're going to be moving home in the next year or two, and
therefore we said age equal to the minimum of 1 over T30.
So when T is just one month or six months this number here is very small and so it
multiplies in here to give it a very small CPR, in the earlier life of a
mortgage pool. And M is the monthly multiplier, so it's
approximately equal to one. There's a different for each month,
January through December, they're all close to one, but some of them are less
than one, and some of them are greater than one, and this is just used to
reflect the fact that. People tend to prepay at different times
of the year. For example, there tends to be more
activity in the housing market in spring and summer, for example, than in winter.
And so, in winter, you have a number, in the winter months, you have a number less
than one. And maybe in the spring, summer months,
you might have a number greater than one. So this is the monthly multiplier, and
then bmk is what's called the burnout multiplier.
So for example, in this case, we could say bmk equals .3 plus .7 minus m k minus
one over m zero. So for example, this is equal to one when
K equal one and that's equal to .3. When k is well approximately n close to
the end of the mortgage. And what this is reflecting is the
following. So early on in the life of the mortgage.
This burnout multiplier is equal to one. But it decreases linearly through the
life of the mortgage and falls to .3 at the very end.
And this is just used to reflect the fact that people who can prepay tend to prepay
early. For various reasons, there are some
people who have houses who can't prepay or who don't want to prepay and those
people, if they haven't prepaid early in the life of the mortgage, then it turns
out that they become increasingly less likely to ever prepay.
And so that's the burnout multiplier. Now as I said at the beginning of this
slide, don't worry about the details. I just want to give you an example of how
some people might go about modeling this conditional prepayment rate.
By the way, another comment on the spreadsheet in the work sheets in the
excel work book that we've looked at so far, we always assumed that the CPR was
deterministic. We used the so called PSA benchmark, I
mentioned that in the real world. Of course the conditional prepayment rate
will be random. It will depend on what's going on in the
economy. And certainly that randomness is
captured, for example, over here. The ten year spot rate, at time k, is
random. So certainly the refinancing incentive
will also be random. So, again just emphasizing the fact that
in the real world the cash rolls are associated mortgage back securities are
indeed random. How about choosing a term-structure
model. Well, we also need to specify
term-structure model IE, a model like the binomial model, the black term
and[UNKNOWN] model or[UNKNOWN] model for example that we use to price fixing and
derivatives. In order to fully specify the mortgage
pricing model the term structure model is used to; one.
Discount all of the Mortgage Back Security cash-flows in the usual
risk-natural pricing framework but also to compute the refinancing incentive
according to 15. Whatever term structure model is used it
is important that we are able to compute the relevant interest rates analytically.
For example, r10 in the prepayment model of Richard and Roll.
Such a model would need to be calibrated to the time structure of interest rates,
in the market place, as well as the liquid interest rate derivative
securities like caps, floors, swaptions, and so on.
The actual pricing of mortgage backed securities then requires Monte Carlo
simulation. It is very very computationally
intensive. Analytic prices are not available.
So the point I want you to take away from the slide, and this module more generally
is that the pricing by mortgage by security is, is very complicated.
It takes and awful lot of work and an awful lot of modeling and it is very
computationally intensive. You don't have to worry about the details
I've mentioned here. As I've said before I just want you to
understand the big picture of what is going on.
Finally let me say a little bit about the financial crisis.
The so called sub prime mortgage market played an important role in the financial
crisis of 2008 and 2009. Sub prime mortgages are mortgages that
are issued to homeowners with very weak credit.
The true credit quality of the homeowners was often hidden.
And the mortgages that were given to these sub prime.
Homeowners were often ARMs. Remember an ARM is an adjustable rate
mortgage where the mortgage rate is reset periodically.
While the adjustable rate mortgages that were given to sub prime homeowners often
had what were called teaser rates. In other words the initial mortgage rate
was very, very low. And that was used to attract new
homeowners to encourage them to take out mortgages that maybe they should not have
been taking out in the first place. Because the very initial low rates made
them easy to service, easy to make the payments in the earlier life of the
mortgage. But after two or three years, these
teaser rates were replaced with much higher coupon rates, and so become much
harder for the subprime homeowners to pay these higher coupon rates.
The financial engineering aspect of the MBS-ABS market certainly played a role in
the crisis. Particularly when you combine this with
the alphabet soup of CDO's and ABS-CDO's. And so on.
We actually will discuss CDO's later in this course.
So the financial engineering aspect certainly played a role.
The securitization that took place became very, very complicated and it became very
difficult to understand some of these very complicated securities.
And also because these securities became so complicated, the models that were used
to price these securities also became very difficult to understand.
And people started to lose trust in these models as well.
So certainly financial engineering played a role in the financial crisis.
But it has to be said there are many other causes as well including the moral
hazard problem with mortgage brokers. What is the moral hazard problem.
Well the moral hazard problem is the problem of getting people to behave the
right way. If they're given incentives to behave in
the wrong way, they may well end up behaving poorly.
So, for example, some of the mortgage brokers that sold these mortgages to
sub-prime home, homeowners, they weren't careful about checking the paperwork and
the credit worthiness of these homeowners.
And the reason they weren't is they weren't going to be holding these
mortgages themselves. They were going to be selling these
mortgages on to other groups. Other agencies other banks who would hold
the mortgages via mortgages bank securities.
So they didn't have to worry about. The sub prime homeowners for example
defaulting on their mortgages that would not be their problem and so that is the
moral housing problem that I am referring to.
Rating agencies also face moral housing problems.
the problem at rating agencies is that they were rating mortgage backed
securities and asset backed securities more generally.
For the banks that were creating them. So they were being paid by banks to do
the ratings. Well obviously there is a moral hazard
problem there as well. There was inadequate regulation,
inadequate risk management, and poor corporate governance in general.
So there were many, many parties to blame for the financial crisis.
Certainly some of the financial engineering behind the very exotic and
complicated mortgage banks securities and asset bank securities share some of the
blame. But certainly I would say these other
sources here were very much responsible as well.
