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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Our last class
Yi is running from his home in New Jersey due to snow. So he couldn't fly in. But actually, now
I'm learning a lot. It's a good way to run
the classes going forward. I think. We may employ it next year. So Yi will present CV
modeling for about an hour. And then Jake, Peter and myself,
we will do concluding remarks. We will be happy to answer
any questions on the projects or any questions whatsoever. All Right? So Yi, please. Thank you. YI TANG: OK. I'm here. Hi everyone. Sorry I couldn't make it in
person because of the snow. And I'm happy to
have this opportunity to discuss with you
guys counterparty credit risks as a part of our
enterprise-level derivatives modeling. I run a Cross Asset Modeling
Group at Morgan Stanley. And hopefully you
will see why it's called Cross Asset Modeling. OK, counterparty credit
risk exists mainly in OTC derivatives. We have an OTC derivative trade. Sometimes you owe your
counterparty money. Sometimes your counterparty
owes you money. If your counterparty owes you
money, on the payment date, your counterparty
may actually default, and therefore, either
will not pay you the full amount it owes you. The default event includes
bankruptcy, failure to pay and a few other events. So obviously, we
have a default risk. If our counterparty
defaults, we would lose part of our receivable. However, the question is before
the counterparty defaults, do have any other risks? Imagine you have a case where
your counterparty will pay you in 10 years. So he doesn't need
to pay you anything. Then the question
is are you concerned about counterparty risks or not? Well, the question is yes,
as many of you probably know, it's the mark-to-market
risk due to the likelihood of a counterparty
future default. It is like the
counterparty spike widens, even though you do not need a
payment from you counterparty. If you were to sell,
a derivative trade to someone, then someone may
actually worry about that. So therefore the
mark-to-market will become lower if the
counterparty is spread wider. This is similar to a corporate
bond in terms of economics. You own a bond on the
coupon payments date, or on the principal date,
the counterparty can default. Of course, they can
default in between also. But in terms of
terminology, this is not called counterparty risk. This is called issue risk. So here comes the important
concept credit valuation adjustment. As we know the
counterparty is a risk. Whenever there's a risk, we
could put a price on that risk. Credit valuation
adjustment, CVA, essentially is the price of
a counterparty credit risk. Mainly mark-to-market
risks, of course, include default risk too. It is an adjustment to the
price of mark-to-market from a counterparty-default-free
model, the broker quote. So people know,
there's a broker quote. The broker doesn't know
the counterparty risk. A lot of our trade models do
not know the counterparty risk either, mainly because
of we're holding it back, which I will talk
about in a minute. Therefore, there is
a need to actually have a separate
price of CVA to be added to the price
for mark-to-market from counterparty
default free model to get a true economic price. In contrast, in terms of
a bond, typically there's no need for CVA because it is
priced in the market already. And CVA not only has important
mark-to-market implications, it is also a part of
our Basel III capital. Not only change your valuation,
but could impact your return on capital. Because of a CVA risk,
the capital requirements typically is higher. So you may have a bigger
denominator in this return RE, return on capital
or return on equity. CVA risk, as you may know, has
been a very important risk, especially since
the crisis in 2008. During the crisis, a significant
financial loss actually is coming from CVA loss,
meaning mark-to-market loss due to counterparties'
future default. And this loss turned
out to be actually higher than the
actual default loss than the actual
counterparty default. Again, coming back
to our question, how do we think in terms
of pricing a derivatives and price the CVA together
with the derivatives. First of all, it adds
some portfolio effect the counterparty can
trade multiple trades. And the default
loss or default risk can be different depending
on the portfolio. And when people use a
trade-level derivatives model, which is by default what people
would call a derivatives model, typically you price each
trade, price one trade at time. And then you aggregate the
mark-to-market together to get a portfolio valuation. So when you price
one trade, you do not need to know there may
be another trade facing the same counterparty. But for CVA or counterparty
risk, this is not true. We'll go over some
examples soon. This is the one
application of what I call enterprise-level
derivatives, essentially focusing on
modeling the non-linear effects, non-linear risks in a
derivatives portfolio. Here's a couple of examples. Hopefully, it will help you
guys to gain some intuition on the counterparty
risks and CVA. Suppose you have an OTC
derivatives trade, for instance like an IR swap. It could be a
portfolio of trades. Let's make it simple. Let's assume the trade
PV was 0 on day one. Of course, we assume
we don't know anything about the counterparty
credit risk. We don't know
anything about CVA. This is just to show how
CVA is recognized by people. So to start with
again, the trade PV was 0 on day one, which is
true for a lot of co-op trades. And then the trade PV became
$100 million dollars later on. And then your counterparty
defaults with 50% recovery. And you'll get paid
$50 million of cash. OK, so $100 million
times 50% recovery. If the counterparty
doesn't default, you eventually would
get $100 million. Now he defaults, you get
half of it, $50 million. The question is have you
made $50 million dollars or have you lost $50 million
over the life of the trade. Anyone have any ideas? Can people raise your
hand if you think you have made $50 million? Can I see the
people in the class? I couldn't see anyone. PROFESSOR: How do I raise this? YI TANG: OK, no one thinks
you made the $50 million. So I guess then,
did you all think you have lost $50 million? Can people raise their
hand if you think you have lost $50 million? OK, I see people. Some people did not
raise your hand. That means you are
thinking you are flat? Or maybe you want to
save your opinion later? OK, so this is a common
question I normally ask in my presentation. And I typically get two answers. Some people think
they've made $50 million. Some people think
they've lost $50 million. And there was one case, someone
said OK, you know they're flat. Now, this would look like
a new interesting situation where no one thinks
you made $50 million. I mean, come on, you have $50
million of cash in the door. And they don't think you
have made $50 million. You have a $0 from day one. Now, you have $50 million. OK? All right, anyway
so for those of you who think you have lost
money-- I don't know if it's a good idea [? Ronny-- ?]
can someone tell us why do you think you lost $50 million? You went from 0 to
positive $50 million. Why do you think you
lost $50 million? Are we equipped to allow
people to answer questions? PROFESSOR: Yeah, I think
if someone presses a button in front of them. YI TANG: OK, so people choose
not to voice your opinion? AUDIENCE: It is because
you have to pay to swap and you have to pay
$100 million to someone on the other side of trade? YI TANG: OK, very good. So essentially, you
are saying hedging. That was what you
are trying to get to? So you have a swap
as 0 and you have an offsetting swap as a hedge. Is that what you
are trying to say? AUDIENCE: No. I'm saying that if you're
the intermediary for a swap, then you have to pay $100
million on the other end. So if you're receiving 50 and
paying 100, you have a loss. YI TANG: That's good. Right, so intermediary is right. And that's similar to
a hedge situation also. So that's correct. That's the basically
the reason for a dealer. Essentially, we are
required to hedge. We're very tight on the limit. We actually would
lose $50 million maybe on the hedge fund. When our trade went from 0
to a positive $100 million, our hedge would have gone from
to 0 to negative $100 million. In fact, we receive only half
of what we need to receive. And yet, we have to pay
the full amount that we need to pay on the hedge side. Essentially, we
lost $50 million. But that's where the CVA
and CV trading, CV risk management would come in. Again, CVA is the price of
a counterparty credit risk. And you know, if you hedge,
the underlying trader or whoever trades swap, if
you hedge with the CV desk. Theoretically, you
will be made whole on a counterparty default. So
you would receive $50 million from counterparty,
and theoretically you receive $50 million
from the CV's desk if you hedge with CV desk. Now, the second part is
how do we quantify CVA. How much is the CVA? CV on the receivable,
which we typically charge to the
counterparty, essentially is given by this formula. MPE means mean
positive exposure, meaning only our receivable
sides when the counterparty owes us money, and times
the counterparty CDS par spread, times duration. The wider the spread
the more likely the counterparty will
default, the more we need to charge on the CVA. And the same thing is
true for the duration. The longer the
duration of trade is, there's more time
for the counterparty to default so we charge more. Very importantly,
there's a negative sign. Because CVA on the receivable
side, is our liability. It's what we charge
our counterparty. And there are some
theoretical articles, they don't include
the sign, that's OK for theoretical purposes. But practically, if
you miss the sign things will get very confusing. Now, here is more
accurate formula for CVA. You know how the MPE
side, on the asset side. So we can see to start
with, there's an indicator function where this capital T is
the final maturity of the trade or counterparty portfolio. This tau is the counterparty's
default time, first default time. And if the tau is greater
than this capital T, essentially that means
a default happens after the counterparty
portfolio matures. And therefore, we don't
have counterparty risk. So that's what this
indicator is about. If the counterparty defaults
before the maturity, that's when we will have
counterparty credit risk. And there's a future evaluation
of the counterparty portfolio right before the
counterparty default. And this is how much collateral
we hold against this portfolio. So the net receivable,
the net amount, where the future value is
greater than the collateral, is our sort of
exposure, how much the counterparty would owe us. And this 1 minus R essentially
is the discount rate. So 1 minus R times the
exposure essentially is the future loss
given default. And beta essentially is a
normal mock money market account for defaulting,
and this is the expectation in the risk-neutral measure. It looks simple. But if you get to the details,
it's actually very complex maybe because the
portfolio effect and this option-like payoff. If you recognize this
positive sign here, essentially you recognize
this is like options. And so again, here is about some
details of non-linear portfolio effects. First of all, we talk
about offsetting trades. In the previous example,
you have one trade and went from 0 to $100 million. Counterparty defaults, you get
paid $50 million, essentially, you lost $50 million. But what if you have
another trade facing the same counterparty? Well, that's offsetting. When the first trade went
from 0 to $100 million, the offsetting trade can go
from 0 to negative $100 million. And therefore if the
counterparty were to default, you're going to have
a 0 default loss. That's just one example
of portfolio effects because I'm offsetting trades. So therefore, in
order to price CVA, you've got to know all
the trades you have facing the same counterparty. This is very different
from a trade-level model where you only need to
know one trade at a time. There's also
asymmetry of handling of the receivable, meaning
assets versus the payable, meaning liabilities. And that's where the
option-like payoff comes about. Typically, roughly
speaking, if we have a receivable
from our counterparty, if the counterparty
were to default we're going to receive
a fraction of it. So we would incur default loss. However, if we have a
payable to our counterparty, if the counterparty
were to default, we more or less need
to pay the full amount. We don't have a
default gain, per se. So this asymmetry is the reason
for this option-like payoff we just saw previously. And as you know,
a counterparty can trade many derivative
instruments across many assets, such as
interest rate, FX, credit, equities, a lot of
time also commodities and sometimes also mortgage. And then my group is
responsible for the modeling of the underlying
exposure for CVA for capital as well
as for liquidity, because multiple
assets are involved and we need to
model cross assets. So therefore, we named our
group Cross Asset Modeling. Furthermore, it is not only we
have option-like payoff, which is non-linear, we have
an option essentially on a basket of a cross
asset derivative trades. And the modeling becomes
even more difficult. So that's when the
enterprise-level will come in. And the enterprise-level
model, which we'll touch upon
even more later on, will need to leverage
trade-level derivative models, and therefore, will
need to do a lot of martingale-related
stuff, martingale testing, resampling, interpolation. So here's a little bit more
information on the CVA. We have talked about
assets or MPE CVA, essentially for our
assets or receivable. In this formula, we
have discussed already the first one. There is also, theoretically,
a liability CVA. Essentially, it is the CV
on the payable side, when the bank or when us having
a likelihood of default. And this is a benefit
for us, all right. So the formula is
fairly symmetric, as you can recognize, except the
default time or default event is not for the
counterparty but for us. OK? And then the positive sign
here became negative sign, essential to indicate this is
a payable negative liability to us. This is an interesting
discussion first to default. We talked about how if the
counterparty were to default, we more or less pay the
counterparty full amount. So argument can be used
on the receivable side. So if we have a receivable, and
if we were to default first, roughly speaking
the counterparty would pay us close
to the full amount. And there, some people start
to think about OK, when we price CVA, we've got to
know, among counterparty and ourselves, which
one is first to default. But my argument is that we
do not need to consider that. And I have some
reference for you guys to take a look if you are
interested in this topic, but I'm not going to
spend much time because we have lots to go over. Now, here's another example. You have a trade, same
as the previous trade. The trade PV was 0 on
day one, and the trade PV becomes $100 million later on. This time of course
the counterparty risk are properly hedged. Then the question is do
you have any other risks. Does anyone want to try to tell
us do you see any other risks? There are actually
several categories of risk we will have. I wonder if anyone
would like to try to share with us your opinion. Sorry, I couldn't hear you. Yes? AUDIENCE: Some form
of interest rate risk. YI TANG: Interest
rate risk, OK, fine. OK fine, this is a market risk. Yes, you're right there
is interest rate risk, but I did mention here that
the market risks are properly hedged. So that means this interest
rate risk of the trade will be handled by the hedge. What other risks? AUDIENCE: Is there
a key man risks? So if the trader that made
the trade leaves and doesn't know about the-- YI TANG: Ah, OK AUDIENCE: --portfolio? YI TANG: That's a good point. Yeah, there is a risk like that. Yeah. Any other risks? OK. Let's go over this then. I claim there is a cash
flow liquidity funding risk. OK? Our trade is not collateralized. And then I claim we need funding
for uncollateralized derivative receivables, meaning we are
about to receive $100 million in the future. We don't have it now. And I claim we actually need
to come up with cash for it in many cases, in most
cases, not every trade. Anyone have any idea of why when
you are about to receive money, you actually need to
come up with money? This comes back to the hedge
argument similar to CVA. Essentially, if you
were to hedge your trade with futures or
with another dealer which are typically
collateralized. That means when you are about
to receive $100 million, essentially you are about to
pay $30 million on your hedge. In fact, you had to be futures
that maybe mark to market, that means you need to
actually really come up with $100 million cash. The same is true for
collateralized trades. And there that's
where the risk is. Because when you need to
come up with this money and you don't have it,
what are you going to do? You may end up like Lehman. And there's also a
contingent on the liquidity risk, meaning how much liquidity
risk is dependent on the market conditions, how much
interest rates changed, how much other market risk
factor changes like that. And depending on the
market condition, the liquidity may
be quite different and you may not know beforehand. So that's the another challenge. [INAUDIBLE] If you turn
the argument around, applying the argument
to the payable and if you have
uncollateralized payable, essentially you would have a
funding or liquidity benefit. So one interesting thing to
manage this liquidity risk essentially is to use
uncollateralized payable funding benefits
to partially hedge the funding risk in
uncollateralized derivatives receivables. There are a lot of other
risks, for instance, tail risk and equity capital risks. Now here is one more example I'd
like to go over with you guys on the application of CVA. This is about studying put
options or put spreads. If you trade stocks
yourself, you may have thought
about this problem. I mean, either you can
buy the stock outright or you sell put possibly
with a strike lower than the current price. With that, you more or
less have a similar payout. Some people may
argue OK, if you see put, if your stock comes down,
you're going to lose money. But you're going to lose
money if you were to hold the stock outright also. One of these strategies
that if you sell put, if the stock
is not put to you, and you're not participating
the up side when the stock price increases significantly
then you are not going to capture that price. But of course, one
thing people can do is [? you continue to ?]
sell put so they become like an income trade. So it's an interesting strategy. Some people say that selling
put is like name your own price and get paid for trying it. And that's why we have this
famous trade, Warren Buffett, Berkshire, sold long dated puts
on four leading stock indices, in US, UK, Europe,
and Japan, collected about four billion premium
without posting collateral. Without posting collateral,
that was very important. This is something I
actually was very involved in one of my previous jobs. This happened about, I
think, around 2005, 2006. It's one of the biggest trades. And I was told when I
was involved with this, this was one of the biggest
cash outflow in the derivatives trades at that time,
because Warren Buffett collected the premium
without posting collateral. If he had agreed
to post collateral during the crisis
of 2008, he's going to post many billions of
dollars of collateral. And one reason he had more
cash than other people was he's very careful
[INAUDIBLE] and I think I put a reference
if you are interested and then you can essentially
see the [INAUDIBLE] link [INAUDIBLE]. And what's interesting
is that, there were quite a few dealers who
are interested in this trade, but they know the size. And in a long-dated
equity option, it is not easy to handle,
but I think a lot of people were able to handle. To me, some people were not able
to trade or enter this trade, not because they could not
handle the equity risk. It's they could not
handle the CVA compounded. First of all, we
know there's a CVA. Essentially, we bought this
option from Warren Buffett. Eventually, he may need
to pay and at that time, he may default. So that's
a regular CVA risk. But this is also
a wrong-way risk, meaning a more severe risk. You can imagine when
the market sells off, Warren Buffet would
actually owe us more money. Do you think in that scenario he
will be more likely to default or less likely to default? He'll be more likely
to default. That's where the term
wrong-way risk comes in. When your counterparty owes
you more and more money, that's when he's more
likely to default. And that's even harder to model. And there's a
liquidity funding risk which can also be wrong-way,
because as a dealer you may need to come up
with a billion or two cash to pay Berkshire. Where do you get the money from? Typically, people
need to issue a debt to fund in a [? sine ?] secure
way and essentially, you'll pay for quite a
spread on your debt. That is essentially the cost
of your liquidity of funding. So what we did was,
essentially, we charged Warren Buffett
CVA and wrong-way CVA, charge of the funding costs,
some wrong-way funding costs. Another challenge, of course,
is that some dealers, I suspect, they could've priced
CVA, but they do not have a good CV trading
desk risk management to deliver risk management
of CVA and funding. Once you have this
position at hand, you have counterparty risks. But how do you hedge it? You charge Warren Buffet x
million dollars for the CVA. If you don't do anything,
when their spread widens, you're going to have
a lot more CVA loss. So you need to risk manage that. Of course, you can do
that with any hedge. But at any hedge, if we
drill down to details, you suffer a fair
amount of gap risk. It's not like a bond. If you own a bond, you
can buy a CDS protection on the same bond. More or less, you are hedged
for a while in a static way. But for a CVA, it's not. The reason for that is the
exposure can change over time. One thing we tried at
the time, essentially we sort of structured
like a credit-linked note type of trade. Essentially, you go to
people who own or would like to buy Berkshire's bond. Essentially, you
should tell them OK, we have a credit asset
similar to Berkshire's bond. If you feel comfortable with
owning Berkshire's bond, you may consider our asset
which pays more coupon. And the reason we were
able to pay more coupon is we were able to charge
Berkshire a lot of money. And there's also a tranched
portfolio protection thing that's involved,
but I'm going to skip that for
the sake of time. So then the question is the
we charged a lot of the money from Berkshire. Why would he want
to do this trade? What would they think? So here's my guess. As you know, they have
an insurance business. Then they wanted to explore
other ways to sell insurance. So selling puts essentially
is spreading insurance on the equity market. They sold like 10,
15 year maturity puts at below their spots. So then people can
think, OK, what's the likelihood of a
stock price coming down to below the current
stock in 10, 15 years. Well, it happens, but
it's not very likely. And they do have a
day one cash inflow. So essentially, I think one
way Berkshire was thinking is that they thought
low funding costs. If you read Warren
Buffett's paper, essentially he's saying
it's like 1% interest rate on a 10 year cash, or
something like that. And it's very important to
manage your liquidity well. They do not have any cash
flow until the trade matures. So that's how they avoided the
cash flow drain during 2008, even though they did suffer
unrealized mark-to-market loss. But what's interesting is
that during 2008, 2009, Berkshire did explore
the feasibility of posting collateral. This started with no
collateral posting. But then they wanted
to post collateral. They actually approached some
of the dealers saying oh, I want to post some collateral. Why is that? There's no free lunch. So what happened was they were
smart not to post collateral, but during the crisis
their spread widened. Everyone's spread widened. So Berkshire's spread widened. Then Warren Buffett
owed more money. So guess what? The CVA hedging would
require the dealer to buy more and more
protection on Berkshire. When you buy more
and more protection on someone, that will
actually drive that person's, that entity's, credit
spread even wider. So Berkshire essentially saw
their credit spread widening a lot more than they had hoped
for, than they had anticipated. And later on, they found out
it was due to CVA hedging, CVA risk management. That actually affected
their bond issuance. When you have a high credit
spread from CDS market, essentially the cash
market may actually question may actually follow. And whoever would like
to buy Berkshire's bond would think twice. OK, if I have to buy
this bond, if I ever have to buy credit
protection, it's going to cost me a lot more
money because of the spread widening. So therefore, I'm going
to demand higher coupon on Berkshire's bond, and that
drives their funding cost high. So they explored in
different to post collateral. Another thing of course is a
very interesting thing to ask. Berkshire thinks
they're making money and the dealer
thinks they're making money, which is probably true. But then the question
is, who is losing money or who will lose money. Anyone has any ideas? I think there's probably
a lot of answers to this. My view is that essentially
whoever needs to hedge, whoever need to buy put. If the market doesn't decline as
much as much as you hoped for, essentially you'll
pay for put premium and do not get the benefits. Here's an interesting
CV conundrum. Now, hopefully by
this time, you guys fully appreciate the CVA
risks and the impact of CVA. In terms of risk itself,
in terms of magnitude, as I mentioned earlier
being the crisis, 2008 crisis, which
[? killed ?] among easily billions of dollars loss for
some of the firms due to CVA, and that's more than
the actual default loss. Now given you know
the CVA, so if you trade with counterparty
A, naturally you'll say you want to think OK,
I want buy protection to hedge my CVA risk, to
buy credit protection on A, from counterparty B. If you
trade with counterparty B, you would have CVA
against counterparty B. You would have a credit
risk against counterparty B. So what are you going to do? If you just follow
the simple thinking, essentially you may
think oh OK, maybe I should buy credit protection
on B from counterparty C. But if you were to do that, then
you have to continue on that. It becomes an infinite series. Infinite series are OK
I'll say theoretically, but in practice I
feel it's going to be very challenging to handle. So what would be
a simple strategy to actually terminate this
infinite series quickly? Yeah this also has theoretical
implications for CVA pricing. Sometimes we say, OK, arbitrage
pricing is really replication, use hedge instruments. Now, you have to use an infinite
number of hedge instruments. That's going to impact your
[? replication ?] modeling. So the way we would
do it practically is to buy credit protection
on A from counterparty B fully collateralized,
typically from a dealer. So however much money you owe
from counterparty B right away, they're going to
post collateral. In a way, it's more or less
similar to a futures context settling. That minimized the
counterparty risk [INAUDIBLE]. So you can cut off this
infinite series easily. Here, I'd like to
touch upon what I call enterprise-level
derivatives modeling. We mentioned trade-level
derivatives models. That is essentially, is
just a regular model. When people talk about
derivatives model, usually people talk
about trade-level models. Essentially, you model
each trade independently. Your model is price,
mark-to-market or its Greeks sensitivity. Then when you have a
portfolio of these trades, essentially you
can just aggregate their PV, their Greeks,
through linear aggregation. Then essentially you get
the PV of the portfolio. But as we have
seen already, that doesn't capture the
complete picture. There are additional risks
that require further modeling. One is non-linear
portfolio risks. So essentially, these risks
cannot be like a linear aggregation of the risks of
each of the component trades in the portfolio. The example we have
gone through is CVA, funding is of similar
nature, capital liquidity are also examples. The key to handle
this situation is to be able to model
all the trades in the market and the market
risk factors of a portfolio consistently so that you
can handle the offsetting trades properly. Of course, we need to
leverage the trade-level model essentially to price each
individual trade as of today, as of a future date. What's interesting is
that there's also feedback to the trade-level models. For instance, when we
price a cancellable swap of a very public trade. Now this cancellable swap we
trade with a counterparty, let's say assumed
uncollateralized, we trade with a counterparty
that's close to default. You know the trade-level
model doesn't know about this
counterparty, about default. The trade-level will give
you independent, the exercise boundaries, when do you need
to cancel the swap, independent of the counterparty
credit quality. That invites a question,
when the counterparty is close to default, even if your
model says OK, you should not exercise based on the
market conditions, but shouldn't we consider the
counterparty condition, credit condition. If the counterparty
were close to default, if you cancel the swap
sooner essentially you'll eliminate or reduce
the counterparty risk. This is actually interesting
application and feedback between a trade-level model and
the enterprise-level models. So what we did was, in
some of my previous jobs, what we did was actually figure
out the counterparty risk in these trades,
the major trades. Then essentially, we just
tell the underlying trader, if you were to
cancel this trade, we have a benefit because
we're going to reduce the CVA or even zero out CVA. So the CV trader would be able
to pay the underlying trader. So therefore, the
underlying model actually can take this as
input rather than as part of the exercise
condition modeling, knowing if you cancel
earlier you potentially can get additional benefits. This model may eventually
be able to tell you to handle the risks more
properly, market risks together with counterparty risks. This is roughly the
scope and the application of the enterprise-level model. This is actually a fairly
significant modelling effort as well as significant
infrastructure and data effort. Essentially, it
requires a fair amount of martingale testing,
martingale resampling, martingale interpolation
and the martingale modeling. The reason for that is
you have a trade model, and each trade model can model
a particular trade accurately, and there's certain market
modeling, simulations of the underlying
market or great PDE. But when you put a portfolio
of trades together, now the methodology you use for
modeling one trade accurately may not necessarily be the
methodology you need to model all the trades accurately. Some of these require PDE
and some require simulations, but you need to
put them together. Typically, we use simulation. And that introduced
numerical inaccuracies. And the martingale
testing will tell us are we introducing
a lot of errors, martingale resampling
essentially would allow us to
correct the errors. As you know, the
martingale is a foundation of the arbitrage pricing. Essentially,
martingale resampling will actually be able to enforce
the martingale conditions in the numerical procedure,
not only theoretically. Martingale
interpolation modeling are other important interesting
aspects if we have time we can [INAUDIBLE] There are
different approaches for how to do it
in a systematic way and still remain
additional ways. I'd like to quickly go
over some of the examples of martingale and
martingale measures. I may need to go through this
quite quickly due to the time limitations. But hopefully, you guys have
learned all these already. This will hopefully be more
like a review for you guys. So essentially, we are
talking about a few examples. What's the martingale
measure for forward price, forward LIBOR, forward
price, forward FX rate, forward CDS par coupon. I would hope you guys would
know the first few already. The for CDS par
coupon in my view is actually fairly challenging. For simplicity, I'm not
considering the collateral discounting explicitly. That adds additional challenges
but still we can address that. So under the risk
neutral measure, essentially for
this Y of t being the price of a traded asset
with no intermediate cash flow. Essentially, that is y_T
over beta(t) is a martingale. This is essentially the
Harrison-Pliska martingale no-arbitrage theorem. It says for two traded assets
with no intermediate cash flows, satisfying
technical conditions, the ratio is a martingale. There's a probability
measure corresponding to the numeraire asset. Therefore, naturally
we have this composite. The forward
arbitrage-free measure essentially corresponding to a
numeraire of zero-coupon bonds. Naturally, we can
find this Y_t and P(t, T) ratio is a martingale. Again, it's just a ratio
of two traded assets with no intermediate cash flow. From the definition
of the forward price, essentially the forward
price is a martingale under the forward measure. Forward LIBOR-- this is the
forward LIBOR-- essentially, it's a ratio of two
zero-coupon bonds. So naturally, we know
it's a martingale under of the numeraire asset. So essentially it's a forward
measure up to the payment on the forward LIBOR. So this is the
martingale condition. Similarly, we can do this
argument of the forward swap rate. Essentially, a
forward swap rate is, we can start with, like
an annuity numeraire. And since the forward swap
rate, you essentially know, is the difference of
two zero-coupon bonds divided by annuity. And therefore we
can conclude based on Harrison-Pliska theorem the
forward swap rate essentially is the martingale under
the annuity measure, with this annuity
as the numeraire. The same argument goes
for the forward FX rate. Mainly the idea is or the
pattern you probably have seen is, for any quantity you see
if you can find two assets and then use these two
asset ratio to represent this in a quantity. So the forward FX essentially
is a ratio like this. This is nothing more than
the interest rate parity. From the spot you
grow both [INAUDIBLE]. You start with spot, you
grow the domestic currency and then you grow
the foreign currency. You get FX forwards. And FX forward is a
martingale measure under the domestic
forward measure. This is a simple technique to do
change of probability measure. It's roughly how I remember
change of probability measure and Radon-Nikodym derivatives. You essentially start
with, again, martingale, assuming this is martingale
under a particular measure corresponding to
the numeraire asset. And then this quantity
is also a martingale under a different
measure corresponding to a different numeraire asset. One key point is when you change
probability measure essentially you change the
numeraire corresponding to the probability measure. And therefore essentially
the important thing is we know the PV or the
mark-to-market, of a traded security is measure-independent. It doesn't matter
what mathematics you use if the traded
security is going to match the market price. And therefore, you can
price this security under one measure
or one numeraire. And then you can price
again with another measure, another numeraire. They've got to be the same. Then naturally, you see
this simple equation as the starting point to
do the change of measure. If you just simply change the
variables, then essentially you get your change
of measure as well as Radon-Nikodym derivative. And if you worked
on the BGM model, you'll probably recognize
this change of measure which is used for the BGM
model under the old measure. Now here's the subtlety,
credit derivatives. Naturally, people would think
OK, since the forward swap rate is a martingale
under the annuity measure, naturally people would
think OK, then forward CDS par rate, it's like a
forward swap rate. It's got to be a martingale
under the risky annuity measure. So that's quite intuitive
except there's one problem. If the reference
credit entity has zero recovery upon
default. Then, this risky annuity
could have a 0. And now we're
talking about we want to use something that could
be 0 as our numeraire. How do we resolve the
technical mathematical problem. So that actually
very interesting. Schönbucher was the first person
who published a paper on this model. I was just trying to
do some work myself when I was working on BGM model. I thought oh, it would've been
nice to expand the BGM model to the credit derivatives. But then immediately I
stumbled with this difficulty where when the
recovery is 0, you're going to have a 0
in your numeraire, in your risky annuity. So Schönbucher, essentially,
his idea was let's focus on survival measures, meaning we
have a difficulty if a default happens and the recovery is 0. Now his idea is let's
forget about that state. Let's not worry about that. One immediate
question people will ask, if that's the
case, the probability measure, physical
probability measure or risk-neutral
probability measure, and this survival probability
measure are not equivalent because the survival probability
measure knows nothing about the default event. So they are not equivalent. That's, essentially,
you actually transform one mathematical
difficulty to another one. Luckily, the second
one turns out to be actually easier to solve. So the starting point is again
using Harrison-Pliska theorem. Essentially, you
just need to identify like a numeraire asset,
and the denominator assets. You identify two assets. You make a ratio and then
those are a martingale. So essentially this is forward
swap rate and forward annuity. If we have this indicator of
the default time of j-th credit name, greater than this
t, essentially this is like the premium leg of CDS. That's a traded asset. So therefore, we have a
martingale [INAUDIBLE] like this. The subtlety as you
probably can envision is going to come in when we
do the change of probability measures. OK, so we have talked
about how are we going to find the martingale
measure of a CDS par coupon or forward CDS par rate. This is a starting point
of martingale model. Essentially, for
any quantity you want to model you try to
find its martingale measure. Once you find this
martingale measure, you can do a martingale
representation. And then often times you need
to a change of a probability measure. So that all the term
structure functions, a consequence of a
variables are modeled in a consistent
probability measure. So finding the martingale
measure is the first point. Survival probability
measure, essentially, he just defined this with. You can define this
Radon-Nikodym derivative. Once you define
that essentially-- if you remember the
previous formula-- you will have a martingale
condition like this. [INAUDIBLE] The
probability measures are not equivalent
anymore, but yet they can still do change of
probability measure. You need to
separately model what going to happen when
the default happens if you want to use this model. Now, I'd like to move onto
the second part, martingale, martingale testing
and martingale resampling and interpolation. Martingale testing essentially
given the previously model formula's conditions. Those are, by the
way, just examples. There are a lot more. Essentially, you
know that's what it should be
theoretically you just test in your numerical
implementation and see if those
conditions are satisfied. That's the martingale test. Martingale resampling is we
know most likely if you were to test, we're going to fail. This is not necessarily
for enterprise-level models but even for trade-level
derivatives models. A lot of times, I think the
martingale conditions are not exactly satisfied. So one way to do that, is
to correct that, correct this error. The rationale is
essentially because of a numerical approximations. Whatever quantity
we model essentially is not a true quantity. The true quantity
we model essentially is some function of what
we have in our model. So therefore, you expect
a certain function. Sometimes you can have a
linear, log-linear function. This X_0 is what we
have in our model, and then X is what
we need to satisfy the martingale condition. Essentially, in this
[? Purdue ?] case is very simple. You first of all,
use the mean and then you would adjust the deviation. So therefore, given any
quantity X_0, you can have, you can force it to
be any given mean. This mean, in our case,
will be determined by the martingale condition. The next interesting thing
is martingale interpolation. Oh, I have a typo here. Sometimes you have an interest
rate model, for instance, you model LIBOR. Your LIBOR, you can
have different tenors. When you have a yield curve,
you know, at any given time, there's a term structure. In the model, a
lot of times we can model a few selected points. But what if your model
requires a term structure, a term that not in your model. So what people normally
do is you do martingale, you do interpolation. So you have a 1-year LIBOR
and you have a 5-year liable. And then you need a 3-years. What do you do? You interpolate, for instance. But interpolation
doesn't automatically guarantee martingale
relationships. The martingale
interpolation has a goal of automatically satisfying
the martingale relationships, so we're particular
with our interpolating. Actually, it turns out
to be a [INAUDIBLE] The starting point is the
martingale condition that I wrote out on the slide. Essentially, this s and
t are the calendar time. And this capital T is really
like a term structure. You have a 1-year rate, 2-year
rate, 5-year interest rate, those term structures. How do we interpolate such
that after interpolation the corresponding martingale
relationships are satisfied. So here's what we do. We start with, let's
say, capital T_1. Capital T_1, that's
a point we model. We assume that one is
properly martingale resampled and satisfies the
martingale condition. This is a martingale
for T capital 2. That also satisfies the
corresponding martingale conditions. Our goal is to figure out T_3. How do you do interpolation
for the term T_3 such that this T_3 will
satisfy its own corresponding martingale condition. If you do simple linear
interpolation using T as independent
variable, essentially, you are not going to achieve that. So the key is we need the choose
the proper independent variable for the interpolation. Essentially, it's the previous
time or time 0 quantity. So time s is before time t. Imagine time s will be 0. So using the
corresponding quantities at time 0 as the
independent variable, essentially, you
can achieve that. It's still linear
interpolation, it's just to use a different
independent variable. Essentially, you can
show that very easily. This is just simple algebra. If you take the expectation,
this one being martingale, this little t will become s. Then if you do expectation
here, the little t will become little s. And therefore if you combine
these two, a lot of terms will actually cancel. Essentially, you will be
left with this martingale at time s and T_3, meaning
this is the martingale target of this particular term. And that turns out to be the
expectation of this quantity. So it's a very simple
linear-- simple algebra. You guys can figure
it out if you want to. So this one
essentially guarantees the interpolated
quantity automatically satisfies all the conditions
of the martingale target. Of course, you need to
know the martingale target. If you don't know,
that's a different story. Then you need to
do something else. Specifically, time
0, for instance, is what the market tells us. Often time we do a
big time assumption. So whatever assumption on time 0
you make, in you dynamic model, you automatically satisfy the
needed martingale condition. This is just a
brief introduction of how we do the
martingale modeling. This LIBOR market model,
as you guys probably have learned already,
there's different forms of BGM as the initial form. And then Jamshidian
came with another form. And in terms of a
general martingale model, what we'll
do typically is we start to find the
martingale quantity. And we know a forward
LIBOR is a martingale in its own forward measure. Then we know we can use
martingale representation. Under certain
technical conditions, the diffusion process can be
represented by Brownian motion. Then we can assume
log-normal just for example. We don't have to,
we can use CEV, we can use [INAUDIBLE]
stochastic volatility. The starting point
is martingale, identifying the
martingale measure and then perform
martingale representation. Essentially, you get this
stochastic differential equation. They need to change measure
or change numeraire. Because this one essentially
says for particular LIBOR, you have a Brownian motion
and a different measure. So that has limited usage. A lot of the derivative
trade, IR trade essentially, it's [INAUDIBLE]
the entire yield curve. So you need to make sure
you model the entire yield curve consistently. So therefore you have to
change the probability measure so that everything is specified
in the same common measure. Of course, you can have
a choice which one you want to use as common measure. Through a simple
change of numeraire, essentially, we can get a
stochastic equation like this. We have a Brownian motion. Right, we have a Brownian motion
with a correlation like this. So this is essentially a
market model in a general form, with full dimensionality
meaning one Brownian motion per term of a LIBOR. So that's the full
dimensionality. PROFESSOR: Yi? YI TANG: And then you
need to do-- Yeah, hi. PROFESSOR: Can you wrap up
because we need a bit of time for questions. YI TANG: Oh you need me to end. It's all right I can
actually wrap up now. If you want to. PROFESSOR: Sure. OK, any conclusions? YI TANG: Well, OK. The conclusion is
the following thing. There is a need for
enterprise-level models to handle non-linear
portfolio effects and we need to leverage
our trade-level models. By doing so we do employ
martingale testing, martingale resampling,
interpolation. And not only we
need that for CVA, but we also need that for
funding liquidity capital risks which are very critical risks. And people have started
paying more and more attention to these risks, especially
since after the crisis. Because of time
limits, I'm not going to be able to finish
another example. But if you like, you can take a
look on page 22 of the slides. Hopefully, Vasily can
still get it to you guys. Thank you guys. PROFESSOR: Thank you Yi. We will publish the slides
probably later tonight so please take a look. So to wrap up, let me see. I want to bring up, PROFESSOR 2: That's OK. Probably if it's the course
website, that's fine. PROFESSOR: I did add
a few topics which were used last year for
final paper for interest in the document which
is on the website. So take a look. Basically, the themes there were
mostly Black-Scholes or more advanced models or manipulation
of Black-Scholes equation. There was a very interesting
work on statistical analysis of commodity data. So if somebody's up for it,
that would be very interesting. And there were a few numerical
and Monte Carlo projects. So any questions? PROFESSOR 2: Yeah,
so actually we were planning to give
you a bit more time to ask your questions. But since we have
five minutes, I think maybe I'd like to ask
you to just think about what we learned this term. So Peter can add
on what we think in on the mathematics and
also those applications, and in conjunction while you're
doing the final paper, just focusing on the new things
you think that you learned and what did you like to
explore in the next stage of your research. So I think probably we
don't have a lot of time for lots of questions. But if you have
any questions, this will be a good
opportunity to ask about the paper or the course. Peter you want to
make some comments? PETER: Sure. I'd just like say that
I think this course was a very challenging course
for most of you and that was, I guess, our intention. And I really respect all
the hard work and effort everyone put into the class. And in terms of
the final paper, we will be looking
at your background and look for insights
that demonstrate what you've learned in the course. And I've already
reviewed several papers. I'm very pleased
with the results. So I think everyone's
done a great job. This course, I
think, is intended to provide you with
the foundations of the math for the financial
applications as well as an excellent introduction and
exposure to those applications. I think you'll find this
course valuable over the course of your careers,
and look forward to contributing
insights with questions you might have
following the course. I'm sure the other
faculty feel the same way. We want to be a good resource
for you now and afterwards. PROFESSOR: Very, very well put. So please feel
free to contact us. And please stay in touch. All the contact details
are on the website. We plan to have a repeat
of this class next year. So please, tell your friend
or stop by next year, which will be the next fall. It will not be exactly the same. We will try to make
it slightly different, but it will be close. PROFESSOR 2: If you have
any suggested topics, you feel you haven't
been exposed to and would like to know more,
send us email if you can. I think one of the
values of this class is we can bring in
pretty much everyone from the frontier
in this industry to give you some insights
of what's going on. PROFESSOR: Please take
a review on the website, this is important. And that's all. PROFESSOR 2: OK. Thank you for your
participation this semester. [APPLAUSE] PROFESSOR: And thank
Yi for the pleasure.
