26.對手方信用風險介紹 (26. Introduction to Counterparty Credit Risk)

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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.