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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.

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