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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Anyway,
welcome today. Stefan Andreev is our guest
speaker from Morgan Stanley. And as I understand
you have a degree, a PhD Degree in
chemical physics. STEFAN ANDREEV: In
chemical physics, yes. And maybe I should go here. [LAUGHTER] PROFESSOR: And now he's
in the world of finance. And we're here to benefit
from your experience. STEFAN ANDREEV: Thank you very
much for the introduction. Yeah. I went to school at
Dartmouth College undergrad, and then up the street
at Harvard for my PhD. And then I transitioned
from science to finance. And for the last
eight years, I've been working at Morgan Stanley,
working with Vasily Strela, an instructor in the course. So today, what are we
going to be talking about? Well, to give you a big-picture
view of where our topic fits within the grand scheme
of finance-- in general, there are really two
big areas in my view. Well, there's probably
more, but these are kind of most famous,
I would say, areas where quantitative
skills can be-- are very valuable in finance. And the two areas are--
one area is statistics, predictions, which
is essentially, say given some historical
behavior in the market, how do we predict what
will happen in the future? And that's certainly
a huge industry. People have made a ton of money
applying quantitative concepts to that. But that's not what we're
going to talk about today. What we're going to talk
about is another very big area called pricing, which
is pricing and hedging of complex instruments. And that area is really
about essentially when you have a complex
product that you don't really know the price of, but you know
the prices of other products. And then you can use
the other products to essentially replicate the
payoff of your complex product. Then you can use
mathematical techniques to essentially say, look,
the main statement is hey, because I can
replicate my payoff, and using products that
I know the price of, then that means that
I can say something about the price of
my complex product. I can basically price it. And not only can
I price it, but I can also-- when I
give the price, I know that I can eliminate
any uncertainty from owning the product by executing
a hedging replication strategy-- at least
theoretically speaking. So that's the area we're
going to focus today. And our main focus is going to
be on FX-- foreign exchange-- interest rates, and
credit-- and in particular about credit-FX hybrid models. We're going to be talking
about essentially what happens, why do we need
credit-FX hybrid models, and going through an
example of a simple one, and how to apply it. In particular-- and there's the
mathematical techniques we're going to be using--
as I said, we are going to be talking about
the risk-neutral pricing, which is essentially replication. And we're going to talk about
how to use jump processes-- which you might have seen in
other parts of your studies as Poisson processes-- to
describe certain behaviors of price behavior that you
cannot really describe very easily using pure diffusion
Brownian motions that you probably have seen
so far in the course. And why do we care about that? Well, there are certain
financial applications where this is important. And in particular,
something that happened in the last few years--
the sovereign crisis in Europe. And also, it has happen
not just last year. This happened many times in
other parts of the emerging markets. And given the emerging
markets as my background, I've worked on these
kind of models. And this is, when you
have Greek bonds in Euros, and there's a potential for
Greek default. And as we know, as you might have
read in the news, there was really a
big worry about what will happen to the Euro
currency if there is a spate of sovereign defaults. And in fact, Euro currency
did-- in anticipation of the possibility of
default-- it actually did depreciate for while
back in 2011 and 2012. Now it's pretty much back
where it was before that, but it certainly--
there was a fear in the market-- which was also
very, very obvious in terms of option prices-- that Euro
currency could depreciate significantly if, in fact, a
disorderly default did happen. Now it didn't happen,
so that's good. But in other emerging
markets in history, it has happened before. So it's not really
an empty question. So foreign exchange-- how do
we describe it in math finance? Well, we think of
it as the price of a unit of foreign
currency in dollars. In our presentation we're
going to denote the spot FX rate, which is the current
rate of exchange, by S. And here is a sample graph
of euro-USD FX rates. You can see it looks
like a random walk. It's very well described
in normal circumstances as random walk. So one very fundamental property
that connects FX and interest rates is the so-called
FX forwards interest rate parity, which says if I have
a certain amount of money-- in this example, $5 million,
and I can invest it, there's two ways I can
utilize this money. One way is to just
invest it at a dollar kind of risk-free rate. And we're assuming here
we have a risk-free rate-- the standard assumption. Or we can do something
like we can take the money, exchange it into,
say, euros, invest it using the euro
risk-free rate, and then exchange it
back into dollars. And this is essentially used
to price FX forward contracts. So FX forward contracts
are a contract that allow you to say, look,
I'm going to agree with you, then in one month's time,
I'm going to, say, give you 4,108,405 euros, and you're
going to give me back $5,170,000. It's essentially an agreement;
it's a derivative contract. And if you see the-- if you
have this forward contract, you can lock in, essentially,
through conversion in euros. So you can lock in an
effective dollar interest rate. So FX forwards can be
essentially described fully by knowing the interest rates
in each currency and the spot FX rate. Conversely, you can infer
foreign interest rates knowing the FX forwards. They're very connected. Yes? AUDIENCE: In this example,
there's no mispricing, so you get that same amount. Is that the idea? STEFAN ANDREEV: In this
example, there is no mispricing. You get back the same amount. So we are assuming, essentially,
there's no arbitrage. We're not assuming,
but we're given-- if the prices were, indeed,
if this interest rate-- 4.6% in Euros-- interest rate was
4.6, in dollars it's 3.4. And here are the current spots,
which is 127 and the forward, 125-- if these were, in
fact, the observable mark quantities in the market, then
there would be no arbitrage. And there is-- you're basically
indifferent whether you invest the money in dollars,
or you go the way of exchanging into Euros, and
investing in Euros, and then back into dollars. So in this example, the way
I've presented, worked it out, there is no arbitrage. Now, if some of these numbers--
say the interest rate in Euros were 4% instead of 4.6, and
all the other quantities were the same, then, in fact,
there would be arbitrage. And you could make
money by borrowing money in dollars and investing. I mean, the purpose
of this slide is really to
illustrate kind of hey, if there's no arbitrage,
how one would actually compare-- how one
would actually look for arbitrage in this example. This is-- again, this is
a little bit of definition what are compound
interest, interest rates. We're going to talk about
instantaneous risk-free rates. We're going to, again,
say they're risk-free. So basically, we
know for sure we're going to get our money back. You can think of
risk-free rates as the one that treasuries pay in
real life, or the one that Federal Reserve
guarantees on deposits. There's various examples
of risk-free rates. And while, in practice,
different risk-free rates can actually be different, so
they're not really risk-free. But in our world right
now, in our model, we're going to
assume that there is such a thing as a risk-free
rate for every currency, and it's unique. And now, as we talk about our
dynamics of the FX process, what's really focused
on here is an FX-- we're making an FX model. And we want to see-- in
the previous example, we saw if you're given a FX rate
and given some interest rates, here's what the
FX forward really has to be in order
to have no arbitrage. Well, now we are
trying to describe. We tried to describe or define
a process for the FX currency. Essentially, this kind
of no-arbitrage condition leads to having
certain constraints on what the stochastic
differential equation has to be. So in this particular
case, the constraint is that the drifts
of the process has to be the differential
in interest rates. So if one currency pays more
than the other currency, obviously, people would want
to invest in that currency. So that in order for
no arbitrage to exist, there has to be an expectation
that the currency that pays more would depreciate
in the future, otherwise it would be an arbitrage. So if it doesn't depreciate,
if you can kind of say, hey, this currency
won't depreciate, then you can just always
invest in that currency that pays higher interest rates and
make money-- which, in fact, many people do. But again, that's a-- they're
taking a certain risk. They're taking the risk that
the currency will depreciate. So what do we actually want? What we want is to say--
we want to essentially-- and for the arbitrage
conditions from before-- which is to say that my forward rate
has to be essentially the spot rate-- well, this
condition here has to be observed, essentially. And what does that mean? It means that my forward
rate has to be my spot-- AUDIENCE: [INAUDIBLE]
each other, you mean? STEFAN ANDREEV:
What did you say? AUDIENCE: [INAUDIBLE] set
those equal to each other? STEFAN ANDREEV: Yes. My forward rate has to equal
to the spot rate times, essentially, the interest
rate differential. If that is true, than
in the previous set up, there will be no arbitrage. And why is that? Well, that's because
the amount of money I earn on the domestic
leg is e to the rd. The amount of money I earn on
the foreign leg is e to the rf, but then I multiply
by the forward, and that has to equal
to the e to the rd. So this is a standard. This is a the most
basic dynamic FX model that people use in industry. It's referred to as the
Black-Scholes FX model. And the stock price-- you've
seen stochastic models before. Usually, you see the stock price
when people talk about options. In that case, this drift is just
the risk-free interest rate. Well, here in FX, it's a
differential of interest rates. Otherwise, it's very similar. So FX has some
interesting properties. So we're gonna talk
about the game. Before we go to the
game, one question: can FX exchange rate
ever be negative? What do you guys think? Can the dollar-euro
exchange rate be negative? Any ideas? No, it's hard, because
what does negative mean? It means I have to pay you
money to give you euros. Why would you? You have to pay me
money to give me euros. Nobody would do that. It can be 0, potentially,
if dollars are worthless or something, but it
cannot really be negative. So that's one reason why
I wrote my SDE as a kind of a log-normal process. You recognize this by
the form dS over S. So the changes in the FX are
proportional to the value of the FX. So the process can
never become negative. OK, so it can never
become negative, but how big can it get? And the answer is,
it can get very big. I mean, we have currencies,
notably some currency like Zimbabwean dollars,
that traded-- I don't know, I mean, I actually don't know
what Zimbabwean dollars trade at, but I think it's
somewhere in the billions of Zimbabwean dollars
per dollar, so something. It's a really extreme example. It can really get extremely big. So there is now
really upper bound, while there is a lower bound. So the distribution, as
you can imagine has a skew. It's not symmetric
around the average. It's limited on the lower side. It can go very high
on the high side. And log-normal distribution
has that property. Have you guys seen a
log-normal distribution? You've talked about this stuff
in the course before, right? So let's go back to our game. So our game is, we
have assumptions. My assumptions are
not to be realistic, but to make it simple. Let's assume that our dollars
and euros exchange rate is one, so we can exchange
one Euro for $1. Clearly not exactly the case,
but let's make that assumption. And we also assume
that the FX forwards is 1, which basically
means that the interest rates in both
currencies are the same. And now let's say
I'm going to make you a bet that-- now dollars and
euros is a volatile process. Right now, it's one,
but in the future, it could be different from
one-- could be higher or lower. So if dollar-euro FX process
is more than one in one month, then you give me money. And then if it's less
than one in one month, then I give you money. And we're going to have
two payoffs, so two games. I don't know why
it says "bet B." Should say just "bet." I'm sorry about that. That in the payoff
A, basically you're going to give me $100
if I win, and I'm going to give you $100 if you win. And in payoff B, you're going
to give me 100 euros if I win, and I'll give you
100 euros if you win. And the question is, which
game would you prefer to play, or do you not care? So in each case, you kind
of win, lose same number. So I want to see hands. Who wants to play game A? Come on guys, wake up. Who wants to play game A? I mean if you don't
know, you just-- lets say you like Euros better. You can say this is not
really graded, so it's OK. OK, nobody really
knows what to play? Like how about game B? Anybody wants to play game B? OK, you guys want to play? Three people for game--
four-- game A nobody still? Same person for game A and B? All right. OK, two people-- so now-- AUDIENCE: Behavioral
science says people are reluctant to
lose-- more reluctant to lose. STEFAN ANDREEV: That's
true, that is true. However, so people
are reluctant to lose. And I said, look, the FX
forward in one month is 1. So you can actually-- that's the
market price that in one month, FX forward is 1. And this-- our bet--
kind of the strike is 1. So our bet level is 1. So you can kind
of say, well, this looks like kind of fair game. So I don't expect
to win or lose much, but I'm just reluctant to do it. And I can get that feeling. That's the risk
aversion aspect of it. But if you were
forced to make a bet, question is, which
one would you prefer? So I understand you
might not want to play. But I'll say, OK,
so you guys don't seem to be in the mood to play. That's fine. Let's look at some scenarios. So let's say in one month,
dollar-euro goes to 1.25. In bet A, I lose $100. In bet B, I lose 100 euros. So bet A-- actually,
you lose 100 euros, not-- so bet A for you, you
are $25 better than bet B. And in the second case,
if dollar-euro is 0.75, you make $100, or you make 100
euros in bet B. In that case, you also-- bet A is $25 better. So it doesn't
matter what happens. Bet A seems to be
the better case. So if you're, like, our dear
professor here, then you don't like to lose,
then you probably are going to choose
bet A, I assume, right? That's a better deal. And that's kind of
strange, though. I mean, like both
payouts were symmetric-- so it's 100 euros,
100 euros, $100, $100. Why is it one is
better than the other? Well it's like
what really happens is the units of the bet--
the value of those units depend on whether
you win or lose. So it's not like if I
was betting using acorns, then you get two acorns
or I get two acorns, then actually, it
might be a fair bet. But because I'm betting in
euros and dollars, and the value of these things-- the
relative value changes based on the actual whether
you win or lose, then the game is not
symmetric any more. So the reason I wanted to
take you through this game is because there are a
lot of cases in finance where people make bets. But then the value
of what you get depends on whether
you win or you lose. And that has an effect
on the value of the bet. And in particular, the case
we're going to talk about today is one of these cases,
which is the credit FX. That's why we need to
credit-FX quanto models. To give you an
illustration from finance, let's take Italy bonds. So Italy issues bonds both
in dollars and in euros. Why does it issue
in both currencies? Because Italy has to
issue a lot of bonds. And they need to find as
many investors as they can. And some investors
want to buy euro bonds, and some investors want
to buy dollar bonds. And they want to access
both bases of investors. Now, these bonds
they cross-default, meaning if Italy defaults on
one bond, all of its bonds default together, including
the euros and the dollar bonds. So then there is a
notion of credit spread, which is the measure
of how risky Italy is. So you can take
euro bonds, and you can say, well, how
much premium does Italy pay over German bonds? Let's assume that German
bonds are risk-free, which is the standard
assumption for euros, that the German bonds--
Germany is the main underlying economic force for the Euro. They're kind of risk-free bonds. And Italy pays a certain
spread over euros. Same thing for dollars-- it
pays a certain spread over USA. So if Italy wants
to borrow money, they have to pay a
higher interest rate, just like if you want to
borrow money for student loans, you have to pay higher
interest rate than the Fed. And the size of that
spread is in the market. It determines how risky
of a borrower you are. Well, it turns out that
these spreads are not the same in both currencies. One currency has a higher
spread than other currencies. That's kind of an
interesting thing. So there is two
questions really: when the spreads
are not the same, which currency would Italy
prefer to issue bonds in? And which currency do investors
prefer to buy bonds in? So this is kind of similar to
the previous game we played, because if you're an investor
trying to buy bonds, well, if Italy defaults,
then chances are the euro is not doing so well. So you would lose money. And if you have euro bonds,
you would lose euros. If you have dollar
bonds, you lose dollars. On the other hand, if Italy does
not default and pays you back, then chances are, euros
are not doing that bad. So you would actually be
making euros and dollars. So it's an
interesting-- it's kind of a similar dynamic going on. So there's the same
kind of question that I asked before-- USD,
euros, are equal in both. So what do you think now? Now that we've gone
through an example, maybe we'll have a higher
participation in my pop quiz. Who thinks that USD bonds
have a higher credit spread, and who thinks-- so A.
Vote for A. One, two. So who thinks that euro bonds
will have a higher credit spread? OK, one, all right--
so two to one. I think the two to one wins. All right, I must say, you
guys seem that-- maybe it's the format of the auditorium. People don't like to raise
their hands too much, or maybe they're afraid
that they're being filmed. [LAUGHS] OK. Well, how are we
going to do this? How are we going to
answer this question? Before I give you
the answer, we're going to go through a slide. Well, first we're going to say,
well, FX rates are volatile. There is volatility,
as we said before. So now we're going to-- in order
to compare euro bonds to dollar bonds, we need to really
come up with a strategy to replicate one with the other,
and then look at the price-- look at how much do we need to
buy one to replicate the other. If we're able to come up with
such a replication strategy, then we can
immediately say, hey, if you need 150 euro bonds to
replicate 100 dollar bonds, then that means that the euro
bonds have to be cheaper. That's basically the
replication argument. So you can try to do that
by piecing together bonds, or we can use the powerful
tools of mathematical finance that you've been
learning about, which is all about
replication and pricing. And the three steps
are: we're going to analyze the payoffs
of the instruments, and we're going to
write some model, a model for FX and
for credit, and we're going to price those bonds. And then we're going
to look at the results, and try to understand
the problem intuitively. And that's basically
what we do, pretty much. That's what option quants do
on Wall Street all the time. So here's the answer:
dollar versus euro spreads from the marketplace. So usually what happens in these
kind of questions in finances, you kind of have an
answer, and then you try to compute a model that
explains the difference. So that's what we're
going to do now. Well, the USD spreads are
actually lower-- USD bond spreads are actually lower. Now, so there is
really-- when we're talking about bonds, risky
bonds, there's two states. They are either performing,
or they are non-performing and in default. And
we're going to go here through an example of two bonds. And we're going to use two
zero-coupon bonds, which essentially have zero recovery. And the idea there is really
to make the question simple so we can analyze it better. But you don't lose
a lot of generality by saying zero-coupon
versus coupon. It's not-- the
answer, the intuition would be exactly the same. So let's say we have
two zero-coupon bonds, same maturity, they
pay 100 on maturity. And by the way, bonds-- I don't
know how much you guys have-- I say these things. I'm very familiar with them. Bonds are nothing
more than loans. So zero-coupon bond means I
give you some amount of money, and at some pre-agreed maturity,
you're going to pay me 100. So let's say I give you 80
cents, one year from now, you pay me 100. And I call this a
zero-coupon bond, because you don't pay me
any intervening coupons. There's no interest
payments, but just I pay you less money now, and
you pay me more at maturity. OK, so we know that
bond U pays $100. Bond E pays 100 euros. And let's say we denote the
prices-- price of U is Pu, price of E is Pu. Our spot FX rate, we're
are going to call it St; our FX forward, Ft. Now we're going to have kind
of a simple arbitrage strategy. Well, let's say if we can
sell 100 times Ft dollar bonds and with the proceeds,
buy 100-- we're going to get-- if we
sell 1,000 dollar bonds, we're going to get
this much proceeds. There's the price. And if we buy 1,000
euro bonds-- so we can enter into an
FX forward contract for 100,000 euros for
maturity T at zero cost. All right, so let's see how
this strategy actually pays out. Well, what happens is you--
there's 100,000 euros. You get 100,000 euros for
selling the euro bonds. You pay 100,000 times Ft
dollars, say dollar bonds. There is an FX forward
contract, and at maturity, you can exchange this $100,000 for
100,000 euros using this FX forward contract. You already agreed to do that. So your FX forward
actually exactly hedges-- you can basically use
the proceeds of these bonds to-- you can
exchange the proceeds at zero cost at maturity,
because you have entered into the FX forward contract. So your net payoff is 0. So that means that the
prices of these bonds have to be the same. But what if they're not? What if Ft, which
is in this case is 1, forward contract--
what if the price in dollars is different from exchange
rate times the price in euros? Well, in that case, you can say,
well, there is an arbitrage. And you'll be
right, if you would be able to make money if, in
fact, the bonds performed. But what happens if
there is a default? If there's default, that
wouldn't really necessarily be the case, because
if there's default, these bonds don't pay
anything, and you just have an FX forward contract. And this FX forward
contract is going to be worth something
after default, especially if the
FX rate depends, like jumps, upon default.
So arbitrage, again, is-- so you start with 0 money. You make money if there's
nonzero probability. And let's say in
this particular case, the strategy-- payoff in case of
default with 25% recovery rate is-- you actually
have only-- 25% means you only have a quarter
of the payoff now at maturity, if default occurs. But you have a hedge
for the full 100,000. Your FX forward is
for full 100,000. So for 25,000 of it, you
can use the FX forward to exchange money. For the remaining
75,000 you just have an FX forward outright. So if FX moved against
you, you would lose money. So that's why the strategy is
not necessarily an arbitrage. And that's why the two prices
of the dollar and euro bond are not necessarily
related to each other. They don't have to be equal,
because, in fact, there is a possibility of default. And you cannot really
directly hedge. You cannot really construct an
arbitrage strategy by using FX forwards and the bonds
together so easily. You have to take into account
what happens if default occurs. OK, so give an example. What happens upon FX
when default occurs? Well, one of the
most recent defaults of a country-- of a big country
that has its own currency-- is Argentina, 2001. And when it defaulted, the
Argentinian peso skyrocketed. Here is the graph
of the price series. So if you had an FX forward
contract that essentially-- if you had a position where
you were left with a naked FX forward contract, where you
were receiving pesos and paying dollars in the event
of default, you would have lost a lot of money
when the default happened. It would have really
gone against you. And this is, by the way,
this is a massive move. And the Argentinian
peso still is not recovered from that default.
So can we do better? What do we actually--
what should we be doing when we're hedging this? And the answer
is, again, we have to apply mathematical models
to really try to come up with a replication strategy. So what is the main
features of a model that will help me do this? Well, first I need to model
a credit default, the credit default event. I need to have this in my model. And I need to have
something which says FX has to
move upon default. And then we're going to
construct a complete market. Then we're going to define some
simple dynamics on our exchange rate and on our
defaults, and we're going to try to price for bonds. So how do we do that? Well, what we--
generally, again, how we're going
to use the models, we're going to define an SDE,
like I just defined initially a dS over S of something. And I'm going to solve this
SDE either analytically or numerically. And what's important,
the way we actually use these models
in trading, we're going to look at how the
price of each instrument depends on the
hedging instrument. And that is going to
define my hedge ratio or my replicating strategy. That's really kind
of the main part. It's really hedging and
evaluation and pricing are the same-- right
and left hands. We're really talking
about the same thing. You cannot really
price without hedging. And pricing without hedging
is kind of meaningless, in some sense. Pricing represents the
price of a hedging strategy. OK, how do we-- basic credit
model, how do we model default? Well the standard model
in finance for default is to define the default events. And we say well,
this default event arrives as a discrete event. And it arrives at the time
tau, which is a random time. And we're going to
model the tau, the time, as a Poisson process, which
means that we don't know when it's going to come,
but we know something about the probabilities
of when it's coming. And the Poisson process
has an intensity. The intensity in this case is h. And basically, the
meaning of intensity means the probability of the
default time not the arriving by time capital T is e to
the minus h times capital T minus little t. Little t means now. Let's say we're
saying at time t, we know the default
has not arrived. Here is the probability
the default will not arrive by some time t later. So in our model, we're going
to make a simple assumption. Let's say constant
hazard rate, and we can, since we know the probability
of the default time not arriving after a certain time capital
T-- that's like a cumulative distribution-- we can also find the
probability density the default time happens at
some time capital T, or around some time
epsilon around capital T. It's just the derivative of
the cumulative distribution. And corollary is that
the probability density of the default at
any given time is h, which is essentially the
limit of capital T going to little t. So now in our model,
what happens to FX rate? Well FX rate is going to be
denoted by S. And FX rate right after default would be equal
to FX rate before default times e to the power J.
And J essentially is our kind of
percent devaluation, you can think of it. So it's kind of like
a percent devaluation. So J can go from minus
infinity to infinity. If J is 0, then that means
is there's no devaluation. So you can see the log of
St basically jumps by J. So in a log-normal process,
the log of St is normal, and essentially,
it's just a shift of the normal distribution. OK, so how do we describe this? We define a jump from default
Poisson process with intensity h, as on the board. And our FX dynamics-- and
I apologize for the small script-- is that our d log of S
will have some drift, mu_t dt, and then a jump process J*dN. So this is slightly different
from what you've seen so far. So far you've seen
Brownian motions. This is J*dN. This is now a jump process. Now what we want,
again, we want still our standard no-arbitrage
condition to remain constant. And from before,
we had a condition that expected value of S
of T has to be S of 0 times e to the rf minus rd times T. So
that still has to be the case. And in our case,
we're going to assume that rf and rd are both 0. So in our case, we're
going to ask-- basically zero interest rate
environment to make, again, the model simple. Then we just want the expected
value of S_T to be S_0. So how do we achieve that? Well, we need to
show, essentially, that this mu, the drift, has
to equal to this expression here, h times 1 e
to the J. That's known as the compensator term. And you can think-- you can
imagine this as a formula. Like, if I have a
Poisson process that has a possibility
of jumping up, then in order for that
Poisson process to be on average to be
equal to the initial value, it has to be kind of trending
down most of the time. And then, so that when the
possibility of jump is there, the average of the two can be 0. So that's known as a compensator
term of the Poisson process. OK, so we can go
through and derive how do we get--
what we want to do is, we're going to check that
this form actually does indeed give you that expectation,
does satisfy the condition for the expectation. OK, so again, we start
with dS_t is mu dt. So in our case, it's going to
be h times 1 minus e^j times the indicator function of tau
bigger than T, plus J dN_t. OK, so we're-- not dS_t, sorry. This is d log of S_t. OK, so now what
do we want to do? We want to integrate
this equation. So essentially, what we're going
to do is write integral from 0 to capital T of d of log S_t. We integrate both
sides-- integral from 0 to capital T, h 1 minus e
to the J, tau is bigger than t, times-- here has a dt in here--
times dt plus integral from 0 to t of J dN_t. OK, so then this I just
gives me essentially the log of S_T over S_0. This is just basic calculus. And then here we have-- we
can-- this indicator function just says if tau is
bigger than t, it's 1. If tau is less than
t, then it's 0. That's basically what it is. So I know that
essentially, this is only 1 when t is less than tau. So my integral goes
from 0 to tau now. I can replace this from
an integral from 0 to tau. And I can take out the
indicator function now. Of h, 1 minus e to the J, dt. And then I can say, well,
what if tau is bigger than-- there is also a
possibility here that tau is-- this is tau is less
than capital T. And there's also a
possibility that tau is greater than capital
T. In which case, if tau is greater
than capital T, this integral is there without
any indicator functions. So again, integral from 0
to capital T of h 1 minus e to the J dt times
indicator function, tau being greater
than capital T. So I kind of divided this,
counting both possibilities separately, essentially. And now the second
part, integral from 0 to capital T of J dN_t. Now, N is-- what was N? Well, N of t is
essentially-- it starts out as 0 for t less than tau,
and then becomes 1 for t bigger than tau. So this integral is
just-- J is a constant, so it's just J times N of t. And this is capital T here. And by the way, all these
derivations are posted on the notes, so you don't
necessarily have to worry if you can't-- can I
can move this board up? Not really. So I'm going to
do one more line. I'm going to erase
this top line. So we get to here, and there's
one more step, which is now to actually do the integration. We're going to have
log of S_T over S_0. Well, two things-- now,
if tau is less than T-- so default happened before
capital T-- then what is N_t? N_t is going to be 1. So I can say this equals to h
tau times 1 minus e to the J. This is the first-- this
integral now-- plus J. So this is if tau
is less than T. And then if tau is bigger
than T, then this term is 0. This is a term that's
for tau bigger than T. This is just a constant, so it
just becomes h times capital T, 1 minus e to the J times
indicator function of tau bigger than or equal to T. And we can then
exponentiate both sides. And it becomes-- use the
magic of the blackboard. You can erase. S_T equals S_0 times
the exponential of this. So I have-- this is what my
exchange rate is going to be, essentially, at time capital T.
Now, what was I trying to do? I was trying to do this--
to compute this expectation. With the computed
expectation, now I have to integrate
over the probability distribution of tau. Now remember, probability
distribution of tau is a Poisson process. So we have essentially-- I'll
write it here-- phi of 0, t is just h times
e to the minus ht. That's kind of the
probability density of tau. So now what I need
to do is essentially, the expectation of S_T is just
the integral from 0 to infinity of S of tau, times
phi(0, tau) d tau. So here is my S of T. You
can think of this S of T of tau for time tau. So this is for a given time tau,
I know what my value of S of T is. So I can do this integral. And now we're going to do it. So what is going to
be the first term? So exponential of S_T-- not
exponential, expectation of S_T is going to be--
it's going to be integral from 0 to capital T.
It's going to have two terms. First, I'm going to integrate
from 0 to capital T. And then I'm going to integrate
from capital T to infinity. I'll split this
integral into two parts. And from 0 to capital
T, I have essentially-- h times e to the minus h*tau. And this is my density function. And then I'm going
to plug that in here. So this is for tau
being less than T, so it's basically
this first term. I'm going to divide it by
0 here, to make it easy. So first term is going to be
e to the h*tau times 1 minus e to the J plus J. OK,
and so this is d tau. So this is the first
part from 0 to T. And the second part is
essentially the integral from capital T to
infinity for tau being bigger than capital T.
Now that's actually-- this part here does not depend on tau. It's a constant. So it would be just h,
capital T, 1 minus e to the J times-- what's the
probability of tau being bigger than capital T? That's just the cumulative
probability distribution we saw before, just
e to the minus hT. That's the probability
that tau is bigger than T. OK so it's e to the hT,
1 minus e to the J times e to the minus hT. So I can now simplify
this expression somewhat. You can see that, say, this
term and this term, this term and this term go away. And also this term
and this go away. So I'm left with the integral
from 0 to T of, essentially, h times e to minus h*tau e
to the J times e to the J. So you can think of this as
h times e to the J times e to the minus h*tau
e to the J d tau, plus e to the minus h capital T
times e to the J. So this is-- if I think if h e to the J as
this is the constant in front of tau, this is just a standard
integral of exponential, so this just
becomes, essentially, e to the minus hT e to the J,
minus 1 plus e to the minus hT e to the J. And these two terms are
going to cancel out. And I'm going to have 1. So again, the ratio of e to the
S_T over S_0 just gives you 1. So all this is just
to kind of show you a little bit how you
work with jump processes, and take expectations. It's not-- nothing you
haven't seen in terms of math. It's just slightly different
from Brownian motions. But still the same
idea-- you have dN and you have a compensator term. So this here proves that,
essentially, my drift guess that I started
with, in fact does make my expectation 0. OK, so what have we done so far? We've defined
dynamics for log of S with jump on default,
defined probability density. And now we have to derive
the dynamics of S, price euro bonds, hedge ratios, and so on. OK so log of S dynamics,
we-- I apologize again for the small font-- here
we have the log S dynamics. Applying Ito's lemma, there
is an equivalent-- Ito's lemma you know from Brownian motion,
but there is another one for Poisson processes, as well. And that is-- Ito's lemma
is like the chain rule. So if you know the
process for some log of S, how do you find the
process for S itself? Well, in this case, what's
going to happen is our dS over S is going to be the same drift--
h times 1 minus e to the J, tau is less than T, dT--
sorry, T less than tau-- plus e to the J minus 1, so
J minus 1, dN, dN_t. So that's really the
derivation of the-- that's the final result for
S. Now, how do we get to this? Well maybe I should-- I
can write Ito's lemma. What does it say? Ito's lemma basically
says that if we have dX_t is equal mu dt plus J dN,
then-- and you have a function Y of t, which is
f of X_t, then dY is df/dx mu dT plus f of X_t
plus J minus f of X_t dN_t. So this is the kind
of the term that is kind of an analog of the
convexity term in your Brownian motion Ito's lemma, but
it's now for jump processes. So this f of X_t plus J
and f of X_t-- so what happens, essentially, so you
have some function f, and X_t plus J is what happens
if a jump happens. And X_t is before the
jump, so the effect of the jump on the function. That's what this term is. That's like the convexity term. I think of as a convexity term. I don't know how it's called. Maybe more mathematical
minds here might. So in our case, if you
look at the top equation, our function is just
essentially the exponent. And what happens is when
the function goes up by J is that the exponent goes
e to the J minus e to 0. That's what this term is. OK, so that's how you
write the equation. And now the SDE, solving
the SDE generally means write down what S is. So we have S of t. In our case, it's going
to be S of little t. I'm not going to write it. You have it on the board. I think we're going to get
late, so hurry up a little bit. We're going to the next part,
which is the pricing exercise. So we have two bonds--
zero-coupon, zero-recovery bonds. One pays $1. The other pays one euro. So how are we going
to price this? We have to use our model. We have a model for the FX rate. We have a model for credit. So we price both
bonds in dollars. What is the price in
dollars for each bond? And the ratio of prices
kind of gives you the ratio of the notionals
in your hedge portfolio, if you want to hedge
one against the other. So it's a zero-coupon bond. So I wrote here the
dollar bond price is this. So why do I write it like that? Well, it's a zero-coupon bond. So what a zero-coupon bond
says is at maturity, it pays 1. So we have something
where the payoff at time T is either 1 if tau
is bigger than T, or 0 if tau is less than T. OK, so now what is my price? Well, I know that standard
pricing theory tells me that the price of time little
t is equal to expectation of a price at time big
T. And you can kind of say there is a money
market account. But money market
accounts, in our case, is just 1, because
interest rates are 0. So that's really just the case. That's just true. So now the expectation
of this-- well, that's just equal to the
expectation of an indicator function of tau bigger
than T, which just equals to the probability of tau bigger
than T. So if that's true, we know what that is. That's just the
probability-- that's the cumulative probability
function e to the minus hT. That's why the price
of the bond in dollars has to be e to the minus hT. Euro bond price-- same
idea, except euro bond price in dollars is that. So why is the euro bond
price in dollars like that? Well, the euro bond
price in dollars, again, what is the payoff? Same payoff, except the
payoff is in euros, right? So if I want to do the payoff
of my bond in dollars-- so this, I'm going to call
this the euro bond. But the payoff now, if I
want to do it in dollars, is not really 1. It's 1 times S of T,
and 0 times S of T. That's really my payoff. So then the expectation here
is not just 1, but actually S of T. So now I
have something where I have to take the
expectation of S of T, essentially, at maturity. My bond price in euros is equal
to the expectation of S of T. And what is my
expectation of S of T? Well, it's e to the minus
hT times e to the J. And that's the
expectation of S of T the indicator function of
tau bigger than T, right? So not just-- the expectation
of S of T is S of 0, but the expectation of
S of T times indicator function only in the cases of
tau bigger than capital T. Now, that's not 0. That's basically this-- e to
the minus hT times e to the J. OK, so what can we do? Well, we construct
a-- what we should do is we construct a
portfolio at time equals 0, which is we
sell one dollar bond, and we buy this much
amount here of euro bonds. And the portfolio value at
time equals t equals 0 is 0. Basically, you can take-- so
e to the hT, the first bond, you would get e to the minus hT. And from the second amount
would cost you e to the minus hT to buy. That's how I've chosen
these scaling factors. We start a portfolio
which costs 0. And I should probably--
I'm going to go back here, and going to write
down the notionals, because we lost them. So how many dollar
bonds do we have? We have minus 1. And how many euro
bonds do we have? We have e to the
minus hT times 1 minus e to the J. This is
how many bonds we have. OK, so some time
delta T later, what happens to our bond prices? Well, we know what
the bond prices are. The only thing that changed
was that some time expired. So now instead of capital
T, we have T minus delta T to expiration. So these are the bond
prices if we didn't default. Of course, if we defaulted,
then the bond prices are 0. So obviously, if we defaulted,
since both bond prices are 0, we started with the a
portfolio that's worth 0. If default happened, now we
have a portfolio that's worth 0. So nothing changed, right? So the key part is, OK, now
what if default didn't happen? Would we have the
same price as well? That's what we want to check. And if we have the same price,
both in the case of default and in the case of no default,
then that means we have, essentially, a replicated
portfolio-- a hedged portfolio. OK, so what is the
value of the bonds if default did not happen? Again, we have these
is a dollar bonds here, and these are the euro bonds,
and this is my FX rate. Why did my FX rate move? Well, because default
did not happen, so a jump did not happen. But still I had my drift,
my compensator drift, so FX drifts in the
opposite direction. OK, so the dollar
bonds-- dollar bond that was one bond, minus
1 bond, and the price. So the value of the
dollar bond is just minus e to the minus h T minus delta
T. What about the Euro bonds? Well, the Euro bonds-- here is
the number of bonds we have. This is divided by
S_0, by the way. In our case, S_0 is 1,
so it doesn't matter. Price of each bond,
again, we take that from-- the price of each
bond comes from this formula. And then the FX rate--
multiply by the FX rate. And then when you actually
multiply all these guys out, you end up with,
essentially, the value in dollars of you euro
bond equals, again, the value of your dollar bonds. So we started out with a
portfolio that was worth 0, and then some time
delta T later, it's worth 0 again, both
in the case of default and in the case of no default. So there's no arbitrage. In some sense, not
terribly surprising, because we actually
derived these prices based on the assumption
of no arbitrage. But it's a good check. It kind of tells you,
hey, if I actually follow this model to hedge,
I'm really going to be hedged. And I'm going to be hedged
not just when default occurs, or only if default
does not occur, but I'm hedged in
both situations-- if default occurs and
default does not occur. And you can't really
do that unless you have models that actually
are hybrid models-- that allow you to mix and match--
to basically describe both the current event
and the FX process. So that's kind of
the usefulness. And the hedging strategy
you can see-- it's interesting that the hedging
strategy-- the hedge ratio depends on the credit riskiness. So how much bonds we
bought depends on J. First it depends on h, the
credit riskiness. And it also depends
on J, the jump size. So it really depends. How many bonds you use--
how many Euro bonds you buy to hedge your
dollar bonds, it depends on both the
probability of default and on the jump size. So that's what I mean by it
depends on credit riskiness. It's also dynamic, in the
sense that for a given amount of dollar bonds, the amount
of euro bonds you need to sell is going to vary as FX
and time goes forward. As you can see, if you have
one day before expiration, the hedge ratio of
the two are going to be different than one
year before expiration. So you have to be rebalancing
your portfolio continuously. Which is not--
again, not unusual. If you're hedging an option,
they also have to rebalance. But it's different from, say,
a static replication strategy, where you say, I'm going to
buy x amount of euro bonds, x amount of dollar
bonds, and I won't have to ever worry about it. It's not really the case. Here you're saying, well,
I buy this ratio of bonds, and if default does
not happen, I'm going to have to
readjust my ratio. Because the original
ratio took into account the probability of
default happening. And if default did
not happen, now I have some information--
extra information. And now I have to readjust
my ratio to reflect that. So what happens if
recovery is bigger than 0? And by the way, how much time
do we have-- a quick check? PROFESSOR: We have
till 4 o'clock. STEFAN ANDREEV: OK. So we have about 12
minutes, 10 minutes. OK, Good. So what happens in case the
recovery is bigger than 0? Well, if recovery
is bigger than 0, we can go through this
exercise that we did, again, the pricing exercise,
and see what happens to our bond prices. So let's do this for
dollars and euro bonds, just to give an example of
some of the complexity that can arise when you start making
the model more realistic. Because usually bonds do
not have zero recovery. So then we assume that our
payoff of the zero coupon, zero recovery bonds was 1 if
default doesn't happen, 0 if default happens. Now, it's going to be the
payoff of dollar bond at time T is going to be 1 if
default did not happen, so if tau is bigger than T, and
R if default was less than T. OK, so now when we
price our expectation, it's going to be like this. P of little t would
be just expectation at time little t of-- or
let's say in this case, I'll call expectation
the initial price of 0-- the expectation of P
of capital T, which is equal to expectation
of essentially 1 of tau bigger than T plus
R 1 of tau less than T. Well, what we have
here is essentially-- so you can think we have
this first guy is going to be e to the-- if tau bigger
than T, it's e to the minus hT. And the second guy plus
R times the probability of tau being less than T,
is 1 minus the probability of tau being bigger than T,
so 1 minus e to the minus hT. Which essentially gives you R
plus e to the minus hT times 1 minus R. So that's how you
derive the dollar bond price. And for the euro bond price,
you would do the same thing, except now these will be
multiplied by the FX rate. And now the FX rate-- the
tricky thing about the FX rate is that the FX rate
jumps on default. So it's not going to
be the same number. So in this case, P_T-- this is
for one kind of dollar unit-- it's 1 times S of T
and R times S of T. So now we have P
little T-- this is for euros-- the price at time
0 of the euro bond divided by S_0, that equals to
expected value time 0 of S of T of tau bigger than T plus R S of
T times tau less than T. Well, OK. The first part, S
of T, tau bigger than T, that was like the
zero-coupon bond price. So that's just essentially,
the-- in order to really, I would say, guess
this well, we have to go back to what was
S of T. So if we go back to the equation for S
of T, let me write that. So S of T is S of little t
times e to the hT, 1 minus e to the J plus J times
1 tau bigger than T, this is h tau, tau
less than T, and this is-- if tau is less
than T, and then times e to the hT, 1 minus e
to the J, tau bigger than T. So if default has
not occurred, S of T is S of 0-- in this case, S of
T is S of 0 times this term. And if default has
occurred, then it's S of 0 times this term. So the two terms are the
same, except for the J part. OK, so now when we try
to do this expectation, here we're in the
situation where tau-- where default
has not occurred, so our FX rate is essentially
S_0 times the second term. So we have expectation
of S_0 times-- well, and we're kind of dividing
by S_0, so S_0 drops out. e to the hT times 1
minus e to the J. OK, and when tau bigger than T.
That's the first expectation. And the second one,
the expectation of-- so we put this R times the
expectation of-- now here we have tau is less than T. So
we're going to have our S of T is the first part
only would be true. Second part would
be 1, so that would be the formula-- e to the h tau,
1 minus e to the J plus J times 1, tau less than T. So this e to the J
term that you see here in the euro price, that
comes from this term here. So how do I do this expectation? Well to do this
expectation, again, you have to do an
integral, essentially, over the interval from 0 to
infinity of the probability density. Since tau here is bigger
than T, I'm really integrating from T to infinity. So this here is just a constant. So this first term--
I'll write it here. So you have P_0 over
S_0, the first term would be e to the hT, 1
minus e to the J. And it's going to be
integral from big T to infinity of the partial
differential function, so that is just e
to the minus hT. So this looks like
something we've already done before in
the previous calculation. And then the second
term is R times-- now we're integrating from 0 to tau. So this would be integrating
from 0 to T, e to the h*tau, 1 minus e to the J plus J-- I
can do like this, e to the J. Let's put it like that. And times h times e to
the minus h*tau d tau-- this part being the
distribution function, probability
distribution function. So again, we have
this guy cancels this. And what we're left
with-- first term gives us e to the minus hT e
to the J plus R times h times e to the J times tau. This is, again, an
exponent function. So we have e to the
hT, e to the J minus 1. That's true. Oh, sorry, there's a minus
sign here in front of this. The reason there's
a minus sign is we have minus h e
to the J times tau, and so we have to put
a minus here in front when we do the integral. So there is a minus
here in front. So this thing just basically
reduces to that expression on the board. So that's basically--
so this is how we expand the problem to
having non-zero recoveries. What you could do for your
final paper, if you decide to do a final paper
on this topic, is to extend the model one step
further, and say, in our model, our FX rates jumped, but did
not have any diffusive elements. It was just-- our equation was d
log of S was mu dT plus J dN_t. That was our SDE for log of S. So next step would be
hey, why don't we just add another term, plus sigma dW? So without the
jump, this is just a standard, log-normal process
that you know how to do. Now we add jump, essentially. So you take a
log-normal process. You add a jump process to it. And you repeat
the same things we were going through
so far-- pricing euro bonds, dollar
bonds, and coming up with a replication strategy. This is, for example, a
model that-- we're currently working to implement a model
like that at Morgan Stanley. Our model has non-zero
interest rates. It has dynamic interest rates. So that makes it a
little bit more complex, but overall, it doesn't make
it too much more complex. Having non-zero interest
rates just kind of has an extra drift term. It doesn't really change that
much the mathematics of it. And the reason why we want
to do that is because we want to be able to
price, essentially, the contracts which are credit
contingent, meaning the payoff depends on whether
something has survived or not, whether credit
default has occurred or not. And the payoff is
in units, anything, like foreign currency. A typical example would
be a credit default swap denominated in Brazilian reais. Or that happens-- a
credit default swap on Brazil denominated
in Brazilian reais. Now, common sense is that
when Brazil defaults, Brazilian real is not
going to cost very much. It's not going to be
very valuable, just as we saw on the graph with the
Argentinian peso, which totally devalued, it would
devalue as well. Now Brazil is a very big
economy, strong country. So right now, people are
buying a lot of their bonds. People are investing in it. Still, it has credit risk. And you can buy-- you can
trade the credit risk. You can trade credit
default swaps in dollars. And you can also
enter into contracts that essentially
quanto the credit risk into Brazilian currency itself. And to be able to really
price this, you can do it. We've done it for many years
without having a jump model. But then your hedge
ratios are not very good. And you cannot really
explain the prices you see in the market. So we are essentially
implementing infrastructure to-- we've already
implemented this model or a version of this
model, but we're implementing infrastructure
to kind of really put it in production. As you can see in
this model now, your FX process
depends on credit. So it actually-- calibration
and all these things become a little bit more tricky. Which I don't want to worry
about for your final project, but I think it would be a
very interesting exercise to take something like
that, and basically work out all the steps. It does get a little
bit more complicated, because now you have to--
if you're doing Ito's lemma, you've got to do it both
for diffusive processes and for a jump
process, so you're going to have two terms
in your Ito's lemma. But you've seen them both. They're in your class notes. If you're so inclined,
you can do it. And you can-- once
you solve the model, then you can kind of
check your results. You can actually build a
Monte Carlo simulation, or actually run a
bunch of paths where you simulate both the default
and the diffusive part, and see if your prices
arrived at analytically match with your expectations
computed by Monte Carlo. This will be a good-- it's
always a very good check to see if-- usually, we do this
exercise to check if our Monte Carlo simulations is
correct, because we know that our math is right. But you can also do it to
check the other way around. OK, so in real life,
as we went over-- I mentioned a couple of
times during the lecture-- our models are more complicated. We have stochastic interest
rates, stochastic hazard rates. So currently, we assumed
that our hazard rate, h, is a fixed number. h can be stochastic as well. It can have its
own distribution, and typically that's what we
use in our models-- stochastic effects. So when I say stochastic, both
jump and diffusion processes. And then if you
get really fancy, then you can start putting
correlations between interest rates, FX and hazard rates. So in particular, having
a jump of FX on default naturally introduces
a correlation between credit and FX. When credit occurs, FX devalues. So clearly, there's going
to be a correlation. But there also could be a
correlation between the hazard rates themselves and FX. So it's another
source of correlation. And these correlations would
produce different effects in the market. So basically, you can, if
you have enough data points, you'd be able to
say, well, this model seems like it describes the
market better than that model. Both of them produce
quanto effects, though. And whether we use analytic
solutions or Monte Carlo, they're different approaches to
price derivatives and compute risk. It depends, really, on
how complex your model is. And for certain
markets, you'd rather have a more complex
model that is slower and requires Monte Carlo. And in other places, you
want to have faster, more tractable models that can price
your derivatives analytically. But maybe your
models, they don't have as many features in them. So there's a whole
range of models implemented for various
markets in Morgan Stanley. It's a very big area
of expertise for us. So I think that's it. I think I ran a
little bit over time. I apologize-- five minutes. PROFESSOR: Thank
you very, very much. And we'll thank our
speaker first, I guess. [APPLAUSE] I think there's probably
a question or two that people might have. AUDIENCE: I was wondering
if we could now answer which of the Italian bets was better. STEFAN ANDREEV: Which what? AUDIENCE: Which of the
bets that we initially were considering on the
Italian bonds was better? Could we answer that now? Because we haven't, I think. STEFAN ANDREEV: Yeah. Yeah, let's go back. Which Italian bonds was better? What was that? OK, so let's try to
answer that together. And we can answer it
within our model, right? So in reality, there's
all kinds of factors going into the price. So there's supply
and demand, liquidity in euros, liquidity in dollars. Well, let's say if you're
trying to invest in euros, or trying to invest in
dollars, if I invest in dollars, if a
default happens, I lose essentially-- let's
say the recovery was zero. So I lose all my
money in dollars. I thought I had some
amount of money in dollars. Default occurs. I lost my dollars. Same thing in euros. If I invest in euros, if
default occurs, I lose my euros. So how much do I lose
in a case of euros and in the case of dollars? So if I invested euros, you
say, well, if a default happens, my euros are maybe
not as valuable. So euros are not as
valuable, so I lost my euros, but what I lost was not as
much, because already it's also the value of the lot. Conversely, is we saw, because
of the compensator drift-- remember, if you have a
jump that makes the currency devalue upon
default, the currency will tend to appreciate
if default doesn't happen. Because we want the-- the
expected value of the currency has to be-- that's determined
by interest rates parity, the first thing we talked
about-- the interest rate differential. So that is kind of an
ironclad arbitrage condition that we have to follow. So if you want your FX forward
to really-- the expected value of your FX to remain fixed by
the interest rate differential, and you know that upon default,
your currency would devalue, that means if the
currency does not devalue, it's going to appreciate. Because if a default
does not happen, the currency would appreciate
relatively speaking. So in our case, when
we're buying bonds, we only get paid if
default does not occur. So you would
rather, essentially, buy the bonds in
the currency that's going to relatively
appreciate, essentially. Suppose interest rates
were zero in both cases. You would rather buy the bond
where FX would appreciate it default does not occur. Because if it occurs, you get
nothing in any case, right? But if it doesn't occur,
when you get paid, you want something that would
appreciate versus something that would not. So the dollar,
for example, let's say the dollar doesn't move
versus other currencies when the euro default happens. So you'd rather
get the euro bonds. AUDIENCE: If you want
to estimate recovery, can you use a
bunch-- I mean, not necessarily factors
already in the model, but outside factors like
macroeconomic factors to predict the expected
value of recovery? STEFAN ANDREEV:
Absolutely, yeah. Recovery is something that
we cannot really price, necessarily, because
usually we have bonds. And the bond price-- you can say
we model default, probability of default versus
probability of non-default. But now if you introduce
a second variable, which is the recovery, now you have
essentially both probability of defaults and recovery
amount as variables. And you have only price
as your data point. And you can have
infinitely many solutions. So typically, what
happens is you fix the recovery at something. Now what do we use
to fix the recovery? Well, for sovereign
countries we use 25% and for corporates we use 40%. But these numbers--
everybody knows that they're kind
of just conventions, really, more than anything. We don't really believe that
recovery is really 40% or 25%. It varies a lot by corporation. And there are studies
by credit agencies about how much recoveries--
what are the recoveries for various bonds. And this 25% for
sovereign is based on some study like that
that went over the last 50 years, looked at the
recoveries of sovereigns, of which there are not
that many every year. But if you look at 50
years, there's quite a few. And then they made
some statements-- some recover higher,
some recover lower, but on average,
they recover 25%. If you remember in Greece,
what happened in Greece, how much did bondholders in
Greece get for their bonds? Now, they didn't really
default, technically. Well, they did
default technically, but it was a very
managed process. But they got definitely
less than 25%. I think they got something
on the order of $0.15 on the dollar. So recovery there was, like
I say, was less than 25%. Same for this Argentinian
default I'm talking about, the 2001-- Argentina
is still being sued by creditors trying to
get money back from this. And it's a big
thing in the news. AUDIENCE: [INAUDIBLE] if you
have a claim from Argentina and they fly over, it can be
seized by [INAUDIBLE] funds. STEFAN ANDREEV: Exactly. They tried to do
some settlements. So how much did people recover? Well, it depends who you are. If you took the original deal,
maybe you got $0.20, $0.25, $0.30 on the dollar. Maybe you got $0.20
on the dollar. But now if you hold
out-- if you held out, apparently you got a
little bit more eventually. So it's a little bit
of a fuzzy concept. But it's not something--
you usually make an assumption of what it is. AUDIENCE: And in a
related question, so how would we also
estimate the other constants like the hazard rate and the J? STEFAN ANDREEV: So once
you fix the recovery rate, then you can take
the bond price. And because bond
price theoretically is e to the minus hT, you can
estimate h from the bond price. So if you observe a bond
price in the market, you can say, I'm
going to estimate h. So let's say I'm going to take
some benchmark bonds which I know the price of, and
I'm going to estimate h for each of these bond prices. And I'm going to
create a curve, which is going to be my hazard curve. And then I take another
derivative or bond that I don't know the price of,
and I can use the same curve to price it. So essentially by doing
this, what I'm saying is, I'm going to replicate
my derivative using these benchmark
bonds as much as I can. That's the assumption
that I'm making. AUDIENCE: And how
long [INAUDIBLE] if multiple currencies
are involved, if we are trying to trade with
multiple different currencies, how does the whole model differ? STEFAN ANDREEV: If multiple
currencies are involved, you can-- first you
can-- it becomes tricky. You can say each currency
can devalue X amount. If default happens, you can
have more than one currency being devalued. If you have more
than one currency, if you have more
than two currencies, like three currencies,
there's other identities you have to take care. You really simulate-- if
you have three currencies, there is a triangle
identity that, say, dollar-euros times
euro-yen exchange rate has to equal to
dollar-yen exchange rate. That's kind of an
arbitrage condition. Just like interest rate
FX forward parity-- even stronger in some sense. And so you can basically,
you can write down multiple processes
and price stuff. AUDIENCE: How much
do these equations change when you add in bonds
that are paying coupons? And how do you factor in
duration and all that? STEFAN ANDREEV: Well, you
just-- it's not hard, really. You just, instead of
having this, you just write down all the coupon
payments, when you pay them. And then you just
take an expectation of all the coupon payments. So it's really the same process. You just repeat it
for every coupon. PROFESSOR: Why don't we shut
the formal class over now. But if people have questions
afterwards, we'll [INAUDIBLE]. STEFAN ANDREEV:
Yeah, I'm certainly around to answer questions,
if anybody wants. PROFESSOR: Thank you very much. STEFAN ANDREEV: Thank you.
