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visit MIT OpenCourseWare at ocw.mit.edu. PETER CARR: So, I welcome
comments or questions at any point during this talk. We have an hour and a half,
and I have only 50 slides. So we should be OK. So, this is joint
work with Jiming Yu, who's a colleague of mine in
my group at Morgan Stanley. I head up the global
market modeling team at Morgan Stanley. It's a group of about 70
PhDs, mostly, spread around the world. There's about 30
of us in New York, some in London, quite a few
Budapest, and a few in Beijing. The title of this talk
is Can We Recover? And it's meant as
a triple entendre. So, it could refer to either
the systemic risk arising from the credit crisis, or the
main result in a recent paper by a professor here at MIT named
Steve Ross in the Sloan School, or it could actually be the
academic and practitioner reaction to this
result. So, it's really about two and three. So, it's not about can we
recover from the crisis. There's a professor at Sloan
School named Stephen Ross. And he's very well-known
in academic finance. Your professor was kind
enough to mention that I won Financial Engineer of the Year. And that was two years ago. I was like the 20th winner. He was the second winner. The first winner was another
MIT professor, Bob Merton. So anyway, he wrote a
paper a couple years ago, and it's only now
about to be published. So this is like typical
in academic circles. It takes a long time
for a paper to come out. And this paper is coming
out in Journal of Finance. That's what JF stands for. And Journal of Finance
is the main journal for the academic
finance community. And the title of the paper
is The Recovery Theorem. And that's also the title of
the theorem one in his paper. And that theorem
one we'll go over. And it gives a sufficient set
of conditions under which, what Professor Ross calls
"natural probabilities," at a point in time
can be determined from-- OK mathematically,
from exact knowledge of Arrow-Debreu security
prices, which you probably don't know what they are. But less mathematically,
we'll just say from market
prices of derivatives. OK, so derivatives
you've heard of, I'm sure-- things like options,
for example, on stocks or stock indices, could be on currencies. So, imagine that you
look at Bloomberg. Bloomberg publishes a
whole bunch of prices. And the idea is that you
take this information, and from it you're learning
what the market believes are the probabilities
concerning the future. And so, if the option is
on S&P 500 stock index, then you're learning
from options prices what the market believes
are the likelihoods of various possible
levels for the S&P 500. So, we take this information on
Bloomberg and, truth be told, we use it along with
some assumptions to extract these implied
market probabilities. So, I want to tell you
what those assumptions are. And so, the actual
output of this analysis is a probability
transition matrix. Or, if you do it
in continuous time, you'd call it a-- in
continuous state space, you'd call it a transition
probability density function. So the key word there
is "transition." And what transition
means is you're getting not only the
probabilities going from, say, the current S&P level
to any one of several levels, but even the
probabilities of going from some other level than
we're presently at today to that range of levels. You could say, for
example, the market believes that given
that we're here now with S&P at, say, 1,500,
that the probability of more than doubling is one
half, for example-- which would be really high. But you know, I'm just
picking numbers randomly here. And you can even say that if
S&P were to drop instantaneously to half its level, that
the probability of more than doubling from there
is, say, one-third. So, you can answer
questions like that. That's The output of
this type of thinking. So there'll be three
probability measures that we can be thinking about. And we'll call them
P, Q, and R. And I'd like to tell you what
each of them means. So P stands for physical
probability measures. So the P is for physical. And think of that as
the actual objective reality of future states
for, say, S&P 500. So let's say God knows that,
for example, the probability that S&P is up by the end
of the year is one half. And we, unfortunately, not
being God, don't know that. But let's say the
philosophy is that. There is some sort of
true probability of S&P being up at the end of the year. And let's say I used a half. Maybe it's 60%. If it is 60%, then
the probability of S&P being down at the
end of the year is 40%. And the point is P
is meant to indicate the frequencies with which
S&P 500 in my example takes on various values. Now, there's another
probability measure that people in derivatives spend
a lot of time working with. And that's called risk-neutral
probability measure, and it's often
denoted by a letter Q. So we'll denote it
by Q. And the concept of a risk-neutral
probability measure was also actually proposed
by Steve Ross many years ago. And it's called risk
neutral because when you're working with it, if you
think about how fast prices appreciate over time,
then they grow randomly. But on average, under this
risk-neutral measure Q, the grow at the same rate as
your bank balance would grow. So your bank balance,
let's say, nowadays is growing at best
at the rate of 1%. And when you look at how
fast, historically, stocks have grown, it's actually much
higher, on average, than 1%. It's more like about 9%. So we would call the
difference between 9% and 1%-- we call that 8%
differential risk premium. And let me just pretend
there's no dividends to keep life simple
when I say this. So now, this
risk-neutral measure is kind of a
fictitious probability measure in the
sense that it's not describing the actual
probabilities or frequencies of transitions, it's
more a device, or a tool, or a trick that's handy. And one of its properties that
causes it to earn the name risk-neutral probability measure
is that when you look at how fast, say, S&P grows on
average under this risk-neutral probability measure Q, it would
be growing nowadays at 1%-- so the same as your bank
balance is growing at. So the word
risk-neutral is meant to indicate that the growth
rate under this measure is consistent with investors in
the economy being risk-neutral, meaning that they require
no premium for bearing risk. Now there's a third
probability measure that we're going to be talking
about today that actually you won't find any literature on. And we're going to call it R.
It seems like a natural letter to pick, having already
gone through P and Q. And you can think
of the R as standing for recovered
probability measure. And it's going to be
the probability measure that we get from market prices
as I was talking about earlier. And the operational
meaning of this R measure is it's capturing the market's
beliefs regarding the future. But we allow for the possibility
that the market could be wrong. So we're applying this
to say houses and housing prices in, say,
2005-- it may well be that if we
looked at Bloomberg and got prices of
mortgage-backed securities, that we would extract an
R probability measure that says housing prices
are going to continue on their incessant
upward trajectory. And, you know,
we're going to keep growing at the rate
of, say, 15% a year each year for the next 10
years, or something like that. So, that could be what the
market's beliefs were back in 2005. And we know now that
those beliefs were wrong, if that was what the
market was inferring. So, I want to allow for at least
the theoretical possibility that the market could be wrong. And so, that's why I'm
drawing a distinction, let's say, between the
R probability measure that captures the market's
beliefs and the P probability measure that captures
physical reality. So now, there's a lot of people
in finance who simply cannot accept the possibility that
the market could be wrong. And for those people--
the sort of true believers in market efficiency--
they are free to set R to P every time they
see an R. But I want to allow for the possibility
that what we recover is not physical probabilities,
but simply the market beliefs. And anyway, it's
kind of semantics. It's good semantics if the
probability measure we recover is the one Ross
said we should get. R stands for Ross. So Ross calls the
probability measure that we recover-- he calls them
natural probability measures. And well, let's
say, that suggests that the risk-neutral
probability measures are unnatural, which
I think is fair actually. Because when you hear
the word probability, you tend to think
about frequencies with which events occur. And the risk-neutral
probability measures do not give you the frequencies
with which events occur. What the risk-neutral
probability measures give you is instead prices of
so-called Arrow-Debreu securities. So, let me give you a
sense of what that means. So say I tell you that the
risk-neutral probability of S&P 500 being up at the
end of the year is 40%. Then how should
you interpret that? Well, you should simply
interpret it as this. Imagine that you
can agree now to buy a security that pays $1
just if S&P 500 is up at the end of the year. And usually when you
and I buy things, we buy them in a spot market. So we pay now for things. But sometimes your
credit is good, and you can actually
agree now to pay later. So, we're going to be thinking
that you're agreeing now to pay later some
fixed amount in return for the security that's going
to pay $1 just if S&P 500 is up at the end of the year. And if I tell you that the
risk-neutral probability of S&P 500 being up by the end
of the year is $0.40, what that means financially is
that you agree now to pay $0.40 at the end of the
year for the security. So, you can imagine there'd
be another security that pays $1 just if S&P 500 is
down by the end of the year. And the only possible price
that that security could have in an arbitrage-free
world would be $0.60. Because if you were to
buy both securities, then you get paid a
total of $0.40 and $0.60. So you're agreeing now to pay
$1 at the end of the year. And then having both
securities, either S&P is up, or S&P is down. And so, you collect $1 from
one of them and not the other. So if, for example, the one
paying if S&P is up cost $0.40, while the one paying if S&P
is down only cost $0.50, then there would
be an arbitrage, which we would buy
both securities, agree now to pay $0.90. And then get $1 for sure
at the end of the period. So we'd be up $0.10 by
the end of the year. Question-- AUDIENCE: These are
similar to digital options? PETER CARR: Yes. It's more than similar. They are digital options. Yeah. So, that's right. So, that's another term,
which I'll actually use on the next slide. So, that's exactly right. So, digital options is
just too good a term. So economists, in order to
obfuscate and look smart, call them Arrow-Debreu
securities. So, continuing with
the obfuscation, I want to tell you about a world
with a representative agent. So, economists
are fond of trying to formally model the market. You read the newspaper. Every day, you'll read
something like market thought that stocks were no
longer a good investment. So there was a sell-off. Market is a nice,
short word to capture what people are thinking. And so economists, rather
than say the market, will say there's a world where
the representative agent-- So this representative agent
is a fictitious investor who has all the mathematical
properties that we give an investor, such
as utility, function, and an endowment, and so on. And what makes this
particular investor a representative agent is
that this agent sort of finds that current
prices are such that it's optimal to hold exactly what's
available in the amount that is available. So if what's on offer is,
let's say, some Google shares, and some Apple shares,
and some IBM shares. And if we take the total
market cap of Google, total market cap of Apple,
total market cap of IBM, and, let's say, Apple's biggest. I don't actually know whether
Google's bigger or IBM, but let's say it's
Google, and then IBM. So let's just say Apple's
biggest, then Google, then IBM. Well, this investor
would actually find that it's optimal
for him to have most of his money in
Apple, second most of his money in Google,
third most amount of his money in IBM, that's
the representative agent. So, he's acting in the way
the whole economy is acting. Well, I've been working in
Wall Street now since 1996. I have yet to hear
a trader tell me about a representative agent. Anyway, so although I
understand what the words mean, and even the math, I wanted to
present this material in a way that, let's say, at least
quantitative traders could understand it. So I tried to get away
from representative agents and present these
ideas in the language that at least quants on Wall
Street are familiar with. So, I won't be talking about
a representative agent, and I will be talking instead
about something that's probably not too familiar to you, but
at least quants have heard of. And that would be something
called numeraire portfolio. And it also goes by other names. Another name is growth
optimal portfolio. And it even has a
third name, which is called natural numeraire. And these are three
different phrases that all describe the
same mathematical object. And this mathematical object
is a portfolio-- and more precisely, it's the
value of a portfolio that has some nice properties. So the growth optimal
portfolio indicates one of its properties. This portfolio has a
very nice property, which is that in the
long run-- meaning over an infinite horizon-- the
growth rate of this portfolio is, first of all, random. But second, if you take the
mean of that random growth rate, that mean is actually
the largest possible among all portfolios. So, starting with Kelly in
1956, this particular portfolio with the largest mean growth
rate over an infinite horizon receives a lot of attention. It's actually quite humorous,
some of this attention that it's received. So, Kelly was a physicist
who worked at Bell Labs. And he was actually a colleague
of Shannon's at Bell Labs. So Shannon did his
seminal work at Bell Labs, but actually came
here after that. And his ideas really caught
on-- and especially, I'd say, started the field of
information science, we'll call it-- whatever. But Kelly was applying
these ideas to finance. And certain financial economists
were less than enthused about the application
information of theory to finance. So, in particular, there was a
financial economist here named Paul Samuelson who
championed, I guess, the opposition to this
Kelly criterion it's called. And so, I'll just tell
you a short story. AUDIENCE: Excuse me. PETER CARR: Yeah. AUDIENCE: If I could
just interject-- PETER CARR: Yeah, sure. AUDIENCE: We had mentioned
in an earlier class the book Fortune's Formula. And this book goes into a lot
of background and storytelling about this whole
era and exchanges. PETER CARR: That's true. It's a fantastic book. I read it. I loved it. Especially if you're at
MIT, you should definitely read this book. It talks about a lot of MIT
professors, some of whom are still here, like Bob Merton. It's a quick, easy read. You don't even have to have
a background in finance to really enjoy it. So you can read about
the story I'm going to tell you now in that book. So the story is Samuelson
grew a little tired, I guess, with trying to explain to these
dumb information theorists that this Kelly criterion
was not so great. So he published an
article in a journal called Journal of
Banking and Finance-- that's actually a finance
journal-- where he explained why it wasn't necessarily
such a good idea to hold this portfolio. And in this article, every word
he used was of one syllable, except the very last
word of the article, where he managed to say
that he has-- I can't even do it in one syllable--
OK, so just ignore my multi-syllabic words. But anyway, he
says, I have managed to write an article with all
words with just one syllable, except for this last syllable--
OK, I lost it-- sorry. But anyway, the last
word in his thing was syllable itself, which is
multi-syllabic-- or whatever. So anyway, it was
kind of insane. So, let's move on. So this talk-- it has six parts. And we have an hour to go. So let's say we'll try to
spend 10 minutes on each. AUDIENCE: [INAUDIBLE] PETER CARR: Yes. Well, that's a good question. So, it does have
risk, first of all. It does have a lot of risk. It's not the riskiest, though. So some risk does not carry
with it expected return. And so that's why it's not
the riskiest-- but it's risky. So Samuelson's objections
were precisely what you're getting at, that this
is a fairly risky strategy. So, I'm glad you
brought that up. OK. So there's six
parts to the talk. I'm going to go over what
Arrow-Debreu security prices are-- so again, they're
digital options prices-- and their connection
to market beliefs. I'll talk about this
Ross recovery theorem. So in Ross's paper, which
you can get on SSRN, he does everything in a setting
that's called finite state Markov chains. And so that's
mathematically simpler than what we use in practice. And I totally agree that
when you try and introduce something, you do it in the
simplest mathematical setting. So now that he's
done that, I wanted to do it in a more
familiar setting, which is a diffusion setting. A diffusion has an uncountably
infinite number of states. And I still want
to keep things as simple as possible while
going beyond finite state Markov chains. So I work in a univariate
diffusion setting. So there's only one
source of uncertainty, which is the same as in Ross. And our technique is
to get these results. It's based on something
called change of numeraire. So numeraire is a
technical term, actually, that describes an asset whose
value is always positive. So there are securities whose
values can have either sign. So, swaps are a
classical example. So a swap is a security which
at inception has zero value, actually. And then the moment after
inception, the world changes, and the swap value
either becomes positive or becomes negative. So a swap would not
be eligible to be a numeraire because
of that property that its value is real. On the other hand,
if you take a stock, its price is always
positive-- well, that's debatable actually-- so
let's say let's not do stock. Let's do a treasury bond. A treasury bond-- US
Treasury bond-- its price is always positive. The reason I want to
shy away from stocks is because we take Lehman
Brothers stock, for example. It's price was positive,
then became zero. And actually, because
Lehman's price became zero, Lehman's share
you could not be a numeraire. So when I say that the numeraire
value has to be positive, I mean strictly positive. And so anyway, there's
this literature about how to change
numeraire, how to go from one asset
with positive value to another asset
with positive value. And it's useful
for understanding how this Ross recovery works. So, we apply it when we have
a so-called time-homogeneous diffusion-- and I'll tell
you what that means-- over a bounded state space. So bounded state space
means that the set of values that the
diffusion can take is in some finite interval. So if you're thinking about the
uncertainty being, for example, S&P 500, then the natural
lower bound for S&P 500 would be zero. And you have to accept that
there's a finite upper bound in order to apply our results. Now you know, personally,
I have no problem saying the S&P 500 is
bounded above by 20 trillion. OK, but some economists
have actually said this is ridiculous. and challenged my
work, and stuff like that for that assumption. So, because of those challenges. I have actually been
trying to extend our work to an unbounded
state space, where, let's say, the largest possible value
for S&P 500 would be infinity. And I've found, actually,
that it's not that easy. And so sometimes, I can make it
work, and sometimes I cannot. So, when we get there, I'll
explain some examples that work and some examples that don't. So this last section is kind of
incomplete, this sixth section. And so, basically, I've
got examples that fail, examples that succeed. But I don't have
a general theory. So there'll be
different assumptions in different parts of the talk. But within a section,
there's only one set of assumptions operating, AUDIENCE: Excuse me. PETER CARR: Yeah. AUDIENCE: [INAUDIBLE] the value
of anything is [INAUDIBLE]. [INTERPOSING VOICES] PETER CARR: That's
been my response too. So the universe is bounded. And it's growing,
but it's bounded. So, I agree. You know, I'm on
your side on this. I'm just telling you
what I've been told. Yeah. So, I'm working on it anyway,
just so they can shut up. But, anyway-- AUDIENCE: Actually,
I have some comments on the issue of the numeraire. You'll tell me
how connected this is-- but with the
Kelly criterion, one of the origins of that
is if you have a gambling opportunity where
it's favorable, how much of your bankroll
should you bet on that gamble? And basically, the
Kelly criterion tells you what proportion
of your bankroll you should invest at all times. You should never bet everything. And if you do bet everything,
you lose everything, and you're done. So, the issue with the
numeraire portfolio and never being able to go down
to zero, in the sense that you can never go bankrupt. And so, assumptions of
being able to always rebalance your portfolio-- PETER CARR: So, just
give you a flavor of what this numeraire
portfolio is-- you're betting a constant
fraction of your wealth in every security. So let's just keep it simple. There's only two securities. One is risky, and
the other's riskless. And so you might be betting
putting 40% of your wealth in the risky one, and 60%
then in the riskless one. And that's when you start. So you have $100, and you
put $40 in the risky one, and $60 in the riskless one. And then, time moves forward. And let's say the price
of the risky one changes. Then when you revalue
using the new price, it's unlikely that 40% of your
wealth is in the risky one. So in fact, if the price
went up of the risky one, you'll have more than 40% of
your wealth in that risky one. So you need to sell
some of that risky one. And then the money you get,
you put into the riskless one. And so, every time
the price changes, you need to trade,
theoretically, in order to maintain a constant fraction
of 40% of your wealth invested in this risky asset. So we assume zero transactions
cost when we do this analysis. Because there are positive
transactions cost. One should take
that into account. And there is literature
on how to do that. So, I won't be formally
entertaining transactions cost in this talk. There's work here at MIT,
actually, on doing that. For the question of
how should you invest, it feels like it's a
complication that won't change anything qualitative about--
it'd definitely change how frequently you trade,
but it wouldn't, let's say, it's unclear how it would
change your initial investment across bets. So, let's begin with part one. So we have the digital options,
or also called binary options. That's another term. And they trade, actually, in FX
markets-- so foreign exchange. And they pay one unit
of some currencies, so say dollar-- If
an event comes true. So it might be that you're
looking at dollar/euro. And if by the end of the
year, dollar/euro exceeds 2, then you get $1. Otherwise, you get $0. So there would be a
price in the FX markets. And it would be a spot
price typically-- so meaning you have to pay now for it. Let's let A, for arrow, be the
price today of such a security. And the subscripts
on A are j given i. So, the idea is that you
can think of yourself as in a finite-state setting. There's various
discrete levels of say, dollar/euro that we have
that can be possible today. And there's also various
discrete levels for dollar/euro by the end of the year. And i indicates
the state we're in. So maybe dollar euro is
$2 per euro right now. And j indicates the
state we can go to. Maybe we can go to $3 per euro. So in my example, A_(3|2) would
be the price of an Arrow-Debreu security, given that the current
dollar euro exchange rate is $2 per euro, and it pays $1 just if
dollar/euro transitions from $2 per euro to $3 per euro. So the idea is we
have discrete states. And let's say these
are values that are possible at the
end of the year. And the example I just
went through-- you're getting $1 just if it's $3 per
euro at the end of the year. So the height of that
vertical line is one. Now, I'll just
comment that this is a slightly exotic option,
in the sense that-- let's call it exotic. It's slightly exotic. So in contrast with exotics,
there's this term "vanilla." OK, and it actually indicates
a flavor of ice cream. So, we have this
terminology which you get used to after awhile. And you can't understand when
you talk to a man on the street why they don't understand
what a vanilla option is. So a vanilla option is a
payoff that looks like this-- so it's a hockey stick payoff. And that's the payoff
from a call option. And it turns that there is a
portfolio involving options at three different
strikes that can perfectly replicate the payoff to
this Arrow-Debreu security. And so, here is a payoff from
a single option struck at two. And I'll just say that if
I had changed the strike to, say, be three, then
it would look like that. Now, you can combine
options in your portfolio. So you could, for example,
buy a call struck at two. And then you can furthermore
sell two calls struck at three. So if you sell, on top of that,
two calls struck at three, you end up creating a
portfolio that goes like this. And so, they can go
negative in value. So if you not only buy
one call struck at two, sell two calls struck at
three, but furthermore, buy one call struck
at four, then you end up with this payoff,
which the payoff is called a butterfly spread payoff. Because the picture is meant
to remind you of a butterfly. And notice that if the only
possible values for the FX rate were $1 per euro or $2 per
euro, or $3 or $4, or $5, if that were the world. Then notice that when you
formed that portfolio, the only positive payoff
you can get from it is $1 just if dollar euros at 3. You can synthesize a
Arrow-Debreu security using a butterfly spread. So, this was pointed
out many years ago. So even if the FX
market were, let's say, not directly giving us the
prices of digital options, we could from vanilla options
extract the implicit price of a digital. And what you would learn
from vanilla options is what the market is charging
for the digital, given that, let's say, we're
presently at $2 per euro. And what you would not
learn from these options prices is what the
price of the security will be should we today
have the exchange rate change to some other value. However, you can
make assumptions as have what the
options prices will be were today's exchange
rate different. So, that's commonly
done in practice. So a common assumption,
for example, is that the probability
of transitioning from two to three-- so moving
up by half-- so you're moving up by half of two to
three-- is the same if you were at any other level. So for example, if
you were at four, then the probability
of going to six would be whatever
the probability is of going from two to three. Because if you're at four,
the probability of going up by half of four to six--
that's the assumption. OK, so that's
called sticky delta, and it's a common assumption. So if you make that
assumption, then you can take the information
at just today's level. And like, let's say, you know
all the digitals from two, and you can make
that assumption. Let's say the probability
of a given percentage change is invariant to
the starting level. And then you can,
from that, figure out what the probability of going
from four-- a different level than we're at today-- is to
all these different levels. So you can go from a
vector bit of information that the market is
giving you to a matrix. And that matrix is
called transition matrix. And so, we're going
to, in this talk, assume that somebody's
made such an assumption. And so, you actually
know this matrix. So you actually know,
as a starting point, what the prices are of these
Arrow-Debreu securities or binary options
starting from any level and going to any level. I think in order to get
through my whole talk, I'm going to skip these slides. Because they're
kind of like just being very precise about
what some terms mean that aren't going
to be that important for the overall story. So OK, let's go to this slide. So we think of there being just
a single source of uncertainty X, which could be dollar/euro. And we imagine that we have this
matrix of Arrow-Debreu security prices. We know every number
in this matrix. And we ask what does the market
believe about transitions from any place to any place? What does the market
believe is the frequency of these transitions? Now, suppose that
the number that's indicating the price of the
Arrow-Debreu security going from two to three-- suppose
that number is, say, 0.1. Now what does it mean? It just means that you pay $0.10
today for security paying $1 just if you go
from two to three. That's all it means. Now you can ask what is the
frequency with which you go from two to three? It need not be 10%. There's at least two reasons
why the $0.10 price could differ from the probability of going
from where you are to where you get paid. One such reason is simply
time value of money. So if you were to buy all
these Arrow-Debreu securities, the one paying off--
one for every state, you'll find that the
total cost is less than 1, even though the
payoff, for sure, is one from the portfolio. And that's simply because
of the time value of money. So when you put $1
in the bank today, you actually get
more than $1 back when you pull out at
the end of the year. And if you do the
inverse problem-- how much do you have
to put in the bank today in order to have $1
at the end of the year? It might be $0.95. So that's called
time value of money. And so, just the fact
that you have to pay now for the Arrow-Debreu security. And you only get paid off
at the end of the year. That causes this price
of $0.10 to be lower. So that's just
discounting for time. The interest rates are positive. So that's one effect. Now there's another effect,
which is called risk aversion. So risk aversion is the thought
that even if the interest rate was 0, to abstract away from
the effect I just described, that it still may be the
case that a $0.10 price paid for an Arrow-Debreu security
transitioning from two to three is different from the
probability of such a transition, the real-world
probability of such a transition. Because, for example, it may
be quite desirable to get money in that state, in which case
$0.10 is over the real-world probability. Or it could be the opposite that
maybe it's not desirable to get money in that state, in
which case $0.10 is under the real-world probability. So give you a more
concrete example-- let's say something that is
maybe a little closer to home is-- let's say this
is S&P 500, and I know the values are very
different than the numbers I've indicated here-- but
let's just forget about the actual numbers. So the point is let's suppose
that it's equally likely, in terms of true probabilities,
to go from two to three as it is to go from two to one. So we have two
Arrow-Debreu securities. One struck at one. The other struck at three. And I'm telling you
that it's equally likely that you go up by one as
it is to go down by one. Now you can ask the
question does it necessarily mean that the prices of
these securities that pay $1 are the same? And the answer is
no, not necessarily. And actually, the sort
of standard thinking in financial academic
circles is that for S&P 500, it would cost more to buy this
Arrow-Debreu security than it would cost to buy that
one, even though everyone agrees that it's equally likely
to get paid from each of them. And the reason that it's
thought to cost more to buy this one
than it is to buy that one is because this
one has an insurance value. So the thinking is that on
average, people are long the stocks in the stock
market, and that that means that they're really
upset when the stocks fall. And so they really
like this one that ends up paying should the stock
market fall from two to one, whereas this one, while
it's nice to get money, let's say you're already
fairly wealthy from the fact that you're owning stocks
and the stock market went up. So you'll pay a positive
amount for this security, but not as much as
you pay for this one. So that's called risk aversion. So what we want to do is
go from the prices that are contaminated, let's say,
by time value of money effects and by risk aversion effects. And we want to cleanse
them of that contamination and try to extract what
the market believes are the frequencies
of the future states. So I'll tell you that this
was thought to be impossible before the Ross
paper, and in fact, without making assumptions,
it is impossible. So all Ross did is make
some assumptions that are thought to be fairly
mild by some, including me. And so he essentially,
in essence, showed the power of
some assumptions. That's one way of
thinking about it. So again, let's denote by
R the recovered probability measure which will
tell us the market beliefs about the
frequencies future state. And we don't know
R when we start. What we do know is these
Arrow-Debreu security prices, I'm assuming. And we'll denote
those by A for Arrow. So what Ross's paper does
is it says, you know A. And if you're willing to make
the following assumptions, then you'll know R. So
what are the assumptions? Well, before I tell
you assumptions, I have to tell you some
terminology so that you understand the assumptions. So he'll work with a pricing
matrix A, which we've actually been going through. So that's the Arrow-Debreu
security prices index by starting state
and final state, which we'll call x is starting
state, y is final state. Then there'll be
the desired output from this analysis, which
he calls natural probability transition matrix. So these are the markets
beliefs for every starting value x and for every final value y. And then there'll be something
called pricing kernel, which is literally the ratio
of these Arrow-Debreu security prices to these output
natural probabilities. So, if you want to get
an understanding of what this pricing kernel
is, you can think of it as an attempt to
capture the effects from time value of money
and from risk aversion. So think of it as
a normalization. You start with A,
and A is actually affected by three things. It's affected by the
unknown real world probabilities-- or at least
markets beliefs of them. A is also affected
by a second thing, which is time value of money. And A is affected by a third
thing, which is risk aversion. So if we take A and
divide by P, then we're normalizing for the first
effect, the frequencies. And so we're left with
just the combined effect from time value of money
and from risk aversion. And so, let's say, if interest
rate were zero and people were risk neutral, then we would
actually expect A to equal P. And so this ratio
would be just constant. So Ross talks about a world
with a representative investor. And essentially, this is an
assumption-- this equation you're seeing here. It's an assumption on
the form that a function of two variables takes. So phi, first of all,
is a positive function. So phi is positive, as opposed
to-- so phi cannot take negative values because
both A and P are positive. And phi is a function of
two variables-- x and y. And what this assumption
is doing is it's saying, well, let's put structure
on this function phi because it'll help us to find
it if we put the structure. So this is the first key
assumption actually-- that the function of two
variables x and y actually has the form on the right,
which, for a moment, just ignore the
delta for a moment. And then you can see that
what you have on the right if you ignore delta, if
you think of delta as one, is you have a function of y. And then you have the
same function of x. So it's written in a convoluted
way with this U prime, and c, and all that stuff. But if delta's one, then you
have a fraction whose numerator is a function of y
and whose denominator is the same function, but of x. In essence, what that
does is it reduces the dimensionality of the thing
we're searching for by a lot. So we started by searching for
a function phi of two variables. And we, by this assumption,
reduced the search to a function of
one variable, which is, say, the function
in the numerator, which is the same as the function
in the denominator. So, now let's bring back delta. And delta's a scalar here,
and it's a positive scalar. And so we need to
search for that as well. So in the end, we
reduce the search to a function of one
variable and a scalar delta. So the economic meaning
of, first of all, the function of
one variable is-- it's called marginal utility. And it's meant to indicate
how much happiness you get from each additional
unit of consumption. So it's the typical--
what we think it looks like as a function of
c-- U prime as a function of c is thought to typically
look like that. So it's positive, meaning
every unit of consumption makes you happy. And it's actually
declining, meaning the first unit of consumption
makes you real happy. Then the next unit
of consumption still brings some happiness,
but not as much, and so on. So that's the kind of
function we're looking for. U prime as a function of c. He won't actually find U
prime as a function of c. He'll find the composition of
U prime with a function c of y. Keep that in mind. Then, there's that delta. And that's, again,
a positive scalar. And it's meant to capture
time value of money. And so, that's like the y is the
state at the end of the period. And x is the state at the
beginning of the period. And so, that's why delta's
associated with the numerator, not the denominator. So delta would be
a number like 0.9. And that indicates how
much discount you give to, let's say happiness received
in the future, rather than now. Now, here's a quote from Ross's
paper that is his Theorem 1. That's called the
recovery theorem. And the only thing is
I changed the letters to conform with the
letters I'm using, rather than the ones he used. And that's because
his choice of letters is completely unnatural
to me and most people. So I don't even want to
tell you what he used. So anyway, whereas I tried to
choose letters that make sense. So I used A for Debreu-- So anyway, he says,
you have a world with a representative agent. So that's actually
this restriction that we talked about
on the last slide. And then he says, if the
pricing matrix-- which is the Arrow-Debreu security
prices-- is positive-- which means that
all entries in it are strictly above
zero-- or irreducible-- which means that some
entries have zeros, with the rest being
positive, and there's some structure, which we
need not get into where the zeros are-- then there
exists a unique solution of the problem of finding
P, which P is actually market beliefs. And I've been
calling that R often. So anyway, I slipped a
bit there and called it P. So anyway, that's market
beliefs about the frequencies of future states. He'll also get as
an output the delta, which is the positive scalar
telling you the market's time value of money. And finally, this
pricing kernel phi, which is the ratio
of A to P. So, what you're supposed to realize,
even though he didn't say it, is that as a result-- well, OK. So he did say it actually. You're finding P. I think
that's the main thing. He's actually saying, if
you make these assumptions, surprisingly, there's only one
possible real world or market beliefs that are
consistent with the data and the assumptions made. To give you a sense of what
the importance of this result is-- so prior to
his paper-- I mean, people have been
interested in trying to infer from market prices
what the market believes. But they always
thought that you had to supply some parameters that
capture market risk aversion. So for example,
common approach is to assume that you have a
representative investor. And that they have
a particular type of utility function called
constant relative risk aversion. And there's a parameter
in that utility function. And you had to specify the
numerical value that parameter takes before you could learn the
market's beliefs from prices. And no one ever felt
very comfortable specifying that parameter. So what Ross essentially did
is he managed to essentially do the identification
non-parametrically, where you don't have to
supply any parameters. And so you essentially just
have to buy his assumptions. You don't have to do any work to
actually go from market prices to market's beliefs. OK, so let's skip these remarks. Yeah. AUDIENCE: Can you
elaborate on the fact that risk aversion
does enter in? PETER CARR: Yeah. So the exact
statement is you don't have to supply a
parameter that describes the amount of the
market's risk aversion. Rather you have to
accept this assumption-- and I'll show you--
this assumption about the structure of phi. OK, so if you just accept that
this function of two variables doesn't have the full
amount of degrees of freedom that an arbitrary function
two variables has, it has a reduced
number of degrees of freedom implicit on
the right-hand side. So remembering
that x is actually just a vector of finite
length and so is y, then think of the left-hand side
as having degrees of freedom n squared. And on the right-hand
side, you're looking for the numerator
function is just a vector of length n. And the denominator function
is the same function, so the same vector. And then there's
also this delta. So let's say on
the left-hand side, you're describing something
that without restriction is of order n squared. So let's say n is 10, so it
has 100 degrees of freedom. And on the right-hand
side, you're describing a vector of length
10 along with a scalar-- so 11 degrees of freedom. So you have to
accept that you're willing to before you
place any restriction, it's 100 degrees of freedom. Now you make your restriction. It's 11. You have to accept that. And if you do, then he'll
tell you the 11 entries. That's it. So you don't have
to supply anything. So I haven't told you
how we'll find them. That's probably what you're
asking-- how the hell will you get to 11? OK, so I haven't shown you that. Yes? AUDIENCE: Just really
quickly, the c change as a function of
time and spot price? PETER CARR: c is not
a function of time, to answer your question. And then, the argument
of c could be a price. It's allowed to be a price. OK? So, that's how you
should think of it. So there's a lot of time
homogeneity in everything he does here. So he'll never let anything
depend on time, actually, to answer your question. So, I still haven't
shown you how he did it. He uses Perron-Frobenius
theorem. I don't actually have
slides on how you actually calculate the 11 entries. So I think I just have to
refer you to the paper. But he relies on
something called Perron-Frobenius theorem. And I'm going to show
you how we-- my co-author and I-- actually
calculate the analog of that 11-dimensional unknown. So we're going to work in
a continuous setting, where instead of looking for
a vector and a scalar, we're going to look for
a function, and a scalar, and a function of one variable. So you'll get a sense of
how to do it from ours. And essentially, if you
discretize what we do, you'll get what he did. Let's forget these remarks,
and let's forget these. And so now, we'll
get into some theory about changing numeraire. So this is a backdrop to how
my co-author and I proceed. So again, a numeraire is
a portfolio whose value is always strictly positive. And there is a
well-developed theory in derivatives pricing about
how to change the numeraire. We're going to use that theory
to understand what Ross did. So we start with an economy
with a so-called money market account. And so that's a
theoretical construct that's pretty familiar to most
of us, and it's a bank account. So we're going to be working
now in continuous time. So imagine that time, which is
continuous, is on this axis. And then we're
sitting here today, and we put some
money into the bank. And being poor,
we only put $1 in. So then we ask, looking forward,
how will this money in our bank change? Well, they do still pay
a positive interest rate, and it's awfully small,
but it's positive. And so it'll go up. And they change
the rate actually. So now, maybe it's 0.5%,
but next week, Chase might decide to give you 1%, in
which case it goes up faster. And then they might the
week after give you 2%, it goes up faster. Then they might go back to 0.5%. So that's one possible path
for your money market account balance. And we don't know the future. We know how much we're getting
over this first little bit of time. But they could
actually decide to pay less over the second period,
and then the third, or something like that. OK, so it's increasing
and it's random. So that's the money
market account balance. It's considered as an
increasing, random process. And actually, there's
nothing in the math that requires it to be increasing
if some really cheap bank-- like Bank of America
tried this actually-- charge a negative rate. Then it would actually go
down with a negative rate. But it wouldn't go negative. So it's still counts
as a numeraire. So anyway, that's
allowed, as an aside. OK, so we've got this
money market account. So the growth rate is called
r, and that's just real-valued. And then we also
have risky asset. So we'll have a total
of n risky assets. And then we're
going to say there's no arbitrage between the n risky
assets and the one money market account. The idea is that we look
at Bloomberg's prices for these n plus 1 assets,
we're able to extract the Arrow-Debreu
security prices. That's the idea. What I'm assuming is that
what we're extracting is consistent with the idea
that the uncertainty that's driving everything here
is a diffusion, meaning that the uncertainty has sample
paths that are continuous, but they're allowed
to be fairly jagged. So diffusions actually
have continuous but non-differentiable sample paths. And we're going to assume that. So this is a common assumption. This basically got
its start here at MIT. And diffusions were first
used in a finance context back in 1965 when both
Samuelson and McKean were here. So McKean is a probabilist. He's now at NYU where I
teach, and he's still active. And diffusions are widely used. So they really got
a big boost in 1973 when Black-Scholes and
Merton, who were all here, used the diffusion to describe
the price of a stock underlying an option. And since then, they've
just been used extensively in finance. So Merton, who's
here, really, I'd say, pioneered the use
of them in finance. So there's this
uncertainty X is probably mysterious to you,
hence the name X. So it's like, you
get to choose what it is, is kind of the idea. So this is theory. And it's not trying to be
overly specific so that you can apply it in different contexts. But you'd like to know at
least some examples, I'm sure. So one example would be X
is the level of S&P 500. A different example would be X
is actually an interest rate. So let's say the
benchmark 30 year yield. X could instead be a
shorter-term interest rate, something called OIS--
overnight index swap-- is a possible choice for x. When I apply Ross's
stuff, that's how I choose X, as a
short-term interest rate. In general, let's say I
developed a theory that says the short-rate
of some function of X. And when I actually
apply it, the function is the identity map. The mathematics says that
if there's no arbitrage, then there exists-- as
we're assuming-- then there exists this so-called
risk-neutral probability measure that I talked about
earlier and denoted by Q. It's related but not equal
to the Arrow-Debreu security prices. So if you were to just
imagine that instead of buying these Arrow-Debreu
security prices in a spot market, if you instead bought
them in a forward market where you actually
pay when they mature, then those Arrow-Debreu security
prices in the forward market would be Q. So Q and
A are really close. So the measure A need
not integrate to one, and that's just due to
the time value of money, and that's because you're
paying in the spot market. If you're actually paying
in the forward market, then you don't have to worry
about time value of money. And so then, the measure
Q does integrate to one. So that's why we call it
a probability measure. Under this
probability measure Q, the expected return on all
assets is the risk-free rate. So that's what that actually
says, although you're probably not seeing that this is
literally the expected return-- well more precisely,
it's expected price change. So the expected
price change is-- what that means is
expected price change is the risk-free rate
times the price. That's what that says. So if you divide both
sides by the spot price when it's positive, then
you'll get the expected return is equal to risk-free rate. And we're doing things
in continuous time here. So we're working
with diffusions. And you may or may not
have been introduced to diffusions at this stage
in your mathematical career. But mathematically, one
way to describe diffusion is via the
infinitesimal generator. So this is a
differential operator that's first order in time,
second order in space. And let's just say this is
formally how mathematicians think about this type of thing. What I've drawn here is a
single sample path of diffusion. There's definitely possibility
of other sample paths. These actually are an
infinite number of paths. But they're all continuous
and nowhere differentiable. I want to just kind
of give you a flavor of how you change numeraires. So we started with the
numeraire being the money market account-- this guy. And the idea is
we're going to switch to a different numeraire. What we're mainly
interested in figuring out is what are the drifts of assets
when we measure their values in a different numeraire. So I've kind of given you a
sense of what this is about. So you could hold IBM, and
every time you get a gain, you could put that gain in
your local bank-- Chase-- and see how fast your
bank balance grows as you're putting all your
gains in IBM in the bank. And you'll get a certain
growth rate from that strategy. Now, you could try a
different strategy where you take your gains from IBM. And you actually ship them off
over to a British bank, which is denominated in
pounds, and see how fast that bank balance grows. And there's no reason that
the two bank balances-- the American one and
the British one-- need to grow at the same rate. Because they're denominated
in different currencies. So we're basically
interested to know, given that we know
how fast, let's say, the American bank
balance would grow, we want to know how
fast the British bank balance would grow. And what affects the growth
rate of the British bank balance is the covariance,
actually, between the dollar/pound
exchange rate and IBM. So remember, we're
investing in IBM and we're putting gains
in either an American bank or a British bank. So IBM stock prices in dollars. And so there's no issues
with putting IBM's gains in an American bank. But there's actually
a subtle effect that happens when you put IBM's
gains in a British bank, which the subtle effect is there's
this random exchange rate dollars per pound. And suppose that there's some
correlation, for whatever reason, between dollars
per pound and IBM. So suppose the correlation's
the following form-- every time IBM
goes up, the dollar gets weaker against the pound. So in other words, what happens
is IBM goes up, you go hooray, I'm rich. I got all these dollars. I'm going to go put them
in a British bank account. But suppose, unluckily for
you, every time IBM goes up, the dollar weakens
against the pound. And so, you cannot buy so
many pounds as a result. So contrast that with the
opposite situation where when IBM goes up, the dollar
strengthens as opposed to weakens. Then you can buy lots and lots
of pounds with your IBM gains. So the correlation between
the dollar pound exchange rate and IBM affects how
fast your British bank balance would grow. And that's actually
like the key point. So this would be
well-known to anybody-- especially an FX client. So what we're
actually going to do is find a numeraire
such that the growth rate of the balance
in that numeraire is actually the real-world
drift of the underlying. So the idea is let's
say that I told you at the beginning of this
talk that historically stocks grow at 9% on average. Our starting point here
in this part of the talk is that we're starting from this
risk-neutral measure Q, which, by definition, is the property
that stocks would grow only at 1%. So what we're
actually going to do is go find some numeraire
which will be correlated with the stocks, such
that when we put our stock gains in that numeraire,
we end up growing at 9%, rather than 1%. That's the way we
think about things. And the key is to find
that numeraire that has that property. I'm going to go
fast now-- there's a paper by John Long where he
shows that that numeraire that converts a risk-free growth rate
into the real-world growth rate always exists. And he gave it a name, and he
called it numeraire portfolio. It has another name--
growth optimal portfolio-- that Kelly was talking about. So there's a reference if you're
interested in following up on this material. So the theory says
that there always exists this numeraire
called John Long's numeraire portfolio,
such that if you park your gains in this
numeraire, you end up growing at the real-world drift. And so, let's say
all we got to do to find that real-world drift is
go find this special numeraire. So this part of the talk is
about making some assumptions that lead to an identification
of that particular numeraire-- John Long's numeraire. We're going to continue
to work with diffusions. And now we're going to also
impose time homogeneity like Ross was doing. So let's say when I was just
talking about numeraire, I was allowing
time inhomogeneity. But now we're going to
go time homogeneous. I haven't really been
introducing the notation, but a(x, t) is the diffusion
coefficient of the state variable x. And now it's just being assumed
to be a function of x only. So b^Q(x, t) was the
drift coefficient of x. And now it's a
function of x only. r(x, t) was the function
linking the short interest rate to the state variable x. And now, it's a
function of x only. And finally, sigma_L(x, t)
was the volatility of John Long's numeraire portfolio. And again, that's a
function of x only. So anyway, another
assumption that we're going to impose now in order
to determine uniquely what this numeraire
portfolio value is is to require that the diffusion
that's driving everything live in a bounded interval. So essentially, the
sample paths all have to be bounded below
by some constant, which could be negative,
and have to be bounded above by some constant,
which again could be negative. We make all those
assumptions, and we move on. And so in the end, what
have we been assuming? So we're assuming that there's a
single source of uncertainty X. And that it's a
time-homogeneous diffusion. So that's this
middle equation here. And so that says changes in
X have a predictable part, which is b^Q(X) dt. And they have an unpredictable
part, which is a of X dW. So W there is standard
Brownian motion. And since I'm big
on mnemonics, you might ask why does W stand
for standard Brownian motion? And that's because
W actually stands for Wiener process--
Norbert Wiener being an MIT mathematician. And the W is a standard
notation for this kind of thing. As an aside, when Bob Merton was
here working out all this stuff for the first time
in the late '60s, he knew the standard notation
for standard Brownian motion was W. But it turns
out in finance, the standard notation
for wealth is also W. And he wanted to work on
stochastic wealth dynamics. And so he had to choose should
I use the letter W for wealth, or should I use the letter
W for Wiener process? And he chose W for
wealth, which meant he had to pick a different
letter for Wiener process. And so he actually
chose the letter Z. And you'll have to
ask him why he chose that letter, because it doesn't
stand for anything as far as I know, except that actually
the sample paths of a Wiener process look very jagged,
so if you turn your head, you might be able to see a Z. So another assumption
is that we're going to restrict the possible
dynamics of the numeraire portfolio's value. So we're going to let
L denote the value of this numeraire portfolio. And the mnemonic here
is that John Long invented this concept, so
we're calling it L for Long. Now it's unfortunate that
the inventor of this concept was named Long, actually. Because in finance, the
word "long" indicates that for a security with
a non-negative payoff, if you're long,
then you're going to be receiving that payoff. As you pay money now, you're
going to receive that payoff. It's the opposite
of short, where if you're short a security
with a non-negative payoff, then actually you get
money now and you have to deliver that payoff later. So as it happens, this
numeraire portfolio has multiple positions in it. And the signs of
the positions are allowed to be real-- so
positives and negatives. So it's kind of a misnomer. I say Long's
numeraire portfolio, and everyone thinks
the positions in them are all positive. It's not true-- so
they're real-valued. The kind of problem
here is that we've put the structure on
the value L, John Long's numeraire portfolio, namely
that L is a continuous process, but it's not quite a
diffusion in itself. The only thing you can say
is that the pair X and L are a bivariate diffusion. If you bring this
L over to the side, you can see the coefficients
for dL depend on L and X-- and same thing with
the volatility part. So anyway, we place
the structure. And the idea is that we know,
from looking at Bloomberg, what the risk-neutral drift
of X is-- that's b^Q(X). We know that function. We know what the diffusion
coefficient of X is. That's the function A of X. We know what the
risk-neutral drift of L is-- that's that
function r of X. But we don't know the
volatility of John Long's numeraire portfolio. That's the function
sigma_L of X. And if only we could find
it, we would actually know how to determine
the real-world drift. And remember I was saying
if IBM and you could put an American bank
account, and let's say there was certain growth rate there. And then if instead you
were putting those gains in a British bank account,
you'd achieve a different growth rate. And I was stressing that the
correlation of dollar/pound with IBM was important for
determining that growth rate. And I stand by that. When you're in a
one-factor world, that correlation
can only be one. And so that's what's
happening here. We're in a one-factor world,
and that correlation is one. And the other thing that
affects the growth rate, though, of your British
bank account balance is actually the volatility
of the exchange rate. So what actually matters
is the covariance between the British
exchange rate and IBM. That covariance depends
on both the correlation and the volatility
of the FX rate. So you can think of the FX
rate as here John Long's numeraire portfolio. And so that sigma_L
is sort of the key. It's like we've set things up
so we know the correlation, but we still don't
know the covariance. And that's what's
actually relevant. So as soon as we
get the sigma_L, we'll know the covariance. So we'll be in shape. So we got to find that
volatility function sigma_L. And now I know many
of you have classes, so I'm going to have
to start moving. AUDIENCE: Now, Peter,
people will have access to these slides afterwards. And so, I'm just seeing you've
got another 15 slides left. PETER CARR: Yes,
well actually, you'll be glad to know that five
of those are disclaimers. If I could move along-- AUDIENCE: But the
point is to what-- PETER CARR: The key
is towards the end. Yes, absolutely. We're very close. OK. So I'll be done in two minutes. So basically,
where we are now is we're going to make
one more assumption that the value of John
Long's numeraire portfolio is a function of X and
D. OK then, let's say we've made all our assumptions. And where it goes is
that the assumptions imply that this
value function splits into an unknown
positive function of x, and an unknown positive
function of time. And when you kind
of further analyze, you find that the
unknown function of time is an exponential
function of time. And the unknown function of x
solves an ordinary differential equation of this kind. So this is called a
Sturm-Liouville problem. And it turns out that
Sturm and Liouville were the only mathematicians
I've mentioned in this talk who were not at MIT. And they actually
solved this problem. And one of the
things they show is that when you're searching
for functions pi and scalars lambda that solve
this problem, there's only one solution that delivers
you a positive function pi. And so this is how
you get uniqueness. Remember I was
saying back with 11-- so we're searching for like
a 10-vector and a scalar. Now the 10 vector is a function. And that function is pi,
and the scalar's lambda. So the point is is
that the math implies there's a unique
solution to the problem. So we learn the volatility
of the numeraire portfolio in the end. And then we learn the
drifts of everything you want to know under
the market's beliefs. So that's the gist of it. So then there's been
work on trying to extend to unbounded intervals. And basically, in the
famous Black-Scholes model, this effort fails, whereas
in the less famous but still important Cox-Ingersoll-Ross
model, this effort succeeds. So the sort of
punchline is that when it comes to unbounded state
space, the theory's open. So if there's a grad
student in the room who wants a good dissertation
problem, this is it. OK. So that's all I
wanted to say today. Thanks.
