Alright.
>> Alright.
>> Welcome everybody to
the latest installment
of the BQE.
>> That's the
Brooklyn Quant
Experience in case
you're wondering.
>> And so today
I'm very pleased
to have Professor
Claudio Tebaldi from University of Bocconi
as our featured speaker.
So he's
a tenured faculty member
in Bocconi University,
which is Milano
associate professor
in the field of
Quantitative Methods
for Economics,
Finance and Insurance since 2011
He holds the National Qualification for
Full Professorship
since 2015.
He is a fellow of IGIER and Baffi-CAREFIN
research centers
and has been
a visiting scholar
at UCLA school of business.
His research interests range in the areas of Derivative
Asset pricing and Portfolio management.
His papers are published
in peer-reviewed journals
like review of  financial studies.
Mathematical Finance, JFQA and Journal of Econometrics.
And two of them have
been awarded, one in September 2019
by the Canadian Derivatives Institute as Best Paper in Derivatives of the Northern Finance Association Meeting,
one in January 2007
as  Best Paper of the Swiss Econometrics and Finance Society Meeting.
He is currently serving as Managing Editor of Quantitative Finance.
>> Okay, with that Claudio take it away.
Talking about this research project by Andrea Buraschi from Imperial college Business school.
>> I would say
essential (inaudible)
of this project
the project was network meter
is now taking this class.
Couple of important
things. Yes, I did.
The first part that
we dedicate most of
the first part to
contribution of
this massive
pricing without
specific reference
to a given network,
given economic
well  identified 
economic network.
Later on I will consider
the application of
this theory network,
network and we didn't
find any interbank network
>> Ok?
>> So the key notion
that I want to discuss
is the notion of a cascade risk.
This notion we have to
compare cascade state,
with say a 
risk factors will be
relate risk factors
in general arbitrage
pricing theory.
As a general introduction we  prove that there
is typically
divided into when
you see the first
pricing or say
the first stage
of systematic
versus sideways.
So most of the
literature is driven by
the idea that people
modern systematic
risk factors,
because those are
the only ones that
the rationality based
(inaudible) prices,
you can attach a
price if you can
diversify risk that is
old or firm-specific
or idiosyncratic.
And the main idea
of this paper
is frankly, challenger.
In an economy the separation and
the assumption which is
pools in the arbitrage
pricing theory,
any general
equilibrium theory,
that firm-specific
or idiosyncratic
shock average out.
Okay, so the main
issue/ question,
the same question would be
addressed first in a,
partially believed arbitrage
pricing theory then
Lucas economy where we
are establishing
some conditions are
satisfied here.
Firm specific shocks
average out and
you do not play
any role in pricing.
Is this really is the
fact that these shocks do
not affect the
equilibrium as a  price
is true.  In general
under which
conditions we can say
that you must
surprises every spin.
You're not affected
by these fancy shops.
>> And just to
move quickly to
the review of the results.
>> And we will consider
C as an economy
where many firms,
and these firms will
be really stigmatized
firms in partial
equilibrium.
There will be standard
data generating
process San Ge
non-equilibrium.
They wouldn't be
Lucas microscope,
Lucas freestall,
essentially dividend
producing unique,
identifying two regimes.
And today, now
Nixon came out,
we call stop me,
despite even why
his name way
referred to for
interaction,
a sufficient ego or
cascades in
Angular completed
by the end of the action.
If firm to firm interaction is sufficiently
below a given threshold.
Well, the Lucas
assumption
general equilibrium,
there are idiosyncratic
averaging
out at one holds,
vice versa.
If the firm to firm
interactions sufficiently
stronger are
these averaging
out does not take place.
>> And we would analyze
the specific channels
application channel
And
the indications
are it is variation,
is variation two regimes heterogeneous parameters that remains.
Firm to form interaction has
important implications for
pricing impartially.
I will first discuss
these indications
impartial equilibrium later.
In general equilibrium
can mean implication
is that in
this region is super
critical region where
cascades can take place
and where we want
precisely what they
mean by cascades.
These firm specific shocks.
These shocks
And ideally they say,
and what are those shocks
that we assume
typically
can be averaged out due to 
some sort of law of
large numbers.
On the contrary,
They cluster dynamically and
this creates an
avalanche effect,
contagion 
effect, this
effect has
pricing implications.
Ok, so this more or
less is that
general feature.
And later in the second
part of the book,
we would try to give you
a more economic
description of
what is going to
happen in these two distinct equilibrium when
we think about a network.
that can be
properly  identified
from my point of view.
Let's do the systemic
risk analysis.
Thinking about
these network as
the network of
interaction of
interbank lending,
we can observe
interbank lending,
we can try to assess
the impact of these
network of interaction
within the financial
system into
the expected returns on
the expected return of say,
financial institutions
their stocks.
>> Okay.
>> Yes. Question
say exactly three
firms, in economy.
>> Yes.
>> There are three firm to
firm interactions?
>> Yes. So again, you talking about
single interaction
strength.
Like there's one in the 
minor thing.
>> Ok?
>> So there will be around,
say I go to the equation.
>> So we immediately
specify was the way I
specified interaction,
I will say.
And when I mean
parameter lambda,
which is common through out the firms
let's call it would be more
than a gaseous
matrix or network matrix.
This try this firm to firm specific interaction
So the challenge
the challenge here is to
sufficiently
general to capture,
as Peter was suggesting
here, there is an issue.
That would be a empirical 
issue because we
know that there
are a lot of
interfirm interactions.
>> But then measuring
these interactions,
they strength the
relevance and
the relevance is not that 
easy, right?
>> Okay.
>> So these, let me say
that as I was saying,
there's a lot of literature
on network economy.
Okay? Here is a super
partial description
of the papers that are in
some way are related
to these paper.
These are mostly data
firm financial economics.
>> And that is,
I would say, to
my understanding,
0 papers that
takes seriously
the asset pricing
implications
of these network.
>> You typically here
we're talking about
the general immune
economies of reproduction.
Essentially suddenly
the iMBA Asimov
for these economies
that set up
general equilibrium
economy in terms of
production or in terms
or interbank networks.
>> That's what
they're focused on.
>> The economic
implications
of the emergence of
these networks and
which are the same.
critical conditions
in these
literature
In this literature the
critical issues
the trade off between
diversification
and the possibility
to have compaction
So in practice, you try
to share your risk,
to optimize
the optimal  (inaudible) location
of their instinct economy.
So everybody will
be better off
by improving,
that is sharing.
You establish
connections and
connections by the way,
Drive the possibility
that if somebody does
not pay your debt or
somebody is hit by a shock.
At, the shock will be
transferred through his
connections, right?
So the probablem is
that, generally speaking,
in a general
equilibrium economy,
there is endogenous
creation of a network.
>> This network us do
counter party
counter varying effects.
>> One positive,
even a possibility
to share risk,
well-diversified risk.
on the other possibility
to being hit by
a shock and through one
of these connections.
Ok, so the main idea is
certainly very well-understood
in these days.
We want to use,
we want to model
the propagation
of the shocks
in this network by
using one of the
models start typically
used in the analysis
of virus contagion.
In maybe a real virus or
a maybe computer
was we have
knocked off
networks who are
thinking about
economic networks
like the supply chain
networks for thinking
about computer name  
was that whenever we
are thinking bout networks,
in terms of
people connected.
>> Ok?
>> So there are
models that are
typically used
in biophysics
and we're going to use
this stochastic model
is essential and from
this literature to describe
the propagation
of shocks in economy
on why this is new.
Because in a way,
people so far,
General and finance
international economics
and as trying to
model aggregate
shocks because
of these Lucas assumption.
So in practice the only way
in aggregate shock,
especially this
is when move for
example in credit sphere.
So they aggregate
shock would be say,
a shock to the economy.
And hit uniformly thorugh 
covariation,
it means that essentially
everybody will
be affected in
the same macro shock.
there will be
covariation between the
returns of each firm.
And these, say
general shock,
in general, economic shock.
But what we know from the
data especially,
for example,
if you want to
see what's going
on in our training
portfolio,
we know that
most of the risk,
that is really relevant for
a bank portfolio when you
discuss the way these at
risk that hits the economy
is not through
these common
Shock through
the co-variation
is through direct
interaction.
So if my
>> Supplier and somebody
along the supply chain
is hit by a shock,
it  will have a direct, okay?
>> This direct
effect cannot be
represented by an aggregate factor .
So you are forced to
use a model for
firm-specific shocks.
But this is very far away
from the way arbitrage
pricing theory,
evaluation theories have
been developed since
assuming that if
the economy's
not enough arbitrage
pricing theory,
you typically assume that
its shocks are firm specific.
And by properly selecting
your portfolio,
these shocks at the
role of the shocks or
would be irrelevant
for the,
for the marking
prices. Okay.
>> So let me go quickly
through the model.
>> I would say it's pretty simple
seeing policies
grading done,
purely log-normal
So we are modelling here.
the dividends since we're
thinking about the current government  economy.
So Lucas economy is cashflow,
these are assumed to
be the cashflows.
You're essentially no
capital structure.
We have thinking
modern machinery
moderating the
dividends distributed
to equity holders and
dividends coming from
I will have a common shock,
that regular stand
that Lucas won.
He said not know my shock
continuously distributed.
Continuous pathway shock?
And a small shock,
Firm specific that  is our
super simple short,
because they
differ essentially
can be only two states.
You can think about
states like sick
or healthy or about,
we'll call that x.
When these indicator
HDI is one,
the shock will
correspond to
different going
in this trace.
>> I remember you
then affirming
distress needs.
>> The cash flows that are
produced by the firm are
not sufficient to repay
the debt of these firms.
So it's not stealing
default, you know,
and distress
said you have to
work and to make them
more productive
and to restore
normal distribution
of a (inaudible)
He showed they
transition that
takes this firm from this,
take one from the stress x  ACO is today
H equal to 0  corresponding to this state
where the firm is
normally distributed.
Okay? So we think
that a number of firms
in this economy,
in each one of
these firms is
describe only infinite for
the state of the
distribution to the person
XI zeros bigger than Xi one.
X0 is larger than Xi
counter-intuitive
by recounted the
distress firms right?
Yeah, because yes, a euro.
So from 0 to
one example its called distress rate, right?
That any transition
from one to 0,
it's called the healing 
rate for a reason.
>> I will discuss that.
>> We assume
that the healing rate is
uniform constant
independent from the
state of the economy.
So in practice, when
you go into the stress,
you entering in
a state where
essentially you are
under the control of
All of the authorities
you are in a situation where does
not depend on the general
state of the economy.
It take some time to
making about these data.
So the average time
that their distress
distress solution plan
is going to take place,
we think about
this figure is
quantity as the
state independent.
On the contrary, we believe
that most of the upshot
of interesting option
here is about them.
That is thrice transition
and a you have  epsilon i,
the transition of firm
specific contribution.
You have a second contribution
That is the
critical innovation
depends only interaction.
So depends on the
direct interaction?
That is for us, we've
done the other firms in
the economy through
these delta ij
that represents in practice
a  number a quantity
that gives you increasing,
the life about
this transition
for stress transition ,
even by the fact that
firm j is in distress.
>> Okay?
>> You sum over all the
firms in the economy.
If some data inclination
and this firm,
firm J is in the distress state
that easily arise
mostly diffusibility,
likelihood that
also you, you,
you, you were too
young to school.
>> I'd rather go to
these stress transitions
they are they are
ones and zeros are
yes, yes, yes.
>> State of the economy
is described by these
aggregates shock.
vector, the d vector,
that is a string of
dummy variables,
one for those firms
that are at least
as you let 0.
For this  firms,
our distributing,
normally distributing
dividends
and the key.
A key part of my goal
will be describing
evolution
of HD that we've
called the contagion
process, right?
It's a continuous.
I will describe it.
>> So in the previous slide, whats the meaning of this ratio in simple terms
Ok?
>> So, so in practice,
the way that we want,
we want to work with well identified parameters.
So theta is constant
distress distribution
the inverse of the
distress distribution time.
It's a sort of unit of
measure of time you
want to measure
in your economy.
So you can assume
that it's theta goes to
one extreme or
the parameters
and nothing will be
affected in your economy.
So all of that are
essentially from
now on I will work
with the distress 
parameters scaled by theta
and everything
about you see that
I like or the unit
of measure of time.
Ok. So you will see always
these lambda theta
parameter simply because
then we are testing
well identified
parameter was that
all of lambda theta
These will be will will
come at the very
end of the talk.
What they can
anticipate is that
you can think about
that quantity.
For example, that the level
the average level of
leveraging the economy
So then the higher
the leverage,
we know these from
corporate finance.
The higher you know
the sensitivity of one firm
an exogenous shock
because you have less,
you know, less buffer of
liquidity to
absorb the shock.
Okay, so by the way,
what's the problem that
Big problem is that we
should think
about this now,
but as an endogenous quantity
so Leverage itself evolves.
So we deem a
general adequately
here for the moment
that we think about
this quantity as an
exogenous parameter,
we're going to move this
parameter thinking,
yes, yeah, yes, yes.
>> Alright.
>> That is the only
one identified
was moving lambda and theta
>> Again.
>> Okay.
>> And the delta j. they would be non-negative right?
they would have to
be non-negative?
>> Yeah.
>> Yes.
>> A big bank or we
think about delta Ij now,
like the amount
of money that J,
that I then to j.
>> Alright, so this is us.
>> Well, you see 2345.
This means , you know,
one would be more dependent on five
relatively means lending more money to five
More by five.
is more dependent. Thing 
about these firms
like firms connected
through a supply chain.
Supply chain, you
are have two drivers.
You have a productive
relationship that you have
a financial relationship
here between the ties
both in single numbers
>> Okay?
>> So it is a very reduced
for the description
of the network
>> Okay, so a little bit of
why this process
is interesting
and slightly different
from what we typically have
(inaudible)
because here we have a
simple Markov chain by,
you know, then they
stay access configuration
Here is two to the n,
So it's exponential in
the number of votes.
So you have a large number
of possible states.
And in thinking about
N going to infinity, right,
because the economy will
be large and we will
have many results that hold
in a large economy.
>> Okay?
>> And is these values legit?
>> Well, yes,
we know we will
see that most of the
results for even for my,
for not to be a part
of big economies . But the basic idea is that the results are going
to work on the whole when n
is very large, okay?
And you had this big
debt in the state
that borders the
state of the
vector h. Evolution can be
described in terms
of 2n by 2n matrix.
>> But this An is the matrix that
measure the transition rates.
is in fact the
sparse matrix.
Why essentially, this
is a Markov process.
Acute job Markov process.
>> You know, you
can essentially
describe this processing in
terms of a simple
Poisson processes
and yearning transitions,
(inaudible)
No zero transitional rates,
and those were
the two states,
not the economy differ for
a symbol for a
single firm state.
So in practice
the evolution of
the vector h will
be between two
consecutive states h,
that difference we
beyond in the state
of one symbol for okay?
So there would be no
joint transitions
because the probability,
the joint probability
of having
the same eastern
transitions of
two firms 
according to this formula.
>> Okay?
>> So these will be,
by the way, useful
because it would seem,
if I know the simultaneous
Yes, yes, yes.
Riots, they clustered
but it would be one after the other
>> Yeah. Right. Okay.
>> Then, then, you know,
all the thoughts
in this economy
(inaudible)
the for A for the X may differ
for they will allocate
the location of
the network.
So the risk exposure
that traditional
risk exposure
for these firms
with respect to
the aggregate shortcut that
A11 would be the same.
So essentially the beta
which of these firms  according to
a standard
Lucas Equilibrium will
one and the same.
We are thinking to show
this equation where we
think, for example,
in the case of a bank debt,
all that is that
regulatory authority
that keeps all
the probability.
The firm specific probability
of this price transition.
So in practice
here, that is
our waiting agency or
regulatory authority that
keeps low sufficient low
distress transition rate
that we want them
focused by the way,
when dissolution in the case
of what I have seen is very small
eventually going to
be small as n goes to
infinity with that some power
as power and
notice  will play an
important role then when
if we assume x01 is (inaudible)
on the
configuration where all the firms are
healthy is absorbing for
the dynamics of this chain.
Which means you see if h
is 0 for all the
firms and 0,
when you reach that
h equals zero
configuration,
just stop there and
there is no more for
specific dynamics.
Why this is interesting?
And see it is interesting
that modern shocks that
car for specific NR
transitory, right?
So these shocks are 
the typical shocks
said that according to
the traditional
point of view,
should not take
any role within
the general equilibrium
economy invest
or at what if we forget
that the interaction term,
these shocks should not have
any price of risk, right?
Because you can't
diversify that
plus iterate
macroeconomics,
the only relevant
shops out there,
permanent ones, are
there rather workshops.
And they are firm specific
so diversifying
shocks, right.
>> And if all firms
have helped him yes.
Then they are gonna 
stay healthy right?
Yes. Okay. Let's say
all firms that healthy
without  non-zero then
some firms can become, Yes, yes.
>> So I we saw more
than different from
0 but small by the
basic, you know,
it may be an
important fact,
the fact that in
the limit of n
very large when x i
has to be very small.
Because equity is constrained
the Markov chain will
have its absorbing state.
>> Okay?
>> Then let me as
(inaudible)
financial economists.
Trying to stay in here,
I'm doing a lot of
parametric choice.
I'm giving, I'm, I'm
in practice considering I'm
very well educated guess
on the transitional rates.
So all my friends,
economies, the redox,
what would want,
why I'm choosing
completely asymmetric,
or maybe you want is for
that earlier
slide we saw what
was happening is you have
this generic or
systematic process
because the
Brownian motion,
Yes, and so on.
>> And yes, at
that diffusion
is occurring yet
satisfactory.
>> Hi either 0 or
one is diffusing
as well.
>> Yes, but there's
no connection
between the fusion of
the imaginary factor and the hi,
you are right. That is the limitation of 
the modern day.
>> They aggregate
shock and
the firm specific ones
here as essentially an
independent dynamics.
Okay?
>> So when I say for
i equals one Hz
is 0 at time t?
Yes. And it was 
that dividend? Yes. Yes.
>> Based on the evolution at the fusion
>> Yes.
>> So on again,
the basic idea
is what they,
they think that the permanent
transitory decomposition
in macroeconomics.
So your aggregate
shocks is permanent
and hit 
through the Y.
And you have the Xi.
That is a
transitory dynamics
that are driven by
balance sheet of the firm
that once I could
see that some
of us that yeah.
Yeah. So that dividends
are actually not
0 in the distressed state x i one you can,
yeah, you can put,
zero but will not be 0.
generally speaking
you can show
statistically xi0 is high divedends
Xi1 is low dividend
I mean it can be zero
but lets take more general, so 
in general,
dividends (inaudible) and xi0 is bigger than xi1
And well,
so the more firms
that are unhealthy,
I guess thinking is that lower the ideal dividend
>> Yes.
>> Yeah.
>> So in equilibrium
you will have a lower dividend
them these would
be optional.
And we'll assumption
with feedback.
>> And, and that's
the last line.
That's the evolution
of the HI vector.
Yes, right?
So so the HI age at t
plus delta t yes,
will come from Asia.
>> T, Spanish
individually H of t . Yes.
>> So the matrix now
>> Not through
this matrix a,
that this state
transition matrix was
enemies are saved by
Islam now and
eat up of age.
So once you specify
the distress rates,
the rates for each firm
will depend on
the full vector.
So you see that
lambda depends on
the broad age.
So that possibility
for you to go in,
we stress, depends
on any other firms.
You are connected, right?
>> So this the interaction
that is doing the action
in this movie, right?
>> So here are the
two big assumptions.
>> One is asymmetry,
the fact that we
are considering
interaction for these
phase transitions.
We are not considering
interaction for
healing for solutions by
where we could
do something by
ESA is driven by, you know,
and we know that in
fact these people,
theists and pricing
effect is far
stronger when
they would stay,
then you go in
there and distress,
they then vice versa.
Ok, so this is then
What's the literature
I want to stress,
this is very important.
This is why I
decided to give
this talk here with Peter.
In your instance, equity,
when you want to price.
And here we're thinking
about pricing.
You share the
stocks directly,
this is the 
typically exercise.
But you may also pricing
a cross-section of
equity options or even
joined the shares of
equity and the options written on the shares,
once you learn
learn and when look at jointly
you try to create
portfolios of options.
And unless the
underlying stocks,
you learn that most of the options
are not captured
by many to be
the risk exposures
according to
the standard capital
asset pricing
model assumption.
Capital asset pricing,
because most of
the risk is he then
into these
idiosyncratic shocks.
So they need that there
is a recent evidence.
I must go missing
bigger ESA or
these people thought that
his very recent
relevant shocks,
and this goes back
to your question.
Then move the cross
section or equity options
and zero beta
it means that
>> He's going
doing a Econometrics
convincing.
>> It's a high-frequency
econometric
showing that these are
not neutral shocks
So in practice, the beat of
the shocks is irrelevant.
In a way supports the idea
that we can look at the Age
and we can look at the
I isolated way, okay?
>> Which is in a way
the more challenging.
But going back to
the credit, please,
since we know that you
can use COX processes
or these conditionally
condition of
factor modelling by most of
the option price of
the portfolios of the
banks will be lost.
>> Ok. So does
>> So going back to
our default then,
the stress things.
Dynamics. what
we want from an
dynamic point of view,
representing that
is clustering.
And so what we observe is
that the mode
is whether you
run distress states or
you're going to
run factors there.
diffusion bill factors struggle in
representing the
clustering of the crisis.
>> Okay.
>> So the proposal
is that I think now
it's pretty clear,
it's OK. Now we try to use
these stream of
dummy variables
to represent these
contagion process.
And we want to represent
that cascade risk
for this process.
>> So the idea of
>> Cascade is the
idea of clustering of
a crisis for
this contagion
process H of t right?
So the key definition is the
the definition of cascade
what is a cascading in this economy
So in this economy
we define
so it's easier if
you think about
the economy,
therefore this day then
the challenge is
to generalize.
If you are an
absorbing state
that is super easy
to understand
what's going to be the
new generic cascade
>> Essentially
assume you flip
a set S of, of firms.
>> So you sending
distress or
some exogenous
reason, express
you can measure the time
required and the economy go
back to their
normal distribution
the age =0 state
And the length of
that perturbation of
this fluctuation would be
a good representation of
the severity of
your crisis.
>> Okay?
>> Now the idea
is that you want
to add something,
a definition that
is robust with
respect to
the possibility for
specific shots.
>> Okay?
>> And so we define
the mean time that it
takes a your economy
to go back to the
steady-state.
>> Once said that
you perturb it,
you go out from
the steady-state.
It's a short-term extension
of these absorbing time.
Here, you are
not going back
to a specific
configuration,
you want to go
back to specific time
configuration
with the other.
And the probability that is
the probability that
that configuration
occurs in the steady state,
ok, these are the four
for a  current
chain is defined,
is called the
Khamenei time.
Khamenei time 
of the chain,
it's well-defined,
does a lot of
interesting properties.
So we think about
gets larger than 0,
so our chain will be
recurred and you can
measure is time
define this  time this way will be
essentially once
you start from
an economy where all S set
of firms really
stress and Ts
a which will be
the time required.
to reach the H
the configuration H by
the probability of being
in nature in the
steady state
that I recall you
that the
steady-state vector
is one of the
eigenvectors of
the principal
eigenvectors of
the rate matrix
a corresponding to the eigenvalues 0.
So in fact this is
eigenvector will give me,
this one is called
the Khamenei time
>> And the khameanie time has
interesting properties.
>> First of all, it
does not depend on x.
>> So in practice
that you have
a measure of time
that does not
depend on your exogenous
fluctuation. It is suspected
by means that it depends on
the spectral properties
of transitional rate models
>> And this is why
we choose this
quantity because
we want to have
a quantity that is C bar
with respect to, say,
the representation
of the dynamics
so the mean time does not depend on S but the actual return time
is that what happens?
The expectation
Doesn't by the way,
(inaudible)
yes, but it does not depend
on the original
perturbation.
>> It tends to be
invariant with respect to
things like the whole point
was that the more you critter.
>> The longer it
takes to comeback
is great, great racer.
>> So that problem
is exactly this one.
>> You don't you know here.
>> The idea is that we want
to have at
firm-specific level.
We want to control
the size of
the noise because we know
the arbitrage pricing,
we want that when
the noise is going,
you know, he's going to
0, you can diversify.
So we want to stay
close enough to
a situation where
this noise is small.
>> But on the other
hand, we want weapon,
a representation of the
endogenous child.
>> So we want to know,
given the state
of the economy,
endogenous me out there,
he is prone to being hit
by a severe prices, right?
So by (inaudible),
if S is
if I assume that
all the firms,
in the economy are
hit by these shock
up when it's obvious that
the time I will end,
it will take to
go back to that,
the state will be longer
compared to situation where
S is a small cluster by
what they want more.
>>as this equation work
I prefer a single firm
and this may generate
a cascade that is
larger and gives
rise to an aggregate effect
>> Ok, so I want
to represent
the endogeneity
of the effect.
>> Okay?
>> So I choose this T For
that reason, okay?
>> Because I want
to, I want to have a
major of the exposure to
a cascade of
financing situation
where depending on
the exogenous parameters
essentially the lambda to
variant time t delta, this time is exponential
in the number of firms.
So essentially if when
these exponential,
these means that you have
no way to remove the
impact of this shock,
even start from
a single thing.
>> Okay.
>> So this is not
this is  borrowed from
these contagion
literature that
you force design. In the regional traditional
contagion literature you can even take n going
to infinity in that
simplify your treatment.
>> But the problem
is that we want to
have something  we can
taste on the real economy.
>> So we take this
definition, ok?
We will say that in a state of economy
there are cascades and if the time of return
to the steady state
is longer than a quantity
that is exponential in
the number of firms,
in the economy
And this will be,
the endogenous
characteristics
of the state of the 
economy will not
depend on the severity
of their perturbation
that he's there.
We would seem to be
very small in the limit
that later on
I'm going to 0.
>> Ok?
>> So
in practice in this economy say that
there are cascades
if a single flip may
drive crisis
in the full economy, think about
lambda over theta t the leverage. moving lambda over theta I
will pass from a
situation where,
you know, exhaustion
shocks average out
quickly. This equation where
even a single
firm may drive
aggregate and that he
was the first
year and that is
occurring in
practice extensa?
No, he's asking
the contagion
of literature is that
the transition between
this equation lambda
over theta  is below.
>> The (inaudible) does
not generate cascades.
this equation where there are
cascades and 
this continues,
that is the value of
lambda over theta 
below that value.
The probability of
having cascades in
zero above that value of
the demolition cascade that
is going to infiniy right?
>> So you know,
the traditional
exactly point,
the you know in traditional model
would take n going to infinity
>> We keep n finite
that we think yes,
DG is larger than
eight to DC to n.
>> And when we
say that in an economy
sufficiently large,
it means I take n 0.
If n is larger than n  0,
I will say that they cascade
eg is larger than
the exponential of cpi
>> Okay? Okay.
>> Yes, yes, yes, yes, yes.
>> And here's finally
read it. Yes.
>> So everything is known.
You know, I'm
thinking that he's
out from n going
to infinity by
these various output
for any finite
dimensional nine
different business.
>> This is a mathematical challenge
>> Yeah.
>> Yes. So I understand
the TSH is the mean time that will take you to get
and I'll say yes,
economic content of this
weighted average or
the mean is the
weighted average
of mean pi. Yes.
>> probability is pi a right?
Yes. Yeah. what is the economic
content of that.
>> And those links are
the economic averaging
these mean times for me
to averaging the time
returning to state H,
state H
Its a super good question
we are struggling to give an
economic meaning.
>> This is a meaning tree property of the dynamics
The basic idea here is
that you want to reach
want to wait more this time configuration
is going to happen, right?
And the key problem
is that it does
not depend on
S this is like
some second matrix,
but this has to do with
the meantime
takes it's that,
it's that, the recurs
(inaudible)
of the chain.
That makes these possible.
>> Ok?
>> These are very
technical details
that are really,
you know, we are,
this definition using these
come anytime is new.
This is due to
the fact that
otherwise people
who say that all
the results are depend
on the details of
your selection of
transitional raise. why?
By selecting an ergodic
state, an
ergodic change,
so a change that has
a generic epsilon
i different from 0.
And everything
principle goes
into well defined
steady-state,
even showing in these
very general situation
And we may, you know,
maybe you may find
the Cascades,
this clustering of events.
These are sounds from
an economic point of
view, super important.
It's telling you that
cascades are generic
or you have to
take into account
the possibility your
cascades in a very
generous equation
is not something
specific to
the parameterization
of your model okay
Just to be precise,
when you set x from i to 0,
the supercritical
state is what is
called the situation where
the contagion is intent
is means that you have a probability,
finite probability to reach
him to get sick
even if you are
if you are completely healthy  independent
from your personality
>> Okay.
>> So here's a situation
where the probability,
of final probability of being hit
by a contagion irrespective of your personal state
Okay, you can do whatever
you want on your
balance sheet,
but these may not be
enough for you to
stay safe from the crisis.
>> Okay?
>> Yeah. So in
the Google Once,
I guess in the
subcritical case, yes.
What does it mean?
Why is it that
you never get Cascade
>> The essentially you can  
prove that that,
in a (inaudible)
of short time,
you go back the
steady state
state. Okay.
>> So there's no clues,
there is no existence of 
C or n 0
>> Yes.
>>Key fact  is that you
must find that
c n must be exponentials
because, you know,
essentially you
can send epsilon to 0
in terms of a
positive power
and then minus four.
>> But they will
not be nothing
in reducing these TA.
>> There is an
interesting result.
This shows that this TA as
also an additional
significance compared me,
compared to the TA is
essentially is giving
you the sensitivity of
your state because
more perturbation
from outside.
So if these
exponential in n,
This means that in
practice it will be
infinite in
any realistic situation
between these economies
>> Okay. So then
the return times,
yes, yes, yeah. Yeah.
>> Essentially you may have
perturbations, but
these will be loci,
crisis 1B and
epidemic state.
Than it has implications,
By the way, I
guess is that the,
we expect only aggregate,
Cascades to heat
the rational
investor, right?
So in practice, if
you want similar
equation, think of a cascade
like a raging tree
>> In practice, there
will be a separation.
>> That's separation will correspond to
this question where if
there is one agent that
is going into these,
stress one for
going to stress.
>> These will propagate
at least that one
additional read
directly connected
firm the distress
>> If these number
it is goes to
the reproduction
number is larger than one.
>> You will be in
the supercritical.
>> If these reproduction number
is below that threshold,
it will not be
essentially would
decay exponentially.
Right now I go to
the main course.
He said I want to see
how these cascading effect,
how much time do I have
all the time who
represent that out.
>> That is not so easy.
>>  In practice my exercise 
is the following.
>> I, I didn't place
the traditional 
knowledge in the
arbitrage pricing
theory way that my,
we get a noise and
that these are in
practice, these
contagion process.
So I think about the
traditional capital
asset pricing model
and navigate shock,
and that is the
market shock
outweigh the disadvantages.
Each time there is
a stress at the
price jumps down,
each time there is a
healing the price jumps up
>> And, you know, I
choose a very special
parameterization,
whether very easy,
goes from solution
>> And the parameterization
will be, you see,
one over n, so it will be
spreading uniformly
cross all the firms
>> In this case,
I paid, you know,
all the photos that summer.
>> And then two
sources of up to a
represent the hula contagion process in
to Poisson processes.
>> The number total
number expressed position,
the total number of healing
transitions
a well-diversified
portfolio,
which means a I take 
portfolio where each
firm stock return
as weight one.
>> And this is
a typical that
typically situation
where you
expect that this
quantity averaged
out when n goes to infinity
>> So all these is
that a one over n
And we want to see,
we want to verify
that when lambda
is larger than one and
it doesn't go to 0.
>> Ok?
>> So here we
have presented,
I know where you
see essentially
renormalizing by
their transitional rate
Poisson processes.
We have that the
evolution of
the total number of
distressed firms these IN
stands for
evolves according
birth and deah that process.
It's a very
traditional courses
when n goes to infinity
by the strong national
process, these y, y.
So in practice, these
quantities converge.
>> To one surprise,
you are left with
this equation
where you may not
need semicolons.
>> And the limit
of n going to
infinity goes to a
determining steeply.
>> Okay?
>> So the limit
of n and larger
the probability to be close
blue solution u
stochastic solution
to be close to visit.
An easy socially is
essentially one. Ok?
>> It's very,
it's very large
>> So read precise
probability
whether it's larger
than epsilon goes to 0.
>> So what's going on here?
>> So they don't
kind of state,
that is the steady state.
You find that
this steady state
is characterized
by two solutions.
>> One is, as we
would expect,
0 infected distress
averaged out.
>> Or is question where
I star is larger,
this is one minus
minus one over lambda.
>> Okay?
>> So if you are in the subcritical zone
so one is the
critical point.
>> So if you are in a
subcritical state that
is 0, so no pricey.
>> But if you go into
debt, some supercritical state
I star equal to 0
goes to 0, it is unstable.
So even not infinitesimal
perturbation
out of I star
from 0 and drive
you out and make
your own version of
Unity solution that
gives a nice start.
>> It's a nice time.
>> level infected
over distress.
>> A finite fraction of
infected firms implies
finite contribution
to your expected return.
>> Okay?
>> So these relationship
mentioned he can diversify.
>> Diversify means is
Z is a positive variance
from this portfolio number,
isn't it? Yes. Yes.
>> Okay.
>> So the basic idea
is that even such a simple
set up  up like a campaign,
if you, i, but it
won't be Wusheng
big made that.
>> I think that
actually this cascade
effect, yeah,
e generate to price to alphs
at a price
and these quantity
they are paying
the price of
this yet depreciation
by the amount of,
you know, how
far you are into
the super critical region.
And that you know
ages due to the fact that
the retailer will have
finite period of time.
>> So let me call
it idiosyncratic,
that all these firms
interact with each other.
>> So yes, why wasn't it,
wasn't instead,
just imagine
what we can say is we call
it no cascade
risk right away.
Because on one side
its not systematic
There is no commercial
 doesn't
exactly my firm
is that even
flipping one c?
>> So setting the distress
1for all the
thoughts, regulation.
>> So that firm specific
risk is no. still
you may have an
aggregate effect, right?
>> Yeah.
>> So now the
question is OK.
>> You show ask us that these
happens when n
goes to infinity.
>> What's phenomenon
is known as finite.
>> Ok?
>> So now the tricky part
but then it quickly,
because you're not
really if you are not,
you know, this, is
this a further,
is it unspecified nature?
Because now I reproduce
these resulting a
super-simple campaign,
right?
>> My goal in a state
contingent model
with a lot of
contagious
which is that
pricing measure I
should use because
you're not here.
Everything is based
on a campaign.
And set up when
we assume that
there is any
well-diversified.
So you have to go
through both
factual and one is
for the quasi-steady state.
So if you want to
know what's the price
measuring these
Economy essentially,
to think about that for
some of us in ensuring that
these formalism where
essentially you model
the evolution of
cashflow easting tensor
of a Markov process.
This process is essentially
a process coming from
your contagion process
collision
not reaching 0 state.
>> Okay, full construction
framework
that issue.
>> Finally,
compute a pricing
major that we should call a 
long time, at least.
>> Major, Hey,
these are different
from what you
would expect that based
on a simple rule.
>> Okay? So now
this is the part I
wanted to show you.
Now, you know, you
can complicate
everything as
much as you want,
but then results are
not going to change.
First of all, how
do you want to
represent the
firms that have
different exposure
received by
components of the network matrix.
>> You have this measure
that night, the scene.
>> Try that here is
for that enables,
yeah, popping up qualities
>> You may measure
the systemic nature
of each firm,
the vulnerabilities
of each firm.
Implants of these
two quantities
should quantify how
each firm or match
each firm is
exposed to these
cascade risk
Then we move into
general equilibrium.
So we want to see my party,
the quantity of
these people want to
see also the price.
risk how it 
moves when you
consider that
possibility to
have this cascade risk
And essentially
what you can see is
that each using disparemetrization
each chain corresponds at
which you can represent
the chain star like network
essentially is same for
representing that the
level of distress
that measures then level
of aggregate
probability of
being hit by a cascade and
indicator yard
measures exposure.
And given a fully
analytical characterized solution where
exact expression
in terms of
the characteristics
of the network
your critical level of
leverage over lambda over theta
you can characterize the probability of
exposure of each firm not to
cascade that in the n
one to analytical results.
>> Okay?
>> And so what's
left is see
whether these cascades in
general equilibrium modify
devaluation by agent
and as expected to
additional contributions
is would be the standard lucas theory
So I'm looking at the
price of risk in
an economy that the super
critical word I want,
say is excess risk primia
>> Knew that.
>> And I have to pay you know if
>> I wanted somebody
buys my shirts, right?
So there are
two additional terms
exposure of firms
to these cascade.
>> risk
>> What is very
important is that
the traditional notion of
risk return trade off
your breaks down.
So we know it's
consistent with what
we see in the data.
That is something
that is called
the idiosyncratic
risk puzzle.
You see that firms
that are less
exposed to idiosyncratic
risk
pay high return this is something you can
reproduce because you
understand that the nature
of this risk is different, right?
>> That is the third 
trade-off concerns
systematic risk
or vice versa.
>> If you have
a mostly, say,
systematic component
pictures for my growth.
So what we can do with
this type of
representation,
see that, how stable
is your economy in
relation to the
geometry of an
important if you
want to synchronize
>> Such a name for the UK.
>> For example, you can
verify that if I had
supercritical threshold diversification breaks
down in the basis
prayed an interbank basis
focusing on banks opens.
>> You characteristic using this primea using
the two banks,
Bank framework
two banks that
had the same pricing in
subcritical state  and
spread these prey
that relates a with that
exposure to these shocks
when you're in the
supercritical.
>> Now once they hear
the geometry of the network
they assume you're
thinking about
more concentrated less concentrated networks.
So we'll try some
new relation,
the geometry of
the network.
And what is interesting
is that now it would
be interesting,
but this is definitely
for the next seminar
>> It's about endogeneity.
>> So think about that
are real network. The real network is endogenous
>> What did you put?
>> So lambda over theta would move endogenously
>> Where would you locate?
>> Most likely, most
likely if you push for
diversification, we,
you will rise the lambda over theta
you realize or not match
will be immediately
the optimum point
immediately below the critical point
>> So a small shock
you may drive you up.
>> These may generate
endogenous crises.
And part is pretty
interesting is that
the system will be
close to critical boundary
It means that this
equation will
be a situation
where, you know,
and I've been
trying small shock
may drive you into
a severe crisis.
Example helped
me come to me.
These are spanning set
in about the company.
>> You are close
to the critical boundary.
Assume that b is
willing to push,
push, push scroll.
>> So they've reduced
the regulation.
So they, it allowed
local banks to rise the amount
of credit that they show.
>> So what is the amount of
interaction among
these two rises
enough, you know,
a small shop and we
divide the system into
this supercritical is going to happen
is that even a local shock
may hit a global economy
think about this as an international crisis
driving prizes
over worldwide.
>> Okay?
>> So going to
generally believe you
what you would expect.
You would expect
that they are risking
your pricing instead
let skewness of consumption
>> This is the
case, you see,
this is the evidence.
>> What is going to
happen in this crisis?
You see what is
going to change.
That is the skewness
on consumption.
And here what that
means in times of
very sacrosanct.
And that's all you see,
that in these days you have
a cross sectional
distribution of
these depending
on the characteristics
of the firm,
which is our
standard price.
>> And it's a good
point to stop. Okay.
>> So I guess you could use
the model to get right.
And also very price
assets in the way they
perceive numbers is that
if the situation is
such that you cannot
diversify even
and reverse,
it nonetheless has
the that goes 0,
then there is a there
is a higher expected return
than the opposite
case when you cannot reverse
>> Yes.
>> Yeah. Yes.
>> Will be so as far a there 
distress risk premium.
There will rise be
the key point is
that that is the
exposure effects.
So more you are 
exposed to cascade risk 
additional preimia, because you know
you are paying an insurance
with respect to be 
hit by cascade,
my what is key is that
that it's a
predictable HI of t
So in practice, when is the
state where you are
most exposed
supposed to this,
which is a state
where you believe
you are in a good,
state that because
the aggregate economy
is in a good state,
that but still
there is a lot
of you are very close
to the critical point.
So a lot of firms
in R and V,
So you believe, okay,
I mean the outside
of the cycle,
but any smokey will
drive you into the
supercritical and
this will generate
a fails
for everything.
So this is more or
less what the why this is 
No impact,
because the impact will
be more severe when
a part of the economy is
arrested and this is just from
They've probably
Mathematica, if you try to
write down this stochastic
discount factor
to move from the
pricing manager plumage
that why we call
it still possible to
reduce this economy,
economy multiple factors,
because here is the
number of sources.
of risk that is
enough because each firm
can contribute also
to propagating risk
So the general
expression of
passing is Km factor is
related with this quasi
stationary measure.
>> And that is it
cannot in any way.
The wow factor model
statement here,
let me draw that.
>> A classical factor model is
a number that says yes,
yes, yeah, absolutely
picks that.
>> So here is the same
as capacity, right?
>> And this is what
creates the cascade?
>> In fact, did
sometimes, you know,
the number of as move for
smoothers service
is going to
decrease as sometimes a,
even a small shop, we
derive a huge, ie,
the rise in the number
of firms that suffer.
>> Reasonable choice
is that there's
no path that the avalanche
formation has not been yet.
>> So let's say there are infinite number of firms
>> And then you thinking 
the number of factors at a point of time is the number of firms that are unhealthy
>> Yes.
>> Yes.
>> Okay.
>> Okay.
