19. Black-Scholes Formula, Risk-neutral Valuation

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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So let's start with
a simple but quite illustrative example. So suppose you're a bookie. And what a bookie does–
he sets bets on the horses, sets the odds, and
then pays money back. Probably collects a fee
somewhere in between. So suppose he is a
good bookie and he knows quite well the horses,
and there are two horses. He knows that for sure one
horse has 20% chance of winning and another horse has
80% chance of winning.

Obviously, the
general public doesn't have all of this information. So they place a bet
slightly differently. And then there is $10,000
bet on one horse and $50,000 bet on another horse. Well, bookie is sure that he
possesses good information. So he– suppose he
sets the odds according to real-life probability. So he sets it four to one. What would be possible
outcomes of the race for him? Monetary. So suppose the first horse wins. Then what happens? He has to pay back $10,000
and four times more. So he pays out $50,000. And he receives $60,000, right? So he can keep
$10,000 out of it. OK. So what happens is the other
more probable horse wins. Well, he'll have to pay back the
$50,000 and one quarter of it, which is $12.25.

So at the end, he'll
pay 62 1/2 thousand, while he collected
$60,000, out right? So he will– in this
situation, he will lose $2,500. Well, all in all, he
expects to make nothing. So he probably could
collect enough fees to cover his potential loss. But there is certainly a
variability in outcomes. He can win a lot. He can lose some. Now, what if he forgets
about his knowledge about the real-life
probabilities of horses winning or losing and instead sets bets
according to the amount which we are already bet. According to the
market, effectively. So what if he sets
the odds five to one, according to the bets placed? Well, in this situation,
if the first horse wins, he pays back 10 plus
5 times 10, so 60. He is 0. And if the second horse wins,
he pays back 50 plus 1/5 of 50, plus another 10. Again 60. So no matter which horse
wins, he will get 0. We're 100% sure. And if he collects
some fee on top of it, he will make a riskless profit. And that's how, actually,
bookies are operating. So it's a simple example. But it gives us a first idea
of how a risk-neutral framework and risk-neutral pricing works.

So we are, here,
not in the business of making bets on horses. We are actually in the business
of pricing derivatives. So we will talk about the
simplest possible derivatives– mostly derivatives on stocks. But there are more
complicated derivatives, underlying for which could be
interest rates, bonds, swaps, commodities, whatever. So a derivative
contract is some– in general speaking, a
formal pay-out connected to underlying. Usually, the underlying
is a liquid instrument which is traded on exchanges.

And derivative may be
traded on exchanges. Actually, quite a
few equity options are traded on exchanges. But in general, they are
over-the-counter contracts where two counterparties just
agree on some kind of pay-out. One of the simpler derivatives
is a forward contract. So what is a forward contract? A forward contract is a
contract where one party agrees to buy an asset from another
party for a price which is agreed today. Usually, this forward
price is set in such a way that right now, no
money changes hands. Right? And here is an example. Well, suppose there
is a stock which, right now, is priced at $80. And this is the
forward for two years. So somebody agrees
to buy the stock in two years for this price. And not surprisingly, I
somehow set this price such that currently the value
of the contract is 0. And we'll see how I'll
come up with the price.

So this blue line is
actually the pay-out, what will happen at the end. Right? The pay-out,
depending– the graph of F at time T, the
determination time or expiry– how it depends on
the stock price. Right? So obviously, the
pay-out is S minus K, where S is the stock price,
so it's a linear function. It turns out that the counter
price is also a linear function but slightly shifted.

And we'll see how come
it's slightly shifted and how much it
should be shifted. And K is usually referred
to as a strike price. Another slightly more
complicated contract is called a call option. So if previously the
forward is an obligation to buy the asset
for an agreed price, call option is
actually an option to buy an asset at the
agreed price today. You can view it–
a call option can be viewed as kind of
insurance that the– against the asset going down. Basically the pay-out
is always positive. You can never lose money. On the forward,
you can lose money. You agree on the price. The asset ends up being
lower than this price, but you still have to buy it. Right? Here, if the asset ends up
at expiry below strike price or out of the money, then
the pay-out will be 0. If, on the other hand, it ends
up being above the strike price or, it's called, the
option is in the money. Then the pay-out will
be S minus K as before. So in mathematical
terms, the pay-out is maximum of S minus K and 0.

Right? And that's what happens at
expiry time– this blue line. So what is the price
of this option now? Well, obviously it
should be slightly above because even if now
the asset is slightly out of the money–
below strike price– there is some volatility to
it, and there is a probability that we will still end up
in the money at expiry. So you would be
willing– you should be willing to pay
something for this. Obviously, if it's way out
of the money, it should be 0. Right? On the other hand, if it's
way in the money, in fact, it should be just as forward. And in fact, it is. We'll see because
the probability for the asset going back to
the strike price and below will be low. And the Black-Scholes equation
and Black-Scholes formula is exactly the solution
for this curved line, which we'll see in a second.

Another simple contract, which
is kind of dual to call option, is a put option. So put option, on
the contrary, is a bet on the asset going
down, rather than up. Right? So the pay-out is maximum
of K minus S and 0. So it's kind of reversed. Also a ramp function,
at maturity. And here is the current price. Again, even if
it's in the money– if it's way in the money,
we expect it to be 0. If it's way in the
money, we expect it to be slightly below forward,
just because of this counting. OK. So and here are a few–
three main points, which we'll try to
follow, through the class. So first of all,
what we'll see– that if we have current
price of the underlying and some assumptions on how
the market or the underlying behaves, there is
actually no uncertainty in the price of the
option, obviously, if we fix the pay-out. Right? So somehow there
is no uncertainty. It's completely
deterministic, once we know the price of underlying. The other interesting
fact, which we'll find out, is actually
risk-neutrality, meaning that in fact, the
price of the option has nothing to do with the
risk preferences of market participants or counter-parties.

It actually only depends on
the dynamics of the stock, only depends on the
volatility of the stock. And finally, the
most important idea of this class– that
mathematical apparatus allows you to figure out how
much this deterministic option price is now. So let's consider a very simple
example, a very simple market, two-period. So suppose our time is
discrete, and we are one step before the maturity. So right now, our
stock has price at 0. And there is some derivative
f_0 with some pay-out. We'll consider a few of those. Right? Also, we'll add to
the mix a bit of cash. Right? Some amount of
riskless cash B_0. And riskless meaning that
it grows exponentially with some interest rate r. And there is no uncertainty. It's completely– if you
have now B_0, we know then, in time dt, our B_0
will grow exponentially. It will become B e to the rt. So a bond, basically,
zero-coupon bond. Or money market account, rather. If you go to Cambridge
Savings Bank, put $1 in today, then in a year, you'll get
$1 and basically nothing because interest rates are 0.

So in time dt, we will assume
with some probability p, our market can go to the state
where stock becomes S_1– the price of stock becomes S_1. Our bond grows exponentially–
no uncertainty. And our derivative becomes f_1. Or with probability 1 minus p–
only two states, so– our stock becomes S_2. Bond stays the same. And the derivative is some f_2. So let's start with our simple
contract, the forward contract. So one can naively
approach a problem, trying to get the price
of the derivative, using the real-world
probabilities, p and 1 minus p. Right? So we know that the pay-out
is S minus K. That's given. So one would say
that if we know we are one step before the
pay-out, so let's just compute expected
value of the pay-out, using real-world
probabilities, get this value. And actually, what
we are looking here is to set K such that the
price now at time t is 0.

That's usual convention. So we'll then set K
to this probability, to this number, which depends
on real-world probability and obviously depends on
the stock price at expiry. But obviously, we don't know
real-world probabilities. We can guess. We can say, oh, this stock is
as likely to go up then down. Then it's just an average of end
stock prices or something else. But it's all hand-wavy. And actually, we
never will be right. Instead of doing
this– we're kind of following bookie example–
let's try to do something else.

Let's think a little bit. So we have a stock which
is trading at market now for the price S_0. How about we go to the bank and
borrow S_0 dollars right now and immediately go to the
market and buy the stock. So right now we are net 0. We borrowed S_0. We paid it immediately
to buy the stock. So we have stock at hand. Then we'll wait for one period. And at the same
time– sorry– we enter on the short side
of the forward contract. So we agree to sell the
stock for some price K_0. So in dt, in one period of
time, the contract expires. We already have stock. So we just go and exchange
it for K_0 dollars.

Right? But at the same time, we need
to repay our loan which now have become S_0 times e to the r*dt. This is deterministic, right? We borrowed S_0. In time dt, it became
S times e to the r*dt. So what's our net? The net is K_0 minus
S times e r*dt. So suppose K_0 is
greater than this value. Then we made riskless profit. There is no risk in the
strategy which we proposed. So this is good. But why wouldn't everybody
do it all day long? On the other hand, if
K_0 is less than S_0, that's a loss for sure. And if anybody thinks,
as we did– and we assume that everybody can
do it– then nobody would want to enter
it, which means that in order for our
forward to be price 0 now, the strike price has to
be equal to this amount.

And there is no
uncertainty about it. So let's stop and
think a little bit. Well, actually, just
to see how it works. And that's exactly why I
set this K to this number. So by the way, who
can tell me which interest rate does it imply? If our strike– our stock price
is $80, our strike is 88.41. And the expiry is in two
years, approximately. AUDIENCE: 2.5? PROFESSOR: 2.5.

So in two years, it will be 5%. So roughly speaking,
without compounding, it should be 5% of– 80 plus 5%. It would be 84. So 10% for two years. So the interest rate is 5%. Yeah. So yeah. That's actually exactly 5
exponentially compounded. Yeah. Well, in a good world–
probably five years ago, that's how it would work. The two-years interest rates
now, the last time I checked, was, I think, 30 pips. We can check where the
bond is trading now. All right. Give me a sec. Now. Yep. 32 1/2 basis points. 1.6 basis points up,
since the morning. Quite a bit, by the way. So yeah. So right now interest
rates are basically 0. So these two lines would
be very close right now if we were for two
years, in that case.

So coming back to our example. So what's important here? How did we arrive to
this strike price, or to this price of
the forward contract? We, in fact, tried– we
took some amount of stock. In this particular case, it
was the whole price of stock. We took some amount of cash, and
by combining these two pieces, we somehow replicated
the final pay-off. Right? And that's the general idea
of risk-neutral pricing and replicating portfolio. What we will try to do,
in the rest of the class, is take a pay-off and try to
find a replicating portfolio, maybe more complicated, maybe
a dynamic such that at the end, this replicating portfolio
will be exactly our pay-off.

Right? And what would it mean? Well, obviously it would
mean that the current price of the derivative
should be the price of our replicating
portfolio right now. Right? And that's how the
risk-neutral pricing works. So we are still in
this simple situation. But we will try to price
a general pay-off f_1– a general pay-off f. Right? And here's how it goes. So we still will try to form
our replicating portfolio out of the bond, of some amount of
bond, and some amount of stock.

And we'll say that we will
need a S_1 and b of the bond. Right? And we'll try to find a and
b such that no matter what the real-world probability
is, at one step maturity, we'll replicate our
pay-off exactly. And fortunately, in
this particular case, it's very doable. It's just two equations. We use two variables. We should be able to do it. And we can solve it
and find this a and b. Then we'll substitute
them in the formula. Right? Take the current price of
the stock, which we know, and some cash, and
find the current price of the derivative. Right? And this works– it should
work for any derivative. It doesn't matter,
is it forward, call, put, or some
complicated option, as long as it is
deterministic at expiry. An interesting way,
though, to look at it is to rewrite this
formula slightly, in such a way, which does remind
us, taking an expected value, maybe discounting it because
this is expected value at some time in the future. But this probability–
and it is a probability because this number q,
here, is between 0 and 1.

But this probability has
little to do with real world. Right? In fact, it's
something different. But such probability exists. And it's called– the measure
where our stock behaves like this is called a
risk-neutral measure or martingale measure. And in this measure,
as we will see, the value of the derivative
will be just expected value of our pay-out. And that's– yeah. That's what I'm
trying to say, here. So now let's get into
continuous world. Right? In continuous world, we'll
need some assumptions on the dynamics of
our stock underlying.

And let's make an assumption
that it is log-normal. What does it mean
that it's log-normal? It means that the proportional
change of the stock, over infinitely small
amount of time dt, has some drift mu, and
some stochastic component, which is just Brownian Motion. Right? So this dW is
distributed normally with mean 0 and standard
deviation, which is actually square root of dt.

That's how Brownian
Motion works. And that's extremely important,
that the standard deviation of Brownian Motion is
square root of delta t. And that's how it works. And again, we will use this
idea of replicating portfolio. What would it mean in this case? Well, we would like to find
such coefficients a and b, on this infinitely
small period of time dt, such that by combining
small changes in stock, with coefficient a, and
small changes in bond, with coefficient b,
will exactly replicate the change in the derivative–
in the pay-out of derivative– not pay-out. In the derivative. In the change of the derivative,
over this infinitely small time t. Well, to do this, we'll
need to use Ito's formula. Did you talk about Ito already? OK. Cool. That's great. So just to remind you
that Ito's formula is nothing more than the
Taylor rule, actually– the first
approximation up to dt.

But because of the standard
deviation of the Brownian Motion being on the scale
of square root of t, we will need one
more term there. Right? So one term is df/dt by dt. Another is df by dS by dS. And the square of
dS now is actually of order of magnitude of dt. So we'll need a
quadratic term there. All right. So if this is our df, so what
we'll do– we'll differentiate. We'll just substitute it here. Right? We'll substitute it here. We'll substitute df taken from
our dS, which is like this, and dB. Let's not forget that dB–
that B is deterministic. Right? There is nothing
uncertain about it. So dB is actually r*B*dt. All right? Because our B
grows exponentially with interest rate r.

So we substitute everything
into the formula above. This is just our df with
dS expanded and everything. And then when we start
comparing the terms. One immediate thing
to notice– that a has to be equal to df
over dS, for this to hold. Right? And if you compare
the terms near dt, we'll get this expression here. But that's actually even
more the most important part. Then we'll go and use our
knowledge that some part of our equation is deterministic
and basically take f and a*S on one side and leave
the deterministic part, on the other side,
differentiated once again. And left side will
be just r*B*dt. And if we substitute
once again df– and don't forget
that what we learned is that a is equal to df by dS.

Then we collect all
the terms and arrive to this partial
differential equation which connects– which basically
is a partial differential equation for the current
price of a derivative– of any derivative. And how if we solve it,
then we should actually be able to know the
price of the derivative. So now how do we solve this
partial differential equation? Well, for– yeah. So a few observations
about this equation. Well, the first observation
is that any tradable derivative– we made no
assumptions about the pay-off. So any tradable
derivative as any pay-off should satisfy this equation.

The other observation
is as we expected, there is no dependency
on real-world drift or any probability of
it going up or down. The only dependence is on
the volatility of the stock. Right? Not only we found the
value of the derivative– most importantly,
we actually were able to come up with
a hedging strategy. And what does it mean, we came
up with a hedging strategy? Well, we found
coefficients– for any time, we found the
coefficients, a and b, such that we have a
replicating portfolio. So what we could do,
at any point of time, we can hold the derivative–
short derivative and long the portfolio of stock
itself, and some cash, and then know how
much it should be. Here, it's more complicated. We have to dynamically
change these numbers, as time develops. Every time dt we will
have to rebalance. But both parts will replicate
each other perfectly. It's like in a bookie's example.

We can go to a
counterparty, agree for some derivative contract. Probably there will be some fee. And then we'll go to
exchange and buy the stock, and we will get just
cash from the bank. And we'll maintain this
at some amount of stock and some amount of cash. And we'll be sure
that we are hedged. There is no risk in this
combination of the derivative and our hedge. So we will just collect
a fee on the transaction. So that's what actually–
how the business is working. Traders are trading and hedging
their positions immediately. I mean, they do take
some market risks. But you want to take very
little and very directional, very specific market
risks and not everything. So our strategy
allows us to have a hedging portfolio at the
same time– hedging strategy. And now there are more
mathematical but practical consequences that actually,
by certain– not very easy– change of variables, we can
take the Black-Scholes equation and put it back
to heat equation.

Actually, I suggest it as one of
the topics for the final paper, for you to do it or check
it out in the books. Go and understand it. But the good part of it–
that heat equation is well known and well understood. There are many, many ways
to solve it numerically. For simple pay-outs,
for calls and puts, we don't have to
do it numerically, but if the pay-outs
are more complicated or the dynamics is different,
then numerical methods will be needed, for sure. So again, to solve
this equation, we'll need, as for any
partial differential equation, we'll need some boundary
and initial conditions. And these come from
our final pay-out of the option, which we know. We will know what
happens at expiry. And some boundary conditions. For call and put, the
final pay-out we know. Right? So at time T. And the
boundary conditions we discussed, we can
observe them graphically.

So basically for call, as
we said, at current time t, and boundary 0, it should be 0. The price should be 0. And at infinity, it should be
actually the forward price. So it should be just discounted
S minus K. Discounted pay-out. Right? And similarly for put. So given these conditions,
we can solve the equation. And as I said, for call and
put and for simple dynamics– Black-Scholes dynamical or
log-normal dynamics– actually, these equations can be
solved exactly– exactly meaning up to this term, the
normal distribution, which still has to be computed
numerically, obviously. But here are the formulas. They do kind of
look a little bit– and we'll see about it–
there is some kind of expected volume going on. Right? One probability times another.

But these are the formulas. And that's how I drew
the lines on the graphs. And as I said, in
fact, the whole world, instead of solving the whole
partial differential equation, we can approach it from
a risk-neutral position and say that, in fact, the
price of our derivative now is just expected value of
pay-out, discounted, probably, from the maturity. But not in real time
or real-world measure, but in some specific
risk-neutral measure. And how do we find this
risk-neutral measure? Well, the risk-neutral
measure is such that the drift of our stock
is actually interest rate. It's riskless. That's exactly how we saw
it in our binary example. All right? So even in our binary
example, our expected value of our stock, under
risk-neutral measure, meaning using the
risk-neutral probability, was drifting with
interest rate r. So that the same happens
in continuous case.

And that's another
good exercise– and I would accept it as a
final paper– is deriving the Black-Scholes formula
just by the expected value of the call and put pay-out with
the log-normal distribution– terminal distribution. All right. So for more
complicated pay-offs, the life becomes
more complicated. And some finite
differences should be used for more complicated
pay-offs or American pay-offs or path-dependent pay-offs,
tree methods or Monte Carlo simulations.

And that's what was
happening in real life. Yeah. Now, since we have,
actually, plenty of time, I would like to give an example
of how replicating– idea of replicating portfolio works. I give a couple more examples. So OK. Here is a Bloomberg screen
for foreign options– call options on IBM stock. It actually was taken a
while ago– a few years ago. And so here are different
strikes for a call option. The current price of
the stock is $81.14. And here are the
strikes of the call. So obviously, if the option
is way out of the money, meaning the strike is very high
compared to the stock price, the value of the option is 0. If it's way in the money, in
fact, it is just S minus K. So S being $81. And say, the strike being $55. So it's $26.

Right? So there is some difference. But actually, here
it's a bit small because the difference should
be just discounting, as we know. Right? But it's pretty
short-dated options. They are probably a month
long, so there is not much discounting. So it becomes pretty parallel. It's similar here, right? So I mean, this changes by 5. This changes by 5. It's pretty linear. But it becomes non-linear
around the money, around current stock price. Right? So we do observe this behavior. But to tell you the
truth, if you were to– I didn't put implied
volatilities here. But actually, you would
observe that the world is not Black-Scholes,
meaning that– what's the assumption of Black-Scholes. The assumption of
Black-Scholes is that every option, for any
strike, on a given stock, on a given expiry, would
have the same volatility.

Right? So if we went through exercise
of implying the volatility according to
Black-Scholes formula, from the option
price which is traded on the market and
the current price, we would find out that,
actually, the volatility is not constant with strike. Well, it's actually skewed. Well, actually it is smiled. They would find
something like this, which means that Black-Scholes
theory is not perfectly good. Right? So something more
complicated should be done. But in some cases,
we even don't need to do something
more complicated. One example, being
so-called put-call parity. Right? So let's see. Suppose we look at the screen. So we know all prices for all
call options for all strikes. Well, probably will be some
granularity, but we know those. But instead of pricing a
call, we need to price a put. Somehow, we don't know how
the dynamics of our stock looks like. So we have strong suspicion that
it's not exactly log-normal. So there is some
volatility smile.

It's not constant. The world is slightly
not Black-Scholes. So how do we price put? Well, let's see. We'll stare long enough at the
pay-outs of the call and put. So what's the pay-out of
a call with some strike? It looks like this. Right? The pay-out of the put,
with the same strike, would look like this. So what if we take, we
buy a call and sell a put? So this would go like this. Right? Straight line. Looks very much
like forward, right? So if we actually subtract
the stock from here, move it from here,
then it should be– yeah– minus K. Yeah. I think I got the signs correct. Right? And this is just a number. Right? And that's what
happens at pay-out. So if we take this portfolio,
if we action now, buy a call, sell a put, and sell a stock,
we know that at the end, we'll for sure get K in money. Right? So which means that now–
so this is at time t. So right now, it looks, to
me, that if we do write this, and that's just the
current price of the stock, this should be– right? We just need to discount
this price to now, in this amount of cash, which
means that our put, at any time t, is stock minus K.

Right? So if we know all of the
prices for any strike K– if we know price of a call, we
don't need any Black-Scholes or anything. We can immediately tell
everybody how much is a put. Right? So then this relationship is
actually a call-put parity. And that's, again– that's
a replicating portfolio. It's a simple
replicating portfolio. It's static, meaning
that we fixed it now and we don't change
it to expiry. So it's quite good this way. But that's how it works. Another example. So for this, I have,
actually, a picture. So again, we have
the same situation. We have prices of calls. But instead of pricing a call,
we want to price a digital. So what is digital? Digital is such
a weird contract, which pay-out is just
a function– Basically, it's a bet on the
stock to finish above strike price, K. Right? If at expiry, the stock
is above K, you get 1.

You'd get $1. If it's below, you'd
get nothing, 0. Right? So So such an interesting contract. The question is,
can we price it, given that we know
the prices of Calls? And I suggest we use the idea
of replicating portfolio. Any ideas how to do it? It's my typical
interview question. So just pretend that
you are interviewing. Yep? AUDIENCE: You long the call,
and then you short the call, just like smaller
or a higher strike. PROFESSOR: Yep. The call strike. Yeah, you're absolutely right. Good. You've got an offer. Yeah. So here's how it goes. So this is a strike K. Right? So let's buy a call
with strike K minus 1/2 and sell a Call with
strike K plus 1/2.

Right? We just sold. So if we combine these two–
well, actually, if this is 1– yeah. If this is 1, it should
look something like this. Great. So how will it look like? So obviously, here, it's 0. Right? Then it will be like this. Right? And after that, it will be what? AUDIENCE: Constant. PROFESSOR: It will be constant. Right? And because this is K minus
1/2 and this is K plus 1/2, it will be exactly 1. Right? Good. So our pay-out, at the
end, will be like this. So that's good. But there is quite
a bit of slope here. So how can we do
better than this? Well, if we buy it at K minus
1/4, and sell it at K plus 1/4, and just combine those, it
will be exactly the same, but the level will be 1/2.

So we need to buy two of those
and to sell two of those. Right? Well, we might as well go
K minus epsilon and K plus epsilon, so it'll be call price
at strike K minus epsilon, minus call price at K plus
epsilon, divided by 2*epsilon. Right? This 2*epsilon coefficient
needed rescale it back to 1. Right? So in fact, if we go small
epsilon, we need a lot of both. Right? And that's how– that's
the approximation of our digital price. And that's actually how
people on the market do price and hedge,
most importantly, the digital contracts, because
call contracts are liquid, and they are traded on
exchanges while digitals are way less liquid. So somebody would call
again– to counterparty, enter into digital, and
hedge it on the exchange.

These two calls
with a call spread. But now tell me,
is it surprising that– I mean, what
does it remind you? Yeah. So it's derivative
of the call price but with respect to strike. Right? Is it surprising? How did our call
price look like? It's a ramp. Right? If we take a derivative
of this, what will we get? Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: Right. So in fact, if we do something
even more weird with this, and then I'll take a
square or something else, the same will apply. So it's not surprising at all. All right. So that's basically how
the replicate– this idea of replicating portfolios
is extremely powerful. And in fact, that's what
happens in real life. In real life, you have
some complicated derivative which you need to hedge. And how to hedge– you'll
find something else which replicates– to
a certain extent, replicates your pay-off. That's what you'll try to do.

And this will be
your hedge portfolio. Usually, it's dynamic. So you'll have to rebalance. And that's how you
basically reduce the risks..

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