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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..