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In the next series of modules,

we'll actually study the 1-period binomial model.

We'll follow that with the multi-period binomial model.

We'll also discuss replicating strategies.

In fact, that's how we're going to price options within the binomial model.

We'll construct a replicating strategy that replicates the payoff of an option.

And we'll use no-arbitrage pricing then to compute the fair value of the option.

After that we'll discuss European and

American options in the context of the, of the binomial model.

We'll also discuss the Black-Scholes formula, and mention how it can be

obtained by a convergence argument using the binomial model.

Okay, but first of all in this module,

I want to do an overview of some of the questions that we'll be considering.

So here's an example of a binomial model, we're going to be working with

the binomial quite a lot in this unit, and in the unit we cover next week.

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What it means, though, is that an up move, followed by a down move, gives you a price

of ST plus 2 equals ST times u times d.

But that, of course, is equal to ST times d times u,

which is a down move followed by an up move.

In other words, the stock price at time T plus 2 is the same if it had an up

move followed by a down move, as if had a down move followed by an up move.

Okay, and so it's recombining, an up move followed by a down move

gives you the same price as a down move followed by an up move.

And that's why we often call this

a recombining tree, or lattice.

Okay, so this is the binomial for the stock price, and

in any period it goes up or it goes down, we've got a three period model here.

So if the stock price goes up in every period, it ends up with a value of 122.5.

If it goes down in every period, it ends up at a value of 81.63.

We haven't discussed the probabilities of these moves.

For now we'll assume that the probability of an up move is p in any one period.

And so the probability of a down move is 1 minus p.

And that these probabilities are the same at every node in the tree.

So for example, down here the probability of going up to 100 is p.

And the probability of going down to 87.34 is 1 minus p.

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Okay, and of course, we'd be assuming that 0 is less than p is less than 1.

Okay, so that's the stock price, that's the security price,

the risky security price.

We're going to be figuring out how to price options on this stock.

We also have another security in our model that's going to be called the risk free

asset, or the cash account.

Okay, we will assume that's available.

And we'll assume the following.

That $1 invested in the cash account at t equal 0 will be worth r to the power of t

dollars at time t.

So in other words, we're assuming

a growth risk free rate of r per period, okay.

And it's risk free because after t periods,

we know exactly how much we'll have.

We'll have r to the t dollars if we invested $1 in the cash account

at T equals 0.

So this is our binomial model.

We've got the, the stock price, which is described by these dynamics here, a three

period model of the stock price, and we also have our cash account over here.

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I don't know, or at least, we don't know yet.

We'll answer that question pretty soon.

So here's another question, suppose you stand to lose a lot at date t equals 3,

if the stock is worth 81.63.

In other words, if you find yourself down here at date t equals 3,

you're going to lose a lot of money.

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Similarly, maybe you start to earn a lot at date t

equals 3 if the stock is worth 122.49.

In other words if you're up here.

So I've rounded the 0.49 to one decimal place, but if we're up here, we stand to

make a lot of money, and if we're down here, we stand to lose a lot of money.

So suppose you're in that situation,

the question is, could you do something to eliminate this risk exposure?

Is there some way to mitigate your risk, maybe even eliminate it?

And we'll actually see that the answering this question is effectively the same as

answering this question.

And we'll be coming to that in later modules as well.

Okay, so, just to address this particular question here where we say,

should the price be equal to this amount?

Let me give you some evidence for saying why the answer is no.

The option price should not be equal to this quantity.

All right, to do that we're going to come to a very famous example called the St.

Petersburg Paradox, and the St. Petersburg Paradox considers the following game.

A fair coin is tossed repeatedly until the first head appears.

If the first head appears on the nth toss,

then you receive 2 to the power of n dollars.

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Now, you might want to pause the video at this moment and think about this for

a couple of seconds and ask yourself, how much would you be willing to pay to

play this game, if a friend came up to you and gave you this opportunity?

Okay, I'm not sure how much I would be

willing to pay to play this game, but it certainly wouldn't be very much.

And yet, look at the following calculations.

Let's compute the expected payoff of this game.

So the expected payoff is the sum of

the possible payoffs times the probability of those payoffs.

So the probability of receiving a head on the nth toss, well,

to get your first head rather on the nth toss, that means you must get n minus 1,

tails, and then you get one head.

And the probability of this event occurring is,

well you, you get a tail with probability half, so you must get n minus 1 of them.

So that's 1 over 2 to the n minus 1.

And then you get your head with probability a half as well.

And that's equal to 1 over 2 to the n.

So the probability of getting your first head on the nth toss

is equal to 1 over 2 to the n, which is that.

Now remember, you get a payoff of 2 to the n dollars on the nth

toss if that's where the first head appears.

So your payoff at that point is 2 to the power of n.

Well of course, the 2 to the n counts is with the 2 to the n here, and

you're left computing a sum.

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but pretty clearly nobody would be willing to

pay an infinite amount of money to play this game.

Even assuming they had an infinite amount of money to begin with.

So how much would you be willing to, to pay to play this game?

Just to give you an idea, let me ask you this,

would you pay $1,000 to play this game?

In order to break even, or

at least to show a profit, you would have to get, let's see.

So, 2 to the power of 10 is equal to 1024, if I am correct.

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So, this means that in order to break even or to show a profit,

if you paid $1,000 to play this game, you would have to get

nine tails on your first nine tosses,

and only after that point would you actually be assured of showing a profit.

So I personally don't think I'd be willing to play, to pay $1,000 to play this game.

I don't even pay, actually, a much smaller amount to play this game.

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And in fact, Daniel Bernoulli, a famous mathematician,

resolved this paradox by introducing a so-called utility function.

The utility function has the following properties.

It measures how much utility or benefit you're paying from x units of wealth.

So u of x measures how much utility or benefit you obtain from x units of wealth.

Different people of course, have different utility functions.

The utility function should be increasing and concave.

It should be increasing to reflect the fact that people prefer

more money to less money.

And concavity is there to model the fact that getting an extra dollar

when your wealth is say, $1,000, gives you less

additional benefit than getting $1 when your wealth is $0.

In other words, going from $0 to $1 has more benefit

than going from $1,000 to $1,001.

And this idea is captured by using a concave utility function.

So, Bernoulli suggested using log utility function,

the log function is increasing and it's concave.

So, this is an example of a concave function.

It's like an inverted saucer.

Look, it's increasing and it's concave.

So the log utility function is what Bernoulli suggested.

And if we did that with the St. Petersburg game, we find the following.

The expected utility of the payoff is now the sum of the utility of the payoff.

So it's now log of 2 to the n if the first heads occurs on

the nth toss, times the probability of the first head occurring on the nth toss,

which is 1 over 2 to the power of n.

And if you recall, the log of 2 to the n, this is a property of logs,

equals n times the log of 2, well, we get this quantity over here.

And it's quite straightforward to show that this is, in fact, a finite number.

So this is how Bernoulli resolved the St. Petersburg Paradox.

He said that people don't compute values of gains by computing their fair value or

their expected value, but instead everyone has a utility function and

what they would compute is the expected utility of the payoff.

And from there you can determine how much an individual would be willing to pay

to play the game.

Okay.

So given this, you might think that all you need to do

is to figure out the appropriate utility function of an individual and

use it to compute the option price.

Well, maybe, but whose utility function?

The buyer's utility function, the seller's utility function, or

maybe some other utility function in the, in the marketplace?