0:16

In the next series of modules,

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

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

Â We'll also discuss replicating strategies.

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

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

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

Â After that we'll discuss European and

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

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

Â obtained by a convergence argument using the binomial model.

Â Okay, but first of all in this module,

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

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

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

Â 1:50

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

Â of ST plus 2 equals ST times u times d.

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

Â which is a down move followed by an up move.

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

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

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

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

Â And that's why we often call this

Â a recombining tree, or lattice.

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

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

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

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

Â We haven't discussed the probabilities of these moves.

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

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

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

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

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

Â 3:17

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

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

Â the risky security price.

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

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

Â asset, or the cash account.

Â Okay, we will assume that's available.

Â And we'll assume the following.

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

Â dollars at time t.

Â So in other words, we're assuming

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

Â And it's risk free because after t periods,

Â we know exactly how much we'll have.

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

Â at T equals 0.

Â So this is our binomial model.

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

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

Â 6:26

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

Â We'll answer that question pretty soon.

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

Â if the stock is worth 81.63.

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

Â you're going to lose a lot of money.

Â 6:45

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

Â equals 3 if the stock is worth 122.49.

Â In other words if you're up here.

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

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

Â So suppose you're in that situation,

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

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

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

Â answering this question.

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

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

Â should the price be equal to this amount?

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

Â The option price should not be equal to this quantity.

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

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

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

Â If the first head appears on the nth toss,

Â then you receive 2 to the power of n dollars.

Â 8:01

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

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

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

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

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

Â And yet, look at the following calculations.

Â Let's compute the expected payoff of this game.

Â So the expected payoff is the sum of

Â the possible payoffs times the probability of those payoffs.

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

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

Â tails, and then you get one head.

Â And the probability of this event occurring is,

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

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

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

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

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

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

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

Â toss if that's where the first head appears.

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

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

Â you're left computing a sum.

Â 9:41

but pretty clearly nobody would be willing to

Â pay an infinite amount of money to play this game.

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

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

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

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

Â In order to break even, or

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

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

Â 10:15

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

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

Â nine tails on your first nine tosses,

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

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

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

Â 10:52

And in fact, Daniel Bernoulli, a famous mathematician,

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

Â The utility function has the following properties.

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

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

Â Different people of course, have different utility functions.

Â The utility function should be increasing and concave.

Â It should be increasing to reflect the fact that people prefer

Â more money to less money.

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

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

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

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

Â than going from $1,000 to $1,001.

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

Â So, Bernoulli suggested using log utility function,

Â the log function is increasing and it's concave.

Â So, this is an example of a concave function.

Â It's like an inverted saucer.

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

Â So the log utility function is what Bernoulli suggested.

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

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

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

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

Â which is 1 over 2 to the power of n.

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

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

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

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

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

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

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

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

Â to play the game.

Â Okay.

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

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

Â use it to compute the option price.

Â Well, maybe, but whose utility function?

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

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

Â