>> We're now going to extend our results from the one-period binomial model to the

multi-period binomial model. We'll see that our results from the

one-period binomial model actually extend very easily to the multi-period model,

we'll see that our results from the one-period binomial model, actually extend

very easily to the multi-period model. So, let's get started.

Here's a 3-period binomial model, it's actually the same 3-period binomial model

that we saw a while ago when we had our overview of option pricing.

We start off with a stock price of S zero equals $100, we have a gross risk-free

rate of r equals 1.01 per period. We assume that in each period, the stock

price goes up by a factor of u, or it falls by a factor of d.

So, u is equal to 1.07, so stock price goes up by a factor of u to 1.07, or it

falls down to 93.46. Now the true probability of an up-move is

p, and the true probability of a down-move is 1 minus p, but we also saw in the last

module That P, and 1 minus P, don't matter when it comes to pricing an option.

As long as in fact, and this is a subtle point, as long as P, and 1 minus P, are

greater than 0, and there's no arbitrage, we determined that they were Q, and 1

minus Q, also greater than 0. These guys are called the risk mutual

probabilities, and we saw that we can use these probabilities, to compute option

price. For example, in a one-period model, we saw

that we can compute the price of a derivative as being equal to 1 over R

times the expected value using these risk mutual probabilities of the pay-off of the

derivative at time 1. Okay.

So, we're now in our 3-period binomial model.

We want to be able to price options in the 3-period binomial model, and we can easily

do in-, do that using our results from the one-period case.

Because the central observation we want to make, is this multi-period, or in this

case, 3-period binomial model is really just a series of one-period models spliced

together. So for example, here is a one-period

model, here is another one-period model and here is another one-period model.

So, in fact from t equals 2 to t equals 3, there are three different one-period

models, only one of which will actually occur, but there are three possible

one-period models. Likewise, at t equals 1, there are two

possible one-period models, there's this model and there's this one-period model.

And at t equals 0, there's only one one-period model, and it's this one.

So in fact, we see, we've got six different one-period models in this

3-period binomial model. And what we can do is, we can use our

results for the one-period model that we developed in the last module, on each of

these six one-period models, so in fact, that's what we will do.

Okay, so what we have is we saw that if there's no arbitrage, in the one-period

model, we know there are probabilities q and 1 minus q, these are the risk mutual

probabilities that we can use to price an option in this one-period model.

Well the same risk neutral probabilities will occur, or can be used here and here,

and likewise there, and there. Remember each of these one-period models

is essentially identical, the stock price goes up by a factor of u, or it falls by a

factor of d, it's the same u and d in each of these one period models.

It's also the same gross risk free rate r in each of these models.

So in fact, they'll have the same risk mutual probabilities.

Q is equal to r minus d over u minus d. So in fact, since r, d and u are the same

for all of the one-period models, all of the one-period models have the same risk

mutual probabilities, q1 minus q, q1 minus q, q1 minus q, and indeed, it's true also

at time t equals 1. Q1 minus q and of course these are the

true probabilities. Let's erase them, and let's replace them

with the risk neutral probabilities q and 1 minus q.

So in fact, this 3-period binomial model, can be thought of as being six separate

one-period models, if each of these one period models are arbitrage free and we

recall that will occur if d is less than r is less than u.

Then we can compute risk neutral probabilities for each of the one-period

probabilities and then we can construct probabilities for the multi-period model,

by multiplying these one period probabilities appropriately.

Suppose for example, I want to compute some risk neutral probabilities in this

3-period Binomial Model. How can I do that?

Well, let's create some space here and let's get rid of this stuff.

Okay. Let's compute the probability, the risk

mutual probabilities, let's call them Q, of arriving at each of these terminal

security prices. So, how about this point here, what is the

risk mutual probability, of the stock price being equal to 122.5?

Well the only way the stock price can equal 122.5 is if the stock price goes up

in each period. It has to go up in every period.

The probability of it going up in every period is q times q times q and that's, q

cubed. How about at this point here?

What is the risk mutual probability of the stock price being equal to 107 at time t

equals 3? Well in this case, it's actually going to

be 3 times q squared times 1 minus q. Now how do I know that?

Well let's think about it. There are actually 3 ways to get to 107,

one way is to, for the stock price to fall initially, and then to have two periods

where it grows, goes up. A second way is for the stock price to

have two periods up, followed by one period down.

And a third way is for the stock price to go up, then to go down and then to go up

again. So there's three such paths through the

model, where the security price at time, t equals 3 can end up at 107.

Each of those paths requires two up-moves, which occurs at probability q squared and

one down-move which occurs at probability 1 minus q.

So we get q squared times 1 minus q and there are three such paths, so we get 3q

squared one minus Q. Okay, it's the same for 93.46, there are

three ways for the security price to get to 93.46, It can go up and then have two

down-moves. It can have two down-moves and then one

up-move, or it can have a down-move, an up-move, and then a down-move.

So in fact, this occurs with probability 3q times 1 minus q squared.

We have 1 minus q squared, now because we need two down-moves and the down-move

occurs with probability 1 minus q. Finally, the stock price can be 81.63 only

if we have three down-moves in a row and that occurs with probability 1 minus q

cubed. Okay?

You might recognize these probabilities as being the binomial probabilities, okay, so

the binomial probabilities we'll say that the probability will be n choose r times q

to the r 1 minus q to the n minus r. In this case n is equal to 3.

And r is the number of up-moves required. So if r equals 3, then we must have 3

up-moves and we get q cubed. If r equals 1, then it's 3 choose 1 equals

3 and we get this number here, and so on. So now suppose, we want to price a

European call option in our 3-period binomial model.

We're going to assume the strike is $100 and therefore, the pay-off of the option

at time T equals 3 is given to us, here, it's 0 and 0 in the bottom two nodes this

is because the, the strike is a $100, which is greater then the stock price of

these nodes, so you wouldn't exercise and you would receive 0.

If the stock price ends up at 107 you would exercise, you'd get 107 minus 100

which is $7. Likewise up here you would receive $22.5.

And now what we want to do is figure out how much, is this option worth at time t

equal 0. In other words, what's the fair value or

arbitrage free value of this option. Well we can do this simply, by working

backwards using what we know about the one-period model.

So, we know how to price options in a one-period model, we saw this in the last

module, we're going to do this here as well.

So what we can do is, we can start at time t equals 3, okay, and we're going to work

backwards from T equals 3. So what we can do is, we can actually

start with this one-period model here, so let's take a look at this one-period model

and just figure out how much is this derivative security worth at this node

here. This is a one-period model, which pays off

7 at this node, 22.5 at this node, w e can compute the fairer value of the security,

at this node. We can do that using our one-period nodes.

We can do the exact same, for this node, okay, we can come treat this as a

one-period model, compute the fair value at this node and also compute the fair

value at this node. Okay, so by working backwards now we can

assume we know the option price at this node, at this node, and this node, and now

we can do the exact same thing. We can now go from t equals 2 back to t

equals 1. In this case we've got two, one-period

models, here is one of them. We know how much the option price is worth

there, we know how much it's worth here, so we can figure out how much it's worth

here again using our results from the one-period theory.

Likewise, in the one-period model here we can do the same thing, we know how much

the option is worth at this node, we know how much it's worth at this node, it's

already calculated, and we can use our one-period knowledge to figure out its

value at this node. Finally, we can go from t equals 1 to t

equals 0, and again, we want to compute the value of a derivative security with a

pay-off of this quantity at this node and this quantity at this node.

And we can actually compute the fair value of this, again using the risk-mutual

probabilities, to compute its fair value here, which we would call C0.

So that's all you have to do. Right, we can splice our one-period models

together, they're all arbitrage free as we've said, because D is less than r is

less than u, so there are risk mutual probabilities in each of these one-period

models. So what we can do is just work backwards,

starting off with the final value of the option at t equals 3.

Figure out how much it's worth at the nodes at t equals 2, using our one-period

theory. Going from t equals 2, back to t equals 1,

again using our one-period knowledge, and from t equals 1 back to t equals 0.

And here are the calculations. So, I haven't actually done the

calculations here, but there is a spread sheet that you can download with this

module and in the spread sheet there'll be a particular work sheet which will

actually have these calculations as well as the formulas inside the cells which

will do these calculations for you. So, what you'll see is we're actually

calculating these quantities, according to the one-period model.

So for example, let's take a look at this one-period model here.

I know that the 15.48 over here, is equal to 1 over r times q, of 22.5, So q times

22.5, plus 1 minus q times 7. This comes from our one-period theory and

of course q is the risk neutral probability of an up-move, it's equal to r

minus d over u minus d. And of course, in this case, u is equal to

1.07, d is one over u, and r was equal to 1.01.

So, you can actually check these calculations in the spreadsheet, if you

like, you can have the spreadsheet open while you're going through this module,

and you'll see the formulas in each of the cells showing these calculations.

So what we're doing, is we're working backwards, so.

The cell, here, at this point in the spreadsheet, will have exactly this

formula here. Likewise, except it wont have, 22.5 and 7,

it will just refer to the cells, containing 22.5 and 7, and it will be the

same formula repeated throughout, throughout the, the spreadsheet.

So that's how we compute the value of the option and it's fair value at time 06.57.

And it's important to keep in mind that this is the arbitrage free value of the

option. The way we calculated this value is by

using our one-period knowledge and working backwards one period at a time, but in

fact there is a faster way to do it. We can use what we did in the previous

slide, where we calculated these risk neutral probabilities.

Okay, so these are risk-mutual probabilities.

You can easily check, that doing this backwards calculation, working backwards

one period at a time, is actually the same thing as doing it all in one shot.

So instead of doing a calculation coming back from t equals 3, to t equals 2, to t

equals 1, to t equals 0, I can do it as just one calculation, okay?

Where the call price at time 0 c0 equals 1 over r cubed, so this is our discount

factor, it's cubed because it's 3-period, and it's the expected pay-off of the

option, which is ST minus 100, and the maximum of that in 0 under these risk

mutual probabilities here. So, I can do it in one shot!

So, basically working backwards one period at a time you can check is it the exact

same thing as doing it all as just one calculation like this.

Okay. This is risk mutual pricing of the

binomial model, it avoids having to calculate the price at every node.

And by the way, you can compute any derivative security in this model this

way. You can compute the pay-offs here at t

equals 3, and use risk mutual pricing in one shot like this.

So for example, let's create some space here.

So. Okay.

Suppose I want to compute a derivative security, which has pay-offs C, let's call

this, okay so let's call it c of 122.5. So this is the underlying security price

at this node, c of 1 of 7, c of 93.46, and c of 81.63.

Okay, so, this could be the derivative payoff c3, there's some value at time T

equals 3, and, its value depends on the security price at T equal 3, so it could

be a call option a put option or some other funky security.

Then I can calculate this price, As 1 over r cubed times the expected value, using

these risks neutral probabilities of c. Okay, and it's exactly the same margin we

used for the one-period model. I could work backwards one step at a time

to compute the value at each of these three nodes.

Once I have those three nodes, I can work backwards to these two nodes.

And once I've divided these tee-, two nodes, I can work backwards and get the

value here, or I can do all of that in one shot, via this calculation here.

The spreadsheet does it by working backwards one period at a time and you can

see the formulas in there and I'm confirmed that all we're using are the

one-period risk neutral pricing formulas, okay.

Another question, that arises, is down here.

How would you find a replicating strategy? We'll address this question, as well as

defining, what a replicating strategy means, in another module that we'll see

very shortly.