0:00

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â derivative at time 1. Okay.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â with the risk neutral probabilities q and 1 minus q.

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

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

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

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

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

Â by multiplying these one period probabilities appropriately.

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

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

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

Â Okay. Let's compute the probability, the risk

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

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

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

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

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

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

Â cubed. How about at this point here?

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

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

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

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

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

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

Â have two periods up, followed by one period down.

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

Â again. So there's three such paths through the

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

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

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

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

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

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

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

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

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

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

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

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

Â cubed. Okay?

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

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

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

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

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

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

Â European call option in our 3-period binomial model.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â same formula repeated throughout, throughout the, the spreadsheet.

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

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

Â option. The way we calculated this value is by

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

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

Â slide, where we calculated these risk neutral probabilities.

Â Okay, so these are risk-mutual probabilities.

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

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

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

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

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

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

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

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

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

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

Â Okay. This is risk mutual pricing of the

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

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

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

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

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

Â So. Okay.

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

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

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

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

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

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

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

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

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

Â to compute the value at each of these three nodes.

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

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

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

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

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

Â one-period risk neutral pricing formulas, okay.

Â Another question, that arises, is down here.

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

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

Â very shortly.

Â