>> 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.