We're not 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. So let's get started. Here's a three-period binomial model. It's actually the same three-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 S0 equals $100. We have a gross risk-free rate of R=1.01 per period. We assume that in each period the stock price goes up by a factor of u or falls by a factor of d. So u=1.07, so stock price goes up by a factor of u to 107 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-p. But we also saw in the last module that p and 1-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-p are greater than 0, and there's no arbitrage, we determine that there were q and 1-q also greater than 0. These guys are called the risk-neutral probabilities. And we saw that we can use these probabilities to compute option prices. For example, in our 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-neutral probabilities of the payoff of the derivative at time 1. Okay, so we're now in our three-period binomial model. We want to be able to price options in the three-period binomial model. And we can easily 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 three-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=2 to t=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=1, there are two possible one-period models, there's this model, and there's this one-period model. And at t=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 three-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-q. These are the risk-neutral probabilities that we can use to price an option in this one-period model. Well, the same risk-neutral probability 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 also 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-neutral probabilities, q is equal to R-d over u-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-neutral probabilities, q, 1-q, q, 1-q, q, 1-q. And indeed, it's true also a time t=1, q, 1- 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-q. So in fact, this three-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 construct probabilities for the multi-period model by multiplying these one-period probabilities appropriately. Suppose, for example, I want to computes some risk-neutral probabilities in this three-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-neutral 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-neutral 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-neutral probability of the stock price may equal to 107 at time t=3? Well, in this case, it's actually going to be 3 times q squared times 1-q. Now, how do I know that? Well, let's think about it. There are actually three ways to get to 107. One way is 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=3 can end up with 107. Each of those paths requires two up moves, which occurs at probably q squared, and one down move, which occurs at probably 1-q. So we get q squared times 1- q, and there are three such paths, so we get 3q squared 1- 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-q squared. We have 1-q squared now because we need two down moves, and a down move occurs with probability 1-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-q cubed, okay? You might recognize these probabilities as being the binomial probabilities, okay? So the binomial probabilities will say that the probability will be n choose r times q to the r 1-q to the n-r. In this case, n=3 and r is the number of up moves required. So if r equals 3, then we must have three up moves and we get q cubed. If r equals 1, then we have 3 choose 1 equals 3, and we get this number here and so on.
