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 multi- period options the one-period binomial model, actually extend very easily to the multi-period model. So, let's get started. Here's a 3-period multi- period options 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 assume that in each period, the stock multi- period options goes up by a factor of u, or it falls by a factor of d. So, u is equal to 1. 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 multi- period options 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 when binary options open 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 multi- period options expected value using these risk mutual probabilities of the pay-off of the derivative at time 1.
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 multi- period options, 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 multi- period options.
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 multi- period options can use to price an option in this one-period model.
Well the same risk neutral probabilities will occur, or can be used multi- period options 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.
Binomial Option Pricing Model
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. Let's compute the probability, the risk mutual probabilities, let's call them Q, of arriving at each of these terminal security prices.
Multi- period options, how about this point here, what is the risk mutual probability, of the stock price being equal to Well the only way the stock price can equal 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?
the financial encyclopedia.. everything about finance
What is the risk mutual probability of the stock price being equal to 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.
Multi-period, continuous, and compound models
There are actually 3 ways to get toone way is to, for the stock price multi- period options 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 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.
Explore our Catalog
So we get q squared times 1 minus q and there are three such paths, so we get 3q squared one minus Q. Multi- period options, it's the same for 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 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 multi- period options. So if r equals 3, then we must have 3 up-moves and we get q cubed. If r equals 1, then it's multi- period options 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. 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 multi- period options 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 is discipline in trading 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, We can do that using our one-period nodes.
The Multi-Period Binomial Model
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 multi- period options 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 multi- period options 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 multi- period options 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.