Multi- period option. Multi-period option - Oxford Reference


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 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. 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 multi- period option, 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.

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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 multi- period option 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 idea on which people made money 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, multi- period option 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.

Before getting into the depths of an option pricing model, it is important to first understand what an option is.

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 multi- period option 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 multi- period option, 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.

multi-period option

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 multi- period option, q1 minus q, multi- period option 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 multi- period option 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.

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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.

Option Pricing Model

Let's compute the probability, the risk multi- period option 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 Well the only way the stock price can equal It has to go up in every period.

The probability multi- period option 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 at time t equals 3? Well in this case, it's actually going to be 3 times q squared times 1 minus q.

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Now how do I know that? Well let's think about it. There are actually 3 ways to get toone 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.

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  • Multi-Period Option

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. So we get q squared times 1 minus multi- period option and there are three such paths, so we get 3q squared one minus Q.

Okay, 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 multi- period option 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.

Easy Binomial Trees in Excel

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. And now what we want to do is figure out how much, is this option worth at time t equal 0.

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In other words, what's the fair value or arbitrage free value of this option. Well we can do this simply, by working binary option lve 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 multi- period option 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. 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 ideas for internet business with minimal investment, at this multi- period option, 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 multi- period option, so we can figure out how much it's worth here again using our multi- period option from the one-period theory.

Likewise, in the one-period model here we multi- period option 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, multi- period option compute its fair value here, which we would call C0.

So that's multi- period option you have to do. Right, we can splice our one-period models together, they're all arbitrage free as we've said, because Multi- period option 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.

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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.