So now, suppose we want to price a European call option in our three-period binomial model. We're going to assume the strike is $100, and therefore, the payoff of the option at time t equals 3 is given to us here, it's zero and zero on the bottom two nodes, this is because the strike is $100 which is greater than the stock price at these nodes so you wouldn't exercise and you would receive zero. 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. Now, what we want to do is figure out how much is this option worth at time t equals 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, 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 seven at this node, 22.5 at this node, we can compute the fair value of the security at this node. We can do that using our one-period knowledge. We can do the exact same for this node. We can 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. 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's 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. Again, we want to compute the value of a derivative security with a payoff 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-neutral probabilities to compute its fair value here which we would call C_0. So that's all you have to do. We can splice our one-period models together. They're all arbitrage-free as we said because d is less than R is less than u. So there are risk-neutral 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. Here are the calculations. So I haven't actually done the calculations here, but there is a spreadsheet that you can download with this module. In the spreadsheet, there will be a particular worksheet 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. Of course, in this case, u is equal to 1.07, d is 1 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 won't have 22.5 and seven, they will just refer to the cells containing 22.5 and seven, and it will be the same formula repeated throughout the spreadsheet. So that's how we compute the value of the option and its fair value times 06.57. It's important to keep in mind that this is the arbitrage-free value of the option. The way we calculate these values 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. So these are risk-neutral 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, where the call price at time 0 of C_0 equals one over R cubed. So this is our discount factor. It's cubed because it's three periods, and that's the expected payoff of the option which is S_T minus 100 and the maximum of that and 0 under these risk-neutral probabilities here. So I can do it in one shot. So basically, working backwards one period at a time, you can check the exact same thing as doing it all as just one calculation like this. This is risk-neutral pricing in the binomial model, it avoids having to calculate the price at every node. By the way, you can compute any derivative security in this model this way. You can compute the payoffs here at t equals 3 and use risk-neutral 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 it C of 122.5. So this is the underlying security price at this node, C of 107, C of 93.46, and C of 81.63. So this could be the derivative payoff C_3. There's some value at time t equals 3, and its value depends on the security price at t equals 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 risk-neutral probabilities of C_3. That's exactly the same argument 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 we have those three nodes, I can work backwards these two nodes. Once I have the value of these two nodes, I can work backwards to 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 confirm that all we're using the one-period risk-neutral pricing formulas. Okay. Another question that arises is down here. How would you find a replicating strategy? I will address this question as well as defining what a replicating strategy means in another module that we'll see very shortly.
