>> Up until this point we've only seen how to price European options in the binomial model. We are now going to consider the case where the holder of the option has the ability to exercise early, such an option is actually called an American option and we'll see that we can also easily price these securities in our binomial model. So the only difference is that, we must now also check if it's optimal to early exercise at each node. So if you recall the way we priced European options, we actually started at the end. At t equals 3. Here, we've got a three period binomial model. We began at the end, computed the payoff of the European option at t equals 3. And then we worked backwards one period at, at a time, figuring out how much the option was worth at each period. And working way backwards, using our one period results to get the option price at times zero. That's how we computed the European option prices. Well, we can do the exact same thing with American options. We're going to start at the end and we're going to work backwards one period at a time using the risk mutual probabilities in each period to compute the value of the option. The only difference is, now we have to check at each period if it's optimal to exercise early. So when we're working backwards, we're going to compute the value of the option but also check to see if that value is greater than the value of exercising at that point. If it is, we don't exercise. If it's not, we do exercise and we continue backwards. Okay, and we're going to see how this, how this works in practice. There's also a spread sheet that you can use to see the calculations, and to see how we price the American options. So if you like you can go through that spreadsheet or have it open, while we're doing these calculations. Okay, first of all recall that it is never optimal to early exercise an American call option on a non-dividend paying stock. So we saw that in an earlier module, so we're actually going to consider pricing American put options here. So the put option is going to as-, as-, assume an expiration or a maturity of t equals 3. A strike of $100. And we'll assume the risk-free rate, the gross risk-free rate per period is 1.01. Okay, so this is our binomial model that we've been using a lot until now. It starts off at $100 the stock price starts, at $100. It goes up by a factor of u. It goes down by a factor of d in each period. So the payoff of the put option. So remember the payoff of the put option will be the maximum of zero and the strike minus the stock price at time three. The stock price at time three is 81.63, 93.46, 107 and 122.5. The strike is 100. So we will only exercise if the security price is less than 100. So that's why we're going to get zero up here. And we're going to get 100 minus 93.46 down here, and 100 minus 81.63, which is 18.37. So this is the payoff of the American option if we exercise at a time t equals 3. And all we're going to do to price this American option is we're going to work backwards in the same manner as we did with European options, but this time at each node we'll also have to check if it is optimal to exercise at that point or not. Okay so here's an example of how we do these calculations. So we know the value of the American option at t equals 3. It's 0, 0, 6.54, 18.37. So we're going to work backwards one node at a time. So let's actually consider this piece here. And suppose we're down at this node here and we want to figure out how much is the option worth. Well, if it was a European option, we would say the fair value of the option is 1 over r times q times 6.54 plus 1 minus q times 18.37. That's how we computed the fair value of a European option in the one period model. We discount it and use the risk mutual probabilities to compute the expected payoff one period ahead. We do the exact same thing here. Except, we have the ability to early exercise the option. In other words, when we're at this node we have two choices. We can choose to continue. Or we can exercise. The choice is ours. If we choose to continue, the fair value of the option is this guy here, whatever this turns out to be. Let's just call it x. So we'll get x. If we exercise, we'll get 100 minus 87.34. And that's equal to 12.66. However the, the choice is ours. So we will choose to exercise if 12.66 is greater than x. And we will choose to continue if x is greater than 12.66. In this case it turns out that 12.66 is the greater of the two so x is less than 12.66. So at this node we choose to exercise and we get 12.66. So the fair value of the American option at this node is $12.66. And we do the exact same thing at this node and at this node. And as I said earlier, you can see these calculations in the spreadsheet that we've, we've uploaded onto Coursera. Okay. So we can work backwards in each period and we actually find that the fair value of the American option is 3.82. Okay. Not only that, you will see that the only point at which it is optimal to exercise early is down here at this node. And at each of the other nodes the fair value of the option turns out to be the continuation choice. It's only at this node that exercising was optimal, and that's why we've highlight it in a different color here. So that's how we price American options. We just work backwards in the lattice, in the binomial tree as we did with the European options, but with the added complication of having to check at each node whether it was optimal to exercise or not. To give ourselves a flavor of optimal stopping problems, we're going to consider one more example. And this is a simple die throwing game. So consider, for example, the following game. We've got a fair six-sided die. So let's try and draw, a six-sided die. Okay. So you've got the numbers one to six on the six sides of this die. We'll now throw the die up to a maximum of three times. After any throw you can choose to stop. And when you stop you obtain an amount of money equal to the value you threw. So the value you threw is whatever is showing here with you throw the die. So for example if you throw a four on your second throw and you chose to stop, then you'll obtain $4. So if you are risk neutral how much would you pay to play this game? By risk neutral I mean that you just want to compute the fair value of this game using the true probabilities. The amounts of money concerned are so small the risk doesn't enter into the situation. Okay. So how much is this game worth playing? I always like this game because it gives you an example of another optimal stopping problem. It's also a game or a question that many interviewers over the years have liked to ask our students when they were interviewing for jobs in the, in the financial industry. So let's think about this. So the solution is going to be to work backwards starting with the last possible throw. So if we've just got one throw remaining, then the fair value of the game must be 3.5. And the reason is, on that last throw you can get 1, 2, 3, 4, 5 or 6. Each with probability 1 6th. And so the fair value of this game is 1 6th times 1, 1 6th times 2, and so on, and it's equal to 3.5. Now, suppose you've got two throws remaining. What we must do is figure out a strategy which determines what to do after the first throw of these two throws, okay? So that first throw refers to the first of the two throws remaining, so it's throw two in the overall game. Okay, so let's think about this. So, we've got, throw number two. We can get a 1, a 2, a 3, a 4, a 5, or a 6 on that throw. Each of these occurs with probability 1 6th. So if we get a 1, 2, or 3, if you think about it, it would make sense to continue. Why? Well if we continue, we will expect to get 3.5 on our last throw. So why would you stop and get a 1, a 2, or a 3 when you expect to get a 3.5 on your last throw? So in each of these, you would continue, and expect to get 3.5. On the other hand, if you get a 4, 5, or a 6, then why would you continue? Why would you continue and expect to get a 3.5 when you can stop and get a bigger number. So in fact, for each of these numbers, you will stop. And get 4, 5, or 6. So the fair value of the game with two throws remaining, is, 1 6th, times 3.5, plus 1 6th times 3.5., plus 1 6th times 4, plus 1 6th times 5. And so on. So the fair value of the game is 1 6th times 4, plus 5, plus 6, plus 1 half times 3.5 which is 4.25. So that's the fair value of the game if the, if there's two throws remaining. If there's three throws remaining, the exact same idea works. You now consider the first throw. Okay, so there's not much room here, but the first throw, we could start with here. And we would figure out what to do in each of these scenarios. Well, If we throw a 1, 2, 3, or 4, we would actually choose to continue. Why? Because the fair value of continuing with two throws remaining would be 4.25 which is greater than 4. Otherwise, if we get a 5 or 6, we would stop. So you can actually compute the fair value of this game is going to be 2 3rds times 4.5 plus 1 6th times 5 plus 1 6th time 6. And that's the fair value of this die-throwing game. If you could throw the die 1,000 times. What do you think will happen to the fair value of the game then? It should be pretty clear. And I'll let you think about that. The reason I like this game, though, is, it emphasizes the idea of a strategy, or a stopping time. You have to figure out the optimal strategy, what you must do at each point in time. Do you continue or do you stop. With the American option problem you have to figure out do you exercise or do you continue. In the binomial example over here we did it automatically by working backwards. And at each point, at each node we figured out the optimal strategy. Do we exercise or we, do we, do we continue? And we got the fair value of the option, 3.82.