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>> Up until this point we've only seen how to price European options in the binomial

Â model. We are now going to consider the case

Â where the holder of the option has the ability to exercise early, such an option

Â is actually called an American option and we'll see that we can also easily price

Â these securities in our binomial model. So the only difference is that, we must

Â now also check if it's optimal to early exercise at each node.

Â So if you recall the way we priced European options, we actually started at

Â the end. At t equals 3.

Â Here, we've got a three period binomial model.

Â We began at the end, computed the payoff of the European option at t equals 3.

Â And then we worked backwards one period at, at a time, figuring out how much the

Â option was worth at each period. And working way backwards, using our one

Â period results to get the option price at times zero.

Â That's how we computed the European option prices.

Â Well, we can do the exact same thing with American options.

Â We're going to start at the end and we're going to work backwards one period at a

Â time using the risk mutual probabilities in each period to compute the value of the

Â option. The only difference is, now we have to

Â check at each period if it's optimal to exercise early.

Â So when we're working backwards, we're going to compute the value of the option

Â but also check to see if that value is greater than the value of exercising at

Â that point. If it is, we don't exercise.

Â If it's not, we do exercise and we continue backwards.

Â Okay, and we're going to see how this, how this works in practice.

Â There's also a spread sheet that you can use to see the calculations, and to see

Â how we price the American options. So if you like you can go through that

Â spreadsheet or have it open, while we're doing these calculations.

Â Okay, first of all recall that it is never optimal to early exercise an American call

Â option on a non-dividend paying stock. So we saw that in an earlier module, so

Â we're actually going to consider pricing American put options here.

Â So the put option is going to as-, as-, assume an expiration or a maturity of t

Â equals 3. A strike of $100.

Â And we'll assume the risk-free rate, the gross risk-free rate per period is 1.01.

Â Okay, so this is our binomial model that we've been using a lot until now.

Â It starts off at $100 the stock price starts, at $100.

Â It goes up by a factor of u. It goes down by a factor of d in each

Â period. So the payoff of the put option.

Â So remember the payoff of the put option will be the maximum of zero and the strike

Â minus the stock price at time three. The stock price at time three is 81.63,

Â 93.46, 107 and 122.5. The strike is 100.

Â So we will only exercise if the security price is less than 100.

Â So that's why we're going to get zero up here.

Â And we're going to get 100 minus 93.46 down here, and 100 minus 81.63, which is

Â 18.37. So this is the payoff of the American

Â option if we exercise at a time t equals 3.

Â And all we're going to do to price this American option is we're going to work

Â backwards in the same manner as we did with European options, but this time at

Â each node we'll also have to check if it is optimal to exercise at that point or

Â not. Okay so here's an example of how we do

Â these calculations. So we know the value of the American

Â option at t equals 3. It's 0, 0, 6.54, 18.37.

Â So we're going to work backwards one node at a time.

Â So let's actually consider this piece here.

Â And suppose we're down at this node here and we want to figure out how much is the

Â option worth. Well, if it was a European option, we

Â would say the fair value of the option is 1 over r times q times 6.54 plus 1 minus q

Â times 18.37. That's how we computed the fair value of a

Â European option in the one period model. We discount it and use the risk mutual

Â probabilities to compute the expected payoff one period ahead.

Â We do the exact same thing here. Except, we have the ability to early

Â exercise the option. In other words, when we're at this node we

Â have two choices. We can choose to continue.

Â Or we can exercise. The choice is ours.

Â If we choose to continue, the fair value of the option is this guy here, whatever

Â this turns out to be. Let's just call it x.

Â So we'll get x. If we exercise, we'll get 100 minus 87.34.

Â And that's equal to 12.66. However the, the choice is ours.

Â So we will choose to exercise if 12.66 is greater than x.

Â And we will choose to continue if x is greater than 12.66.

Â In this case it turns out that 12.66 is the greater of the two so x is less than

Â 12.66. So at this node we choose to exercise and

Â we get 12.66. So the fair value of the American option

Â at this node is $12.66. And we do the exact same thing at this

Â node and at this node. And as I said earlier, you can see these

Â calculations in the spreadsheet that we've, we've uploaded onto Coursera.

Â Okay. So we can work backwards in each period

Â and we actually find that the fair value of the American option is 3.82.

Â Okay. Not only that, you will see that the only

Â point at which it is optimal to exercise early is down here at this node.

Â And at each of the other nodes the fair value of the option turns out to be the

Â continuation choice. It's only at this node that exercising was

Â optimal, and that's why we've highlight it in a different color here.

Â So that's how we price American options. We just work backwards in the lattice, in

Â the binomial tree as we did with the European options, but with the added

Â complication of having to check at each node whether it was optimal to exercise or

Â not. To give ourselves a flavor of optimal

Â stopping problems, we're going to consider one more example.

Â And this is a simple die throwing game. So consider, for example, the following

Â game. We've got a fair six-sided die.

Â So let's try and draw, a six-sided die. Okay.

Â So you've got the numbers one to six on the six sides of this die.

Â We'll now throw the die up to a maximum of three times.

Â After any throw you can choose to stop. And when you stop you obtain an amount of

Â money equal to the value you threw. So the value you threw is whatever is

Â showing here with you throw the die. So for example if you throw a four on your

Â second throw and you chose to stop, then you'll obtain $4.

Â So if you are risk neutral how much would you pay to play this game?

Â By risk neutral I mean that you just want to compute the fair value of this game

Â using the true probabilities. The amounts of money concerned are so

Â small the risk doesn't enter into the situation.

Â Okay. So how much is this game worth playing?

Â I always like this game because it gives you an example of another optimal stopping

Â problem. It's also a game or a question that many

Â interviewers over the years have liked to ask our students when they were

Â interviewing for jobs in the, in the financial industry.

Â So let's think about this. So the solution is going to be to work

Â backwards starting with the last possible throw.

Â So if we've just got one throw remaining, then the fair value of the game must be

Â 3.5. And the reason is, on that last throw you

Â can get 1, 2, 3, 4, 5 or 6. Each with probability 1 6th.

Â And so the fair value of this game is 1 6th times 1, 1 6th times 2, and so on, and

Â it's equal to 3.5. Now, suppose you've got two throws

Â remaining. What we must do is figure out a strategy

Â which determines what to do after the first throw of these two throws, okay?

Â So that first throw refers to the first of the two throws remaining, so it's throw

Â two in the overall game. Okay, so let's think about this.

Â So, we've got, throw number two. We can get a 1, a 2, a 3, a 4, a 5, or a 6

Â on that throw. Each of these occurs with probability 1

Â 6th. So if we get a 1, 2, or 3, if you think

Â about it, it would make sense to continue. Why?

Â Well if we continue, we will expect to get 3.5 on our last throw.

Â So why would you stop and get a 1, a 2, or a 3 when you expect to get a 3.5 on your

Â last throw? So in each of these, you would continue,

Â and expect to get 3.5. On the other hand, if you get a 4, 5, or a

Â 6, then why would you continue? Why would you continue and expect to get a

Â 3.5 when you can stop and get a bigger number.

Â So in fact, for each of these numbers, you will stop.

Â And get 4, 5, or 6. So the fair value of the game with two

Â throws remaining, is, 1 6th, times 3.5, plus 1 6th times 3.5., plus 1 6th times 4,

Â plus 1 6th times 5. And so on.

Â So the fair value of the game is 1 6th times 4, plus 5, plus 6, plus 1 half times

Â 3.5 which is 4.25. So that's the fair value of the game if

Â the, if there's two throws remaining. If there's three throws remaining, the

Â exact same idea works. You now consider the first throw.

Â Okay, so there's not much room here, but the first throw, we could start with here.

Â And we would figure out what to do in each of these scenarios.

Â Well, If we throw a 1, 2, 3, or 4, we would actually choose to continue.

Â Why? Because the fair value of continuing with

Â two throws remaining would be 4.25 which is greater than 4.

Â Otherwise, if we get a 5 or 6, we would stop.

Â So you can actually compute the fair value of this game is going to be 2 3rds times

Â 4.5 plus 1 6th times 5 plus 1 6th time 6. And that's the fair value of this

Â die-throwing game. If you could throw the die 1,000 times.

Â What do you think will happen to the fair value of the game then?

Â It should be pretty clear. And I'll let you think about that.

Â The reason I like this game, though, is, it emphasizes the idea of a strategy, or a

Â stopping time. You have to figure out the optimal

Â strategy, what you must do at each point in time.

Â Do you continue or do you stop. With the American option problem you have

Â to figure out do you exercise or do you continue.

Â In the binomial example over here we did it automatically by working backwards.

Â And at each point, at each node we figured out the optimal strategy.

Â Do we exercise or we, do we, do we continue?

Â And we got the fair value of the option, 3.82.

Â