The Binomial Replication subject of Option Tutor lets you experiment with the idea of replicating the option using a stock and a bond in the binomial model. When you bring up the Binomial Replication subject, you see the following window toward the bottom of the screen:

Let us look carefully at the bottom half of this picture, where you enter parameters. You can see that the stock price is 50 and the risk free interest rate is 0.1 or 10%. The next field is labeled "uptick," shown as 1.5. This means that the stock can move up to 50*1.5 = 75. The "downtick," set at 0.5, means that if the stock price falls, it will fall to 50*0.5 = 25. You can see that these are the same parameters as above. The exercise or strike price of the option is set at 50 . The face value of the zero-coupon bond is 100 while the maturity of the option one year; the program sets the maturity date of the bond equal to that of the option. This means that the bond value today is 100e^-.1 = 90.484.

The default setting start you out with a put option. Let us replicate the put. If the stock moves to 75, the put will be worth zero since the strike price is 50 and 50-75 < 0. You can see the number 0.00 If the stock price falls to 25, the put will be worth 50-25 = 25. To replicate the put, we therefore need to solve:

75n + 100m = 0

25n + 100m = 25.

This yields n = -0.5, m = .375, so the option price is

(-0.5)*50 + (0.375)*90.844 = 8.93.

Option Tutor lets you try this out with puts and calls for different parameters. The window toward the top of the screen is:

You see displayed a one-period binomial "tree" (or, more precisely, lattice). Each of the squares is called "node" on the lattice and represents a stock price. In the simplest binomial model, there is an initial stock price which can move up or move down. The program also shows you the option value at each node; for example, the initial option price is 8.93 (which is exactly what we just calculated), and the option will either be worth zero in case of an uptick or 25 in case of a downtick.

At the terminal nodes, two numbers are displayed. The top number shows you the option value at that node. The bottom number shows you the value of your stock and bond portfolio. To replicate the option, you need to buy and sell stocks and bonds so that at each node, the top and bottom numbers match.

Let us verify that the numbers we calculated replicate the put. Look again at the lower window:

You buy/sell stocks and bonds as follows. Look at the left part of this window:

First, select either Stock or Bond. You can buy/sell either by clicking on of the buttons marked <<< or >> or by entering a quantity just above where it says OK and clicking OK. The buttons are preset quantities; <<< means sell 1, <<< means sell 0.1, < means sell 0.01, and similarly >>> means buy one, etc. The prices at which you are buying/selling are shown, and your current position is shown under "End." ("Start" is explained below).

For our example, we need to sell 0.5 stocks and buy 0.375 bonds. The easiest way to do this is to make sure Stock is selected, type in -0.5 and click OK. Then, select Bond, type in 0.375 and click OK. (If you make a mistake, simply click Restart and start over). You should now see:

You can see that at each terminal node, the top and bottom numbers match, so that our stock/bond portfolio replicates the option.

Tutor Notes:

• For the simplest (one period) binomial model, the button labeled "Hint" tells you the solution. For other cases, you have to work out the solution. We will soon tell you how you solve the more general cases.
• To reinforce your understanding of this model, you should change the parameters, such as the stock price, the uptick, and so on, calculate the solution, and then verify your solution using Option Tutor.
• You cannot enter arbitrary uptick and downtick values. If you do, (e.g. change the uptick 1.01 and leave everything else the same), you will get an error message. Why? Because it cannot be the case that the interest rate is too high or too low relative to the possible stock values. For example, suppose the interest rate is 75%, the uptick is 1.5 and the downtick is 0.5. This is an invalid combination. Why? Because you should never invest in the stock. If you invest $1 in the stock, the most it will ever be worth is $1.5, and it may only be worth 50 cents. If you invest $1 in the bond, you will get $1.75 for sure. In fact, with these parameters, there is arbitrage; you should short sell the stock and buy the bond.
• At the top of the display, you see a menu titled Options. This lets you change the display colors, the font, etc.. The Calculators menu lets you bring up a calculator where you can do simple calculations and also more advanced options and futures calculators.

Tutor Break: Dynamic Replication

Consider the two-period binomial model taken from Option Tutor:

What happens here is that the stock price can initially move up or down and then move up or down again. This means that there are four terminal nodes instead of 2. (The three period model would have 8 terminal nodes). But in the way we have formulated the model, we only have three possible stock prices at the end. Why? Because the initial stock price is 50. If u is the uptick and d the downtick, then in one period, the possible values are Su and Sd. From Su, the stock price can move to Suu or Sud while from Sd, it can move to Sdu and Sdd. At the end, therefore, the possible prices are Suu,Sud,Sdu,Sdd. But Sud=Sdu so we only have three possible terminal stock values. This happens because we have chosen to make the stock price process multiplicative in that we multiply the current stock price with the uptick and downtick parameters.

The display shows us that the value of the call option is 18.74. How do we replicate the call? (Unfortunately, the Hint button has been disabled!). We do it by repeating the one-period argument at every node, starting at the end. Let us start with the problem after the stock has moved up to 75 (= 50*1.5), corresponding to the following part of the display:

Option Tutor tells you that the call is worth 34.22 at this node, and in the future, will be worth 62.50 if the stock moves up again (to 112.50 = 75*1.5) and zero if the stock price moves to 37.50 (= 75*0.5).

We already know how to figure out that the option is worth 34.22; this is the one-period model all over again, so we need to solve:

112.50n + 100m = 62.50

37.50n + 100m = 0

(Note that the value of the bond at the terminal node is 100).

The solution is n = 0.83333, m = -.3125.

If the stock initially moved down to 25, so that we were at:

then there is nothing to do; the possible future stock prices are 37.50 and 12.5, and the call is worthless in either case. Therefore, we need to hold zero stocks and zero bonds to replicate the call.

Now, let us move back to the beginning:

We have calculated the following: if the stock moves up to 75, the call will be worth 34.22; if it moves down to 25, the call will be worth 0. So let us solve:

75n + 90.484m = 34.22

25n + 90.484m = 0

(The value of bond after 1 period is 90.484).

The solution is n = 0.6844, m = -0.1891

The option price is 50*0.6844 - 81.873*0.1891 = 18.74, as shown.

1.8 Backwards and Forwards, Starts and Ends

There is a subtle backwards and forwards argument here that can be a little confusing. First, note that in order to replicate the call, we had to adopt the following trading strategy:

Initially, hold 0.6844 stocks, -0.1891 bonds.

If the stock moves up, change to 0.83333 stocks and -0.3125 bonds.

If instead the stock moved down, change to zero stocks and zero bonds.

This description goes forward in time. However, to determine the values, we had to start at the end of the lattice and work backwards. The principle is: to calculate the values, you must work backwards. However, when you actually do the buying or selling, you have to move forward in time.

This brings up another subtlety, which explains why there is column marked "Start" and another marked "End," as in:

This is depicting what happens after the first uptick when the stock price moved from 50 to 75. At the beginning, we held 0.6844 stocks, so after the uptick occurred, we still had the 0.6844 stocks. This is why under "Start" it shows that you have 0.6844 stocks. Now, we want to hold 0.8333 stocks, so we need to buy only 0.8333 - 0.6844 = 0. 1489 more stocks. This lets us start with 0.6844 and end with 0.8333. Therefore, when you replicate an option over multiple periods, you need to pay attention to the net trade you have to make at every node so that the ending position at the node ends up being what you want to hold at that node.

Tutor Notes:

1. To see if you understand the dynamic replication problem, try replicating the two period put and then try some three period examples.
2. We motivated the multi-period binomial model as a way to increase the number of possible stock prices. By default, Option Tutor sets the maturity of the option equal to the number of periods (so a two period option lasts for two years):

You can switch this off. This allows you, for example, to have a one year maturity and three periods, so each period is one third of a year.

(C) Copyright 1997, OS Financial Trading System