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CHAPTER 23 -
8. Using the binomial mode, we will find the value of u and d, which are:
u = e/
√
n
u = e.70/
√
12
u = 1.2239
d = 1 / u
d = 1 / 1.2239
d = .8170
This implies the percentage increase if the stock price increases will be 22.39 percent and the
percentage decrease if the stock price falls will be –18.30 percent. The monthly interest rate is:
Next, we need to find the risk neutral probability of a price increase or decrease, which will be:
And the probability of a price decrease is:
The following figure shows the stock price and put price for each possible move over the next two
months:
The stock price at node (A) is the current stock price. The stock price at node (B) is from an up
move, which means:
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CHAPTER 23 -
And the stock price at node (D) is two up moves, or:
The stock price at node (C) is from a down move, or:
And the stock price at node (F) is two down moves, or:
Finally, the stock price at node (E) is from an up move followed by a down move, or a down move
followed by an up move. Since the binomial tree recombines, both calculations yield the same result,
which is:
Now we can value the put option at the expiration nodes, namely (D), (E), and (F). The value of the
put option at these nodes is the maximum of the strike price minus the stock price, or zero. So:
The value of the put at node (B) is the present value of the expected value. We find the expected
value by using the value of the put at nodes (D) and (E) since those are the only two possible stock
prices after node (B). So, the value of the put at node (B) is:
Similarly, the value of the put at node (C) is the present value of the expected value of the put at
nodes (E) and (F) since those are the only two possible stock prices after node (C). So, the value of
the put at node (C) is:
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Using the put values at nodes (B) and (C), we can now find the value of the put today, which is:
Challenge
9. Since the exercise style is now American, the option can be exercised prior to expiration. At node
(B), we would not want to exercise the put option since it would be out of the money at that stock
price. However, if the stock price falls next month, the value of the put option if exercised is:
This is slightly higher than the value of the same option with a European exercise style. An
American option must be worth at least as much as a European option, and can be worth more.
10. Using the binomial model, we will find the value of u and d, which are:
u = e.30/
√
2
u = 1.2363
d = 1 / u
d = 1 / 1.2363
d = .8089
This implies the percentage increase if the stock price increases will be 23.63 percent, and the
percentage decrease if the stock price falls will be –19.11 percent. The six month interest rate is:
Next, we need to find the risk neutral probability of a price increase or decrease, which will be:
And the probability of a price decrease is:
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CHAPTER 23 -
The following figure shows the stock price and call price for each possible move over each of the six
month steps:
First, we need to find the building value at every step along the binomial tree. The building value at
node (A) is the current building value. The building value at node (B) is from an up move, which
means:
At node (B), the accrued rent payment will be made, so the value of the building after the payment
will be reduced by the amount of the payment, which means the building value at node (B) is:
To find the building value at node (D), we multiply the after-payment building value at node (B) by
the up move, or:
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CHAPTER 23 -
To find the building value at node (E), we multiply the after-payment building value at node (B) by
the down move, or:
The building value at node (C) is from a down move, which means the building value will be:
At node (C), the accrued rent payment will be made, so the value of the building after the payment
will be reduced by the amount of the payment, which means the building value at node (C) is:
To find the building value at node (F), we multiply the after-payment building value at node (C) by
the up move, or:
Note that because of the accrued rent payment in six months, the binomial tree does not recombine
during the next step. This occurs whenever a fixed payment is made during a binomial tree. For
example, when using a binomial tree for a stock option, a fixed dividend payment will mean that the
tree does not recombine. With the expiration values, we can value the call option at the expiration
Call value (F) = Max($58,887,320 – 63,000,000, $0)
Call value (F) = $0
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CHAPTER 23 -
The value of the call at node (B) is the present value of the expected value. We find the expected
value by using the value of the call at nodes (D) and (E) since those are the only two possible
building values after node (B). So, the value of the call at node (B) is:
Note that you would not want to exercise the option early at node (B). The value of the option at
node (B) if exercised is the value of the building including the accrued rent payment minus the strike
price, or:
Since this is less than the value of the option if it left “alive”, the option will not be exercised. With a
call option, unless a large cash payment (dividend) is made, it is generally not valuable to exercise
We can value the call at node (C), which will be the present value of the expected value of the call at
nodes (F) and (G) since those are the only two possible building values after node (C). Since neither
node has a value greater than zero, obviously the value of the option at node (C) will also be zero.
Now we need to find the value of the option today, which is:
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