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Unformatted text preview: binomial tress to price American
options (3)
Meanwhile, the current stock price is 34.29
Meanwhile,
(see J27). So, if the investor can make positive
payoff by exercising the option at this node, the
payoff is 50 – 34.29 = 15.71
payoff
So, the investor’s choice is between (a)
So,
exercise the put now = 15.71 and (b) wait for
the next step = 15.54.
Obviously, the investor will choose (a)
Obviously,
So, in each node, we just need to ask the Excel
So,
to compute the maximum of (a) the positive
payoff by exercising the option now and (b) the
expected payoff of waiting
1524 Using binomial tress to price American
options (4)
As a result, we get that the following:
As
 Call option with strike = 50, it’s price = 3.47
Call
 Put option with strike = 50, it’s price = 2.37
Put
Binomial 9step Call Put European 3.47 2.25 American 3.47 2.37 The above matrix highlights the fact that the price
The
of American option is always higher than or equal
to European option (the option to exercise anytime
is valuable)
is
1525 Real example: IBM option (1)
Now, let’s use binomial trees to price IBM options
Now,
and compare it with the current market price
and
The current stock price = 182.39 on Oct. 8, 2011 1526 Real example: IBM option (2)
Now, let’s check the option price of IBM matures
Now,
on Oct. 21, 2011 (10 trading days to go)
on 1527 Real example: IBM option (3)
Now, how to set up the parameters? We use the
Now,
previous 10 days to estimate the following
previous
1. Drift term (riskfree rate = stock return in a riskneutral world) = 0.00502 (r) in daily frequency
in
2. Volatility term = 0.01513 (σ) in daily frequency
3. Dividend rate sets to be 0
4. δt = 1 in daily frequency
4.
5. T = 10 days to be traded by Oct. 21, 2011
6. Step number = 10
See “IBMoptoin” tab
1528 Real example: IBM option (4)
Our calculation implies the price of $25.47, which
Our
is higher than the market price $18.95
is
So, if we believe in our calculation, the market
So,
price is “cheaper” than our estimation. What should
we do?
we
We can either (a) buy the option at the price
We
$18.95 and wait the price to go to $25.47 or (b)
establish a replicate portfolio by buying the stock,
buying a put option at exercise price 165, and
borrowing money from the bank
1529 Exercise 1
Now, compute the price of American put and call
Now,
based on all the same setup, but change the step
number to 15 steps, see the “Exercise 1” tab
number
As shown in the “Exercise 1  ans” tab:
 Call price = 3.44
Call
 Put price = 2.35
Put
Call Binomial 9 steps Binomial 15 steps Euro Call 3.47 3.44 Amer Call 3.47 3.44 Put Binomial 9 steps Binomial 15 steps Euro Put 2.25 2.22 Amer Put 2.37 2.35
1530 How American option prices react to
input changes: stock price...
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 Spring '09

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