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Binomial option pricing replicating portfolio answ...
Binomial_option_pricing_replicating_portfolio_answer_key.docBinomial option pricing – valuing Amgen
Binomial option pricing replicating...
Binomial_option_pricing_replicating_portfolio_answer_key.docBinomial option pricing – valuing Amgen
Page 1
Binomial option pricing – valuing Amgen call option
Suppose that Amgen’s current stock price is $60, and every fourmonth period it can
either go up or down. The standard deviation of Amgen’s stock return is 0.3523. If the
riskfree rate is 0.5% per four months, what is the value of an eightmonth call option on
Amgen’s stock with an exercise price of $60?
Solution:
Our goal is to find C
0
. We first need to figure out how the price of Amgen would change
in each fourmonth period. In this case h = 120/360. In Period 1 it could go up to
60*e
(0.3523*sqrt(120/360))
= $73.54,
or go down to
60/( e
(0.3523*sqrt(120/360))
) = $48.9
In the same way we can find the stock price at the end of the eightmonth period.
Stock price=90.12
Option payoff=30.12
Stock price=60
Option payoff=0
Stock price=39.94
Option payoff=0
Stock price=60
C
0
=?
Stock
price=73.53
Cu=?
Stock
price=48.96
Cd=?
Page 2
1) To calculate Cu, we need to form a replicating portfolio in month 4 by investing M in
the stock and borrowing B dollars at 0.5% for four months in such a way that this
portfolio has the exact same payoffs as the option in month 8. Then, the value of the
portfolio in Month 4 should be exactly equal to the value of the option, Cu.
M*(90.12) – B*1.005 = 30.12
M*(60) – B*1.005 = 0
If you solve for M and B you get M=1 and B=59.70. The value of the portfolio at the end
of month 4 is:
M*(73.53) – B = 73.53 – 59.7 = 13.84 = Cu
2) To calculate Cd, we need to form a replicating portfolio in month 4 by investing M in
the stock and borrowing B dollars at 0.5% for four months in such a way that this
portfolio has the exact same payoffs as the option in month 8. Then, the value of the
portfolio in Month 4 should be exactly equal to the value of the option, Cd. However,
since both option payoffs in month 8 are 0, Cd=0
3) To calculate C
0
, we need to form a replicating portfolio in month 0 (right now) by
investing M in the stock and borrowing B dollars at 0.5% for four months in such a way
that this portfolio has the exact same payoffs as the option in month 4. Then, the value of
the portfolio in Month 0 should be exactly equal to the value of the option, C
0
.
M*(73.53) – B*1.005 = 13.83
M*(48.96) – B*1.005 = 0
If you solve for M and B you get M=0.563 and B=27.44. The value of the portfolio in
month 0 is:
M*(60) – B = 0.563*(60) – 27.44 = 6.34 = C
0
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