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Lily Qiu, Assistant Professor
Economics Department, Brown University
EC1710, Lecture 1921, Spring 2010, page 1
Options, II
1.Binomial Option Pricing
1.1 Let’s start with a simple example: assume S
0
= $100 at the beginning of the year, the price
of the stock can go up to $110 at the end of the year (by a factor of u=1.1), or drop to $90 (by
a factor of d=0.9).
A call option on this stock has an exercise price of $105 and a time to
expiration of 1 year.
The riskfree annual interest rate is 5%.
Let’s use value “trees” to illustrate the payoffs:
Stock
Option
/
/
100
C
0
\
\
Now, we also form a leveraged portfolio.
We buy one share at the beginning of the
year at $100, borrow $85.71 at the 5% to finance the purchase.
Its value tree at the end of the
year:
Leveraged portfolio
/
14.29
\
If we buy 4 call options, we have the same cashflow pattern as the leveraged portfolio,
then it must be that the cost of buying 4 call options is the same as the cost of constructing the
leveraged portfolio:
4C
0
= 14.29 => C
0
= $3.57
REMARK: This method of constructing a portfolio consisting of the riskfree and the
underlying asset, which mimics the cashflow pattern of the option, is called
replication
=>
important concept behind most option–pricing formulas.
We can also create a
perfect hedge
by buying one share and write 4 calls.
Portfolio value
/
$110 – $20 = $90
$100  $14.29 = $85.71
\
$90 – 0 = $90
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View Full DocumentLily Qiu, Assistant Professor
Economics Department, Brown University
EC1710, Lecture 1921, Spring 2010, page 2
The portfolio is riskless => a
perfect hedge
, it earns the riskfree return of 5%:
$85.71 * 1.05 = $90.
The hedge ratio =
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 Spring '10
 Qiu

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