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Unformatted text preview: always get $1500.
Note: the answer doesn’t depend on the ”true” probability of USC
making it to the Rose Bowl. Risk-Neutral Approach
Constructing replicating strategies to price options is cool, but
tedious. If all we care about is pricing, there is a shortcut.
Start with the one-period example:
uS = 75 S0 = 50 dS = 25 and the interest rate is 10%.
Consider the “up” and “down” digital options:
0 and 0
1 C0 1
0 Consider a portfolio (a, b ) where
a is the number of shares of the stock held
b is the dollar amount invested in the riskless bond.
We want to ﬁnd (a, b ) so that
75a + 1.1b = 1 25a + 1.1b = 0. There is a unique solution
a = 0.02 and b = −0.4545. So the price of the digital up option is:
50a + 1b = .5455 = .6/1.1
Similarly, the price of the down digital option is .4/1.1. Using Digital Options Now, consider a security with the following payoﬀ:
CFd = CFu × 1
0 + CFd × 0
1 No free-lunch requires:
PV(CF ) = du CFu + dd CFd = 0.6
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This document was uploaded on 10/28/2013.
- Spring '13