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Unformatted text preview: ere du ( or dd ) is the price of the digital up (or down) option. Simple examples
(25) = 50.
1.1 Call with strike of $50:
1.1 Put with strike of $36:
(11) = 4.
1.1 Risk Neutral Probabilities Since du + dd = 1/(1 + r ) = 1/1.1, we can write:
PV(CF ) dd
du + dd
du + dd = (du + dd ) = qu CFu + qd CFd
qu = du
du + dd qd = dd
du + dd We call qu and qd risk-adjusted probabilities because
they are positive
sum to one. Risk Neutral Pricing Thus, the price of a security equals the expected payoﬀ using the
risk-adjusted probabilities, discounted at the risk-free rate:
PV(CF ) = EQ [CF ]
qu CFu + qd CFd
1+r This is called the risk-neutral pricing formula.
Risk-adjusted probabilities are (normalized for time-value of
money) prices of digital options.
They are diﬀerent from the true probabilities.
The market in general is not risk-neutral. Risk Neutral Pricing Note that in general we are making two errors here:
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- Spring '13