FBE559.slides.07

# 11 11 bond 06 04 1 1 1 11 11 11 call with

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Unformatted text preview: ere du ( or dd ) is the price of the digital up (or down) option. Simple examples Stock: 0.6 0.4 (75) + (25) = 50. 1.1 1.1 Bond: 0.6 0.4 1 (1) + (1) = . 1.1 1.1 1.1 Call with strike of \$50: 0.6 0.4 15 (25) + (0) = = 13.64. 1.1 1.1 1.1 Put with strike of \$36: 0.4 0.6 (0) + (11) = 4. 1.1 1.1 Risk Neutral Probabilities Since du + dd = 1/(1 + r ) = 1/1.1, we can write: PV(CF ) dd du CFu + CFd du + dd du + dd = (du + dd ) = qu CFu + qd CFd 1+r where qu = du = 0.6, du + dd qd = dd = 0.4. 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 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: 1 people discou...
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