673_14_Options_3

673_14_Options_3 - 1 OPTION VALUATION NBA 673 2 Evolution...

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Unformatted text preview: 1 OPTION VALUATION NBA 673 March 10, 2006 2 Evolution of Stock Prices 3 Stock Price Evolution Model • We will model the evolution of prices in the interval [t,T]. We divide this into small intervals of equal length Δ . 4 Stock Price Evolution Model (2) • Three assumptions: (1) r(i Δ ) are i.i.d.; (2) E[r(i Δ )]= μΔ ; (3) Var[r(i Δ )]= σ 2 Δ . • Consequences: (1) E[Z(T)] = μ T; (2) Var[Z(T)] = σ 2 T. (3) Returns are normally distributed. (4) Prices are log-normally distributed. 5 Normal vs. Log-Normal-10-8-6-4-2 2 4 6 8 10 0.02 0.04 0.06 0.08 Normal pdf 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 Log-Normal pdf 6 Black-Scholes Formulas 7 Bernoulli Random Variable • Two possible outcomes, one has probability 0 < p < 1, the other has probability 1 – p. • The outcomes could be interpreted as “yes” & “no”, 1 & 0, “left” & “right”, “up” & “down”. before after “yes”, 1, “right”, “up” “no”, 0, “left”, “down” 8 Binomial Distribution...
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673_14_Options_3 - 1 OPTION VALUATION NBA 673 2 Evolution...

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