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Ch12HullFundamentals5thEd

Ch12HullFundamentals5thEd - Valuing Stock Options:The...

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Fundamentals of Futures and Options Markets , 5 th Edition, Copyright © John C. Hull 2004 12.1 Valuing Stock Options:The Black-Scholes Model Chapter 12
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Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004 12.2 The Black-Scholes Random Walk Assumption Consider a stock whose price is S In a short period of time of length t the change in the stock price is assumed to be normal with mean μ S t and standard deviation μ is expected return and σ is volatility t S σ
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Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004 12.3 The Lognormal Property These assumptions imply ln S T is normally distributed with mean: and standard deviation : Because the logarithm of S T is normal, S T is lognormally distributed T S ) 2 / ( ln 2 0 σ - μ + T σ
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Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004 12.4 The Lognormal Property continued where φ [ m , s ] is a normal distribution with mean m and standard deviation s [ ] [ ] T T S S T T S S T T σ σ - μ φ = σ σ - μ + φ , ) 2 ( ln or , ) 2 ( ln ln 2 0 2 0
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Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004 12.5 The Lognormal Distribution E S S e S S e e T T T T T ( ) ( ) ( ) = = - 0 0 2 2 2 1 var μ μ σ
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Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004 12.6 The Expected Return The expected value of the stock price is S 0 e μ T The expected return on the stock with continuous compounding is μ σ 2 /2 The arithmetic mean of the returns over short periods of length t is μ The geometric mean of these returns is μ σ 2 /2
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