FINA 4110 Slides 13 [Black-Scholes Model]

FINA 4110 Slides 13 [Black-Scholes Model] - Valuing Stock...

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Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 1
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The Black-Scholes-Merton Random Walk Assumption z Consider a stock whose price is S z In a short period of time of length t the return on the stock ( S / S ) is assumed to be normal with mean µ∆ t and standard deviation µ is expected return and σ is volatility t σ 2
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The Lognormal Property z These assumptions imply ln S T is normally distributed with mean: and standard deviation : z Because the logarithm of S T is normal, S T is lognormally distributed T S ) 2 / ( ln 2 0 σ µ + T σ 3
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The Lognormal Property continued [] T T S S T T S S T T 2 2 0 2 2 0 , ) 2 ( ln , ) 2 ( ln ln σ σ µ φ σ σ µ + φ or where φ [ m , v ] is a normal distribution with mean m and variance v 4
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The Lognormal Distribution ES S e SS ee T T T TT () ( ) = =− 0 0 2 2 2 1 var µ µσ 5
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The Expected Return z The expected value of the stock price is S 0 e µ T z The return in a short period tis µ∆ t z But the expected return on the stock with continuous compounding is µ σ 2 /2 z This reflects the difference between arithmetic and geometric means 6
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Mutual Fund Returns (See Business Snapshot 13.1 on page 294) z Suppose that returns in successive years are 15%, 20%, 30%, -20% and 25% z The arithmetic mean of the returns is 14% z The returned that would actually be earned over the five years (the geometric mean) is 12.4% 7
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FINA 4110 Slides 13 [Black-Scholes Model] - Valuing Stock...

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