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MGTC71-10 - The Black-Scholes-Merton Random Walk Assumption...

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1 MGTC71-Lecture 10 The Black-Scholes-Merton Model Ata Mazaheri 2 The Black-Scholes-Merton Random Walk Assumption Consider a stock whose price is S In a short period of time of length Δ t the return on the stock ( Δ S / S ) is assumed to be normal with mean μΔ t and standard deviation • μ is expected return and σ is volatility t Δ σ 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 4 The Lognormal Property (Cont’d) where φ [ m , v ] is a normal distribution with mean m and variance v [ ] [ ] 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 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 6 The Expected Return The expected value of the stock price is S 0 e μ T The return in a short period Δ t is μΔ t But the expected return on the stock with continuous compounding is μ σ 2 /2 This reflects the difference between arithmetic and geometric means
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7 Mutual Fund Returns Suppose that returns in successive years are 15%, 20%, 30%, -20% and 25% The arithmetic mean of the returns is 14% The returned that would actually be earned over the five years (the geometric mean) is 12.4% 8 The Volatility The volatility is the standard deviation of the continuously compounded rate of return in 1 year The standard deviation of the return in time Δ t is If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day?
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