# Ch13 - The Black-Scholes-Merton Model Chapter 13 Click to...

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Click to edit Master subtitle style  7/7/11  Options, Futures, and Other Derivatives, 7th  The Black-Scholes-Merton Model Chapter 13 11

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7/7/11  Options, Futures, and Other Derivatives, 7th International Edition,  The Stock Price Assumption Consider a stock whose price is S In a short period of time of length t, the return on the stock is normally distributed: where μ is expected return and σ is volatility 22
7/7/11  Options, Futures, and Other Derivatives, 7th  The Lognormal Property (Equations 13.2 and 13.3, page 278) It follows from this assumption that Since the logarithm of ST is normal, ST is lognormally distributed 33 or σ σ - μ + φ σ σ - μ φ - T T S S T T S S T T 2 2 0 2 2 0 , 2 ln ln , 2 ln ln

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7/7/11  Options, Futures, and Other Derivatives, 7th  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 44
7/7/11  Options, Futures, and Other Derivatives, 7th  Continuously Compounded Return ( Equations 13.6 and 13.7), page 279) If x is the continuously compounded return 55 = 2 σ σ - μ φ = T x S S T x e S S T xT T 2 0 0 , 2 ln 1

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7/7/11  Options, Futures, and Other Derivatives, 7th International Edition,  The Expected Return The expected value of the stock price is S 0 e μ T The expected return on the stock is μ σ2/2 not μ This is because are not the same 66
7/7/11  Options, Futures, and Other Derivatives, 7th  μ and σ 2 /2 Suppose we have daily data for a period of several months μ is the average of the returns in each day [= E ( S/S )] μ σ2/2 is the expected return over the whole period covered by the data measured with continuous compounding (or daily compounding, which is almost the same) 77

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7/7/11  Options, Futures, and Other Derivatives, 7th  Mutual Fund Returns (See Business Snapshot 13.1 on page 281) 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% 88
7/7/11  Options, Futures, and Other Derivatives, 7th  The Volatility The volatility is the standard deviation of

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Ch13 - The Black-Scholes-Merton Model Chapter 13 Click to...

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