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# LEC10 - ACCG329 Lecture 10 Black-Scholes ACCG329 Merton and...

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ACCG329 Lecture 10: Black ACCG329 Lecture 10: Black -Scholes Scholes Merton and Extensions Merton and Extensions 2009, Semester 1 Egon Kalotay 1 2 The Stock Price The Stock Price Assumption Assumption • Consider a stock whose price is S • In a short period of time of length Δ t, the return on the stock is normally distributed: where μ is expected return and σ is volatility ( ) t t S S Δ σ Δ μ φ Δ 2 ,

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3 The Lognormal Property The Lognormal Property (Equations 13.2 and 13.3, page 278) (Equations 13.2 and 13.3, page 278) y It follows from this assumption that y Since the logarithm of S T is normal, S T is lognormally distributed 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 Continuously Compounded Continuously Compounded Return Return ( Equations 13.6 and 13.7), page 279) Equations 13.6 and 13.7), page 279) If x is the continuously compounded return 4 = 2 σ σ μ φ = T x S S T x e S S T xT T 2 0 0 , 2 ln 1
5 The Expected Return The Expected Return • The expected value of the stock price is S 0 e μ T • The expected return on the stock is μ σ 2 /2 not μ This is because are not the same )] / [ln( )] / ( ln[ 0 0 S S E S S E T T and 6 μ μ and and μ σ 2 /2 /2 Suppose we have daily data for a period of several months μ is the average of the returns in each day [= E ( Δ S/S )] μ σ 2 /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)

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7 Mutual Fund Returns Mutual Fund Returns (See Business Snapshot 13.1 on page 281) Snapshot 13.1 on page 281) y Suppose that returns in successive years are 15%, 20%, 30%, -20% and 25% y The arithmetic mean of the returns is 14% y The returned that would actually be earned over the five years (the geometric mean) is 12.4% 8 The Volatility The Volatility y The volatility is the standard deviation of the continuously compounded rate of return in 1 year y The standard deviation of the return in time Δ t is y 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? t Δ σ
9 Estimating Volatility from Estimating Volatility from Historical Data Historical Data (page 282 -84) 84) 1. Take observations S 0 , S 1 , . . . , S n at intervals of τ years 2. Calculate the continuously compounded return in each interval as: 3.

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LEC10 - ACCG329 Lecture 10 Black-Scholes ACCG329 Merton and...

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