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LEC10_reading - ACCG 329 Lecture 10 Black-Scholes-Merton...

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ACCG 329 Lecture 10: Black-Scholes-Merton & Extensions By Egon Kalotay Macquarie University, North Ryde May 13, 2009 Ver:1.0, Please e-mail me if you find any errors; I lost the original file and had to re-type a lot of it in a hurry!
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1 Introduction This brief set of lecture notes is a tidied up summary of the lecture. They sup- plement but do not replace chapters 13 and 15 of Hull (this week’s readings). In fact, you will see that for some of the topics I simply refer you the relevant pages of Hull. 2 7 Key Concepts For our purposes, we can summarise the important content of this week’s lecture by way of 7 key concepts. There is nothing magic about 7, it’s just the way I chose to consolidate the main ideas. 1. The Distribution of the Continuously Compounded Rate of Return Last week we specified a stock price process that implies a normal return distribution over a infinitely short period of time Δ t , Δ S S ∼ N ( μ Δ T, σ Δ t ) . (1) Given the stock price process, we showed using Ito’s lemma that 1 : ln S T S 0 ∼ N " μ - σ 2 2 ! T, σ T # (2) That is, if x is the continuously compounded return between 0 and T , S T = S 0 e xT , (3) then, 1 More generally, ln S T S t ∼ N h μ - σ 2 2 ( T - t ) , σ T - t i for the return over the interval t to T . 1
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x = 1 T ln S T S 0 and it follows that x ∼ N h μ - σ 2 2 , σ T i ; remembering what happens to the mean and variance in the expression (2) when the return is multiplied by a constant. Note that equation (3) is also useful to see why stock price cannot be negative in a Black-Scholes world (under Geometric Brownian Motion). Why does E ( x ) = μ - σ 2 2 rather than μ ? Because it corresponds to the geometric mean of μ rather than the arithmetic mean (remembering that μ can be thought of as the expected return over an arbitrarily small interval of time). See ‘The Misuse of Expected Return’ by Hughson, Stutzer and Yung, Financial Analysts Journal , 2006.
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