LEC10_reading

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

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Unformatted text preview: 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! 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 weeks 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 weeks lecture by way of 7 key concepts. There is nothing magic about 7, its 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 t , S S N ( T, t ) . (1) Given the stock price process, we showed using Itos lemma that 1 : ln S T S N " - 2 2 ! T, T # (2) That is, if x is the continuously compounded return between 0 and T , S T = S e xT , (3) then, 1 More generally, ln S T S t N h - 2 2 ( T- t ) , T- t i for the return over the interval t to T . 1 x = 1 T ln S T S and it follows that x N h - 2 2 , 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 ) = - 2 2 rather than ? Because it corresponds to the geometric mean of rather than the arithmetic mean (remembering that 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,...
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This note was uploaded on 08/13/2011 for the course ACCG 329 taught by Professor Egonkalotay during the Three '09 term at Macquarie.

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

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