24c_blackscholesdrift

# 24c_blackscholesdrift - Black-Scholes Pricing and Drift At...

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Black-Scholes Pricing and Drift At f rst glance it seems a mystery that the Black-Scholes price does not depend on the drift of the underlying stock. Here we brie F y explain why, reviewing along the way the idea of a Brownian motion. 1 Independent Random Variables We say that two random variables x and y are independent if knowing the real- ization of one of them provides absolutely no information about the reealization of the other. Consider the following random variables whose values (realizations) are given as functions of the four states of nature. Assume each of the four states has probability 1/4. xy z UU 11 1 UD 1 12 DU 113 DD 1 14 The random variables x and y are independent, but the variables x and z are not. If the variable z has taken on the value 2 , it must be that x has taken on the value 1 . 1.1 Expectation The expected value or mean of a random variable is its average value, weighted by the probability of each state. Thus Ex = 1 4 1+ 1 4 1 4 ( 1) + 1 4 ( 1) = 0 Ey = 1 4 1 4 ( 1) + 1 4 1 4 ( 1) = 0 Ez = 1 4 1 4 2+ 1 4 3+ 1 4 4=2 . 5 Clearly the expectation of the sum of two random variables is the sum of their ex- pectations. E ( x + y )= + 1.2 Variance The variance of a random variable is the expectation of its deviation from its mean Var ( x E ( x ) 2 1

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Thus Var ( x )= 1 4 (1 0) 2 + 1 4 (1 0) 2 + 1 4 ( 1 0) 2 + 1 4 ( 1 0) 2 =1 ( y 1 4 (1 0) 2 + 1 4 ( 1 0) 2 + 1 4 (1 0) 2 + 1 4 ( 1 0) 2 ( z 1 4 (1 2 . 5) 2 + 1 4 (2 2 . 5) 2 + 1 4 (3 2 . 5) 2 + 1 4 (4 2 . 5) 2 . 25 Thevar ianceo fthesumo f independent random variables is the sum of their vari- ances ( x + y 1 4 (1 + 1 0) 2 + 1 4 (1 1 0) 2 + 1 4 ( 1+1 0) 2 + 1 4 ( 1 1 0) 2 = 1+0+0+1=2= ( x )+ ( y ) For random variables that are not independent, the variance of the sum can be quite di f erent from the sum of the variances ( z + z 1 4 (1 + 1 2 . 5) 2 + 1 4 (1 + 2 2 . 5) 2 + 1 4 ( 1+3 2 . 5) 2 + 1 4 ( 1+4 2 . 5) 2 < 2 . 25 = ( x ( z ) The proof of these facts will be given in a couple of lectures when we discuss proba- bility theory. 1.3 Standard Deviation The standard deviation of a random variable is the square root of its variance. It is an average deviation from the mean.
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## This note was uploaded on 12/08/2009 for the course ECON 251 at Yale.

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24c_blackscholesdrift - Black-Scholes Pricing and Drift At...

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