# 12 - Copyright c 2008 by Karl Sigman 1 Geometric Brownian...

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Copyright c ± 2008 by Karl Sigman 1 Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is not appropriate for modeling stock prices. Instead, we introduce here a non-negative variation of BM called geometric Brownian motion , S ( t ), which is deﬁned by S ( t ) = S 0 e X ( t ) , (1) where X ( t ) = σB ( t ) + μt is BM with drift and S (0) = S 0 > 0 is the intial value. We view S ( t ) as the price per share at time t of a risky asset such as stock. Taking logarithms yields back the BM; X ( t ) = ln( S ( t ) /S 0 ) = ln( S ( t )) - ln( S 0 ). ln( S ( t )) = ln( S 0 ) + X ( t ) is normal with mean μt + ln( S 0 ), and variance σ 2 t ; thus, for each t , S ( t ) has a lognormal distribution. As we will see in Section 1.4: letting r = μ + σ 2 2 , E ( S ( t )) = e rt S 0 (2) the expected price grows like a ﬁxed-income security with continuously compounded interest rate r . In practice, r >> r , the real ﬁxed-income interest rate, that is why one invests in stocks. But unlike a ﬁxed-income investment, the stock price has variability due to the randomness of the underlying Brownian motion and could drop in value causing you to lose money; there is risk involved here. 1.1 Lognormal distributions If Y N ( μ,σ 2 ), then X = e Y is a non-negative r.v. having the lognormal distribution ; called so because its natural logarithm Y = ln( X ) yields a normal r.v. X has density f ( x ) = ( 1 2 π e - (ln( x ) - μ ) 2 2 σ 2 , if x 0; 0 , if x < 0. This is derived via computing d dx F ( x ) for F ( x ) = P ( X x ) = P ( Y ln( x )) = Φ((ln( x ) - μ ) ) , where Φ( x ) = P ( Z x ) denotes the c.d.f. of N (0 , 1) rv Z . Observing that E ( X ) = E ( e Y ) and E ( X 2 ) = E ( e 2 Y ) are simply the moment generating function (MGF) M Y ( s ) = E ( e sY ) of Y N ( μ,σ 2 ) evaluated at s = 1 and s = 2 respectively yields E ( X ) = e μ + σ 2 2 E ( X 2 ) = e 2 μ +2 σ 2 V ar ( X ) = e 2 μ + σ 2 ( e σ 2 - 1) . 1

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As with the normal distribution, the c.d.f. F ( x ) = P ( X x ) = Φ((ln( x ) - μ ) ) does not have a closed form, but it can be computed from the unit normal cdf Φ( x ). Thus computations for F ( x ) are reduced to dealing with Φ( x ). We denote a lognormal μ , σ 2 r.v. by X lognorm ( μ,σ 2 ) . 1.2 Back to our study of geometric BM, S ( t ) = S 0 e X ( t ) For 0 = t 0 < t 1 < ··· < t n = t , the ratios L i def = S ( t i ) /S ( t i - 1 ) , 1 i n, are independent lognormal r.v.s. which reﬂects the fact that it is the percentage of changes of the stock price that are independent, not the actual changes S ( t i ) - S ( t i - 1 ). For example L 1 def = S ( t 1 ) S ( t 0 ) = e X ( t 1 ) , L 2 def = S ( t 2 ) S ( t 1 ) = e X ( t 2 ) - X ( t 1 ) , are independent and lognormal due to the normal independent increments property of BM; X ( t 1 ) and X ( t 2 ) - X ( t 1 ) are independent and normally distributed. Note how therefore we can
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## This note was uploaded on 10/16/2010 for the course IEOR 4701 taught by Professor Karlsigma during the Summer '10 term at Columbia.

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12 - Copyright c 2008 by Karl Sigman 1 Geometric Brownian...

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