Lecture13

# Lecture13 - Lecture 13 Mariana Olvera-Cravioto Columbia...

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Lecture 13 Mariana Olvera-Cravioto Columbia University February 29th, 2012 IEOR 4404, Simulation Lecture 13 1/15

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Brownian Motion I A Brownian motion (BM) with drift μ and volatility σ is a stochastic process characterized by the following: I B 0 = 0 I B t is continuous (as a function of t ) with probability 1 I B t has independent increments with distribution B t - B s Normal ( μ ( t - s ) 2 ( t - s )) , 0 s < t I A standard Brownian motion has μ = 0 and σ = 1 . I Suppose that ˜ B t is a BM with drift μ and volatility σ and B t is a standard BM, then ˜ B t D = μt + σB t D = μt + σ tB 1 where D = means they have the same distribution. IEOR 4404, Simulation Lecture 13 2/15
Brownian Motion 0 s t N( µ ( t-s ) , σ ( t-s )) 2 IEOR 4404, Simulation Lecture 13 3/15

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Modeling ﬁnancial assets using Brownian Motion I A widely used model for the price of a stock or index at time t , denoted S t , is to assume that it follows a process called geometric Brownian motion (GBM). I GBM has the property that X t = ln S t - ln S 0 is BM, or equivalently, S t = S 0 e X t where X t is BM. I Unlike BM, GBM has the advantage of always being nonnegative, i.e. S t 0 for all t 0 . I Deﬁnition: We say that S t is a GBM with parameters ( μ,σ ) if it satisﬁes the stochastic diﬀerential equation (SDE) dS t = μS t dt + σS t dB t , where B t is a standard Brownian motion, μ R and σ 0 . IEOR 4404, Simulation Lecture 13 4/15
I Let X t = ln S t - ln S 0 and note that for t < s , X s - X t must be a normal r.v. with parameters ( ν ( s - t ) 2 ( s - t )) ( ν and σ to be determined) I Since if B t is standard BM then B t Normal (0 ,t ) , we can write X s - X t = ν ( s - t ) + σB s - t , so X s - X t s - t = ν + σ B s - t s - t and E [ X s ] - E [ X t ] s - t = ν I Taking the limit as s t we get that X t satisﬁes the SDE dX t = ν dt + σdB t and dE [ X t ] = ν dt I In ordinary calculus we would use the formula d ln S t = dS

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Lecture13 - Lecture 13 Mariana Olvera-Cravioto Columbia...

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