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Unformatted text preview: MAFS 5030 Quantitative Modeling of Derivatives Securities Homework Four Course Instructor: Prof. Y.K. Kwok 1. Consider the Brownian motion with drift defined by X ( t ) = μt + σZ ( t ) , X (0) = 0 , Z ( t ) is the standard Brownian motion , find E [ X ( t )  X ( t )], var( X ( t )  X ( t )) and cov( X ( t 1 ) , X ( t 2 )). 2. Show that σ integraldisplay T t [ Z ( u ) Z ( t )] du has zero mean and variance σ 2 ( T t ) 3 / 3. Hint: Consider var parenleftbiggintegraldisplay T t [ Z ( u ) Z ( t )] du parenrightbigg = E bracketleftbiggintegraldisplay T t integraldisplay T t [ Z ( u ) Z ( t )][ Z ( v ) Z ( t )] dudv bracketrightbigg = integraldisplay T t integraldisplay T t E [ { Z ( u ) Z ( t ) }{ Z ( v ) Z ( t ) } ] dudv = integraldisplay T t integraldisplay T t [min( u, v ) t ] dudv. 3. Suppose the stochastic variables S 1 and S 2 follow the Geometric Brownian processes where dS i S i = μ i dt + σ i dZ i , i = 1 , 2 . Let ρ 12 denote the correlation coefficient between the Wiener processes dZ 1 and dZ 2 . Let f = S 1 S 2 , show that f also follows the Geometric Brownian process of the form df f = μ dt + σ dZ f where μ = μ 1 + μ 2 + ρ 12 σ 1 σ 2 and σ 2 = σ 2 1 + σ 2 2 + 2 ρ 12 σ 1 σ 2 . Similarly, let g = S 1 S 2 , show that dg g = tildewide μdt + tildewide σ dZ g where tildewide μ = μ 1 μ 2 ρ 12 σ 1 σ 2 + σ 2 2 and tildewide σ 2 = σ 2 1 + σ 2 2 2 ρ 12 σ 1 σ 2 ....
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This note was uploaded on 02/01/2012 for the course MATH A taught by Professor A during the Spring '11 term at HKU.
 Spring '11
 A
 Derivative

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