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Unformatted text preview: 2 Basic Options 9 b) Due to the fact that the sum of two normal random variables is a normal variable, it is easy to show that the sum of n normal variables, 1 t + + n t , and the limit of the sum as n are normal variables. Because E [ X ( t )] = lim n E " n X i =1 i t # = lim n n X i =1 E [ i ] t = 0 , and Var [ X ( t )] = lim n n X i =1 Var [ i ] t = lim n t = t, X ( t ) is a normal random variable with mean zero and variance t . c) From a) we know that dX = X ( t + dt ) X ( t ) is a normal variable. For dX we have E [ dX ] = E [ X ( t + dt )] E [ X ( t )] = 0 and Var [ dX ] = E h ( dX ) 2 i = E h ( X ( t + dt ) X ( t )) 2 i = E X 2 ( t + dt ) 2 X ( t + dt ) X ( t ) + X 2 ( t ) / = t + dt 2E X 2 ( t ) / 2E [ dX X ( t )] + t = dt, so dX is a normal random variable with mean zero and variance dt . Here we have used the fact that X ( t ) and dX are independent. d) From S = e t + X ( t ) , we have ln S = t + X ( t ). Because ln S t = , ln S X = , 2 ln S X 2 = 0 and dX = dX , using It os lemma, we have d ln S = dt + dX. 4. Suppose dS = a ( S, t ) dt + b ( S, t ) dX , where dX is a Wiener process. Let f be a function of S and t . Show that df = f S dS + f t + 1 2 b 2 2 f S 2 dt = b f S dX + f t + 1 2 b 2 2 f S 2 + a f S dt....
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This note was uploaded on 02/11/2010 for the course MATH 6203 taught by Professor Zhu during the Spring '10 term at University of North Carolina Wilmington.
 Spring '10
 Zhu
 Math

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