(20 marks) (Properties of Brownian Motion) Let AZ = Z(ti+1)Z(ti) = (ti)t o(ti) Z(0) and ~...

 

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(20 marks) (Properties of Brownian Motion) Let AZ = Z(ti+1)Z(ti) = (ti)t o(ti) Z(0) and ~ i=K-1 N(0,1) 0 I(At) = (;)2 i=0 T At K (1) (2) Write matlab code to evaluate I(At). For T = 1, compute the value of I(At). Use 100000 paths, and compute the mean and variance of the result, for K = 100, 200, 400, 800, 1600 timesteps. Vectorize your code so that there is only one explicit for loop over the timesteps. Plot a graph of var(I(At)) versus At. Explain what you see. Relate this to a crucial step in deriving Ito's Lemma. (20 marks) (Properties of Brownian Motion) Let AZ = Z(ti+1)Z(ti) = (ti)t o(ti) Z(0) and ~ i=K-1 N(0,1) 0 I(At) = (;)2 i=0 T At K (1) (2) Write matlab code to evaluate I(At). For T = 1, compute the value of I(At). Use 100000 paths, and compute the mean and variance of the result, for K = 100, 200, 400, 800, 1600 timesteps. Vectorize your code so that there is only one explicit for loop over the timesteps. Plot a graph of var(I(At)) versus At. Explain what you see. Relate this to a crucial step in deriving Ito's Lemma.