only 1 question about financial engineering. please show your work

Financial Engineering MRM 8610, Spring 2016 Homework 3 Daniel Bauer Due Thursday, 02/04/2016 before class. 1. Brownian Motion and Martingales: (1+1+1 points) Let (Wt )t0 and (Zt )t0 be two independent Brownian motions. Use the denition of Brownian motion and the denition of a martingale to show whether or not the following stochastic processes are standard Brownian motions and/or martingales, respectively (both for all three). (1) (a) St (1) = Wt + (1 ) Zt and 0 1 is a constant. (2) = Wt+t0 Wt0 . (3) = W t Zt . t0 where St t0 where St t0 where St (2) (b) St (3) (c) St 2. Path of Bachelier Model: (1+1 point) Consider the Bachelier model for the stock (St )t0 : St = S0 + a t + b Wt , where (Wt )t0 is a Brownian motion and a, b > 0. (a) Download daily data for the S&P 500 index for the twenty year period beginning in January 1994 until the end of 2013 (e.g., from yahoo nance). Use the data to estimate a and b for this model. (b) Use Excel or another spreadsheet software to simulate a (discretized) sample path of a the Bachelier model over the year 2014 using 250 equidistant time steps, and compare it to the realized path. Just hand in the resulting plot. 3. Quadratic Variation 1: (2 points) Let Xt = X0 + ( 0.5 2 ) t + Wt , where (Wt )t0 is a Brownian motion. You are given the following two statements concerning Xt . (a) V ar[Xt+h Xt ] = 2 h, t 0, h 0. (b) limn n j=1 2 X jT X (j1)T n = 2 T , T 0. n Which of them is true? Provide an explanation for your answer. 4. Quadratic Variation 2: (3 points) Dene: (1) (a) St (b) (c) (2) St (3) St Let h = T n = [t], where [t] is the greatest integer part of t; for example, [3.14] = 3, [9.99] = 9, and [4] = 4. = 2t + 0.9 Wt , where (Wt )t0 is a standard Brownian motion. = t2 . and let n (2) (i) n (i) (i) 2 Sjh S(j1)h VT (i) = lim j=1 denote the quadratic variation of the process S over the time interval [0, T ]. Rank the quadratic variations (2) (2) (2) VT (1), VT (2), and VT (3) over the time interval [0, 2.4]. Provide an explanation for your