– only 1 question about financial engineering. please show your workDescriptionSolution downloadThe.

– only 1 question about financial engineering. please show your workDescriptionSolution downloadThe Questiononly 1 question about financial engineering. please show your workFinancial EngineeringMRM 8610, Spring 2016Homework 3Daniel BauerDue Thursday, 02/04/2016 before class.1. Brownian Motion and Martingales: (1+1+1 points)Let (Wt )t?0 and (Zt )t?0 be two independent Brownian motions. Use the de?nition of Brownian motion andthe de?nition of a martingale to show whether or not the following stochastic processes are standard Brownianmotions 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 .t?0where Stt?0where Stt?0where St(2)(b) St(3)(c) St2. Path of Bachelier Model: (1+1 point)Consider the Bachelier model for the stock (St )t?0 :St = S0 + a t + b Wt ,where (Wt )t?0 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 theend 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 modelover the year 2014 using 250 equidistant time steps, and compare it to the realized path. Just hand in theresulting plot.3. Quadratic Variation 1: (2 points)Let Xt = X0 + (? ? 0.5 ? 2 ) t + ?Wt , where (Wt )t?0 is a Brownian motion. You are given the following twostatements concerning Xt .(a) V ar[Xt+h ? Xt ] = ? 2 h, t ? 0, h ? 0.(b) limn??nj=12X jT ? X (j?1)Tn= ? 2 T , T ? 0.nWhich of them is true? Provide an explanation for your answer.4. Quadratic Variation 2: (3 points)De?ne:(1)(a) St(b)(c)(2)St(3)StLet h =Tn= [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 )t?0 is a standard Brownian motion.= t2 .and letn(2)(i)n??(i)(i)2Sjh ? S(j?1)hVT (i) = limj=1denote 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 answer.

 

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