Homework 6: Monday October 7, 2019微分方程代写 Consider the so-called backward differential equation 2  2tFx t + x2 Fx t = 0[1.1]for a function Fx t . Rewrite this equationDue Wednesday October 16, 2019. Total 100 pointsProblem 1. [20 points] 微分方程代写Consider the so-called backward differential equation¶ s2  ¶ 2¶tF(x, t) + ¶x2 F(x, t) = 0[1.1]for a function F(x, t) . Rewrite this equation as an approximate difference equation on a two-step 微分方程代写binomial x, tlattice with all transition probabilities = 1/2, as follows:微分方程代写Then we can approximate(i)Assuming dx = sdt , use values of the function on the lattice to show that approximately微分方程代写(ii)Write down the discrete difference equation that corresponds to Equation [1.1] and explain why it represents abackward  [10 points] 微分方程代写微分方程代写Problem 2. [20 points] 微分方程代写Suppose that y is the continuously compounded yield to maturity (i.e. discount factor is e–yt ) on a perpetual government bond B that pays $1 per year continuously at constant rate (i.e. cash flow is 1dt during time dt) for every year in the future.(i)Showthat B(y) 1(ii)Suppose that the yield y follows the mean-reverting process dy=a(m – y)dt + bydZ ; where 微分方程代写a, m and b are all positive constants and Z is a Wiener process. Use Ito’s Lemma to find an expression (in terms of y, m, a, and b) for the expected total return per unit time (the increase in value dB of bond plus value of the interest earned dt, divided by the value of the bond itself) to the owner of the bond? [15]Problem 3. [15 points] 微分方程代写Consider two Ito processes X and Y that take the form微分方程代写where W(t) is a Brownian processes.For a twice continuously differentiable functions f(t, x, y) Ito’s Lemma for a function of two pro- cesses takes the formShow that the process G(t) = X(t)Y(t) is a geometric Brownian motion.微分方程代写Problem 4. [10 points]Stock A and stock B both follow independent geometric Brownian motions. Changes in any short interval of time are uncorrelated with each other. Does the value of a portfolio consisting of one of stock A and one of stock B follow geometric Brownian motion? Explain your answer.微分方程代写Problem 5. [20 points] 微分方程代写Consider two correlated Ito processes X and Y that take the form微分方程代写where Z and W are two standard Brownian processes with correlation r so that, in addition to the usual Ito expressions for (dX)2 and (dY)2 , we also havedZdW =rdt微分方程代写For a twice continuously differentiable functions f(t, x, y) Ito’s Lemma for a function of two Brownian processes takes the form微分方程代写Show that the process G(t) = X(t)Y(t) is a geometric Brownian motion, and find the drift m andthe volatility sof the motion. .You can use the box rule微分方程代写to compute the drift and variance of a combination of two standard Wiener processes in a sum, and take for granted that the sum of 2 Wiener processes is also a Wiener process.Problem 6. [15 points] Consider this method discussed in class for generating a double Bernoulli set of increments dXand dZ such that dX and dZ each take only the values ±1 and have correlation r :微分方程代写Choose r= –0.5and generate 10,000 successive random steps for dX and dZ.(i)Plotone path for the cumulative value of X as the number of steps n increases, and one path for the cumulative value of  [10](ii)Calculate the correlation between the 10,000 dX’s and the 10,000 dZ’s that you[5]微分方程代写其他代写：web代写 program代写 cs作业代写 analysis代写 app代写 essay代写 assembly代写 Haskell代写 homework代写 java代写 数学代写 考试助攻 web代写 source code代写 finance代写 Exercise代写合作平台：essay代写 论文代写 写手招聘 英国留学生代写

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