Please help me with the problems. Do not have to do the optional part. Please see the attached file.

ESD 40.244 Stochastic Calculus for Finance Homework 2 Due Thursday, 15th June 2017 1) 2) 3) 4) 5) 6) 7) 8) 9) A zero coupon bond expires in 6 months and costs $0.88. What is the annualized yield of the bond? The short rate follows an Ornstein-Uhlenbeck process with k = 1.2, = 3%, = 1%. The current value of the short rate is 1%. What is the distribution of the short rate one year from now? What is the probability that the short rate is negative in 1 year for the process in Q2? What is the distribution of the short rate in the limit as time goes to infinity for the process in Q2? Assume, now, that the short rate follows a Cox-Ingersoll-Ross (CIR) process with the same k and as in Q2 but with = 8%. The current short rate is 1.5%. What are the mean and variance of the short rate one year from now? What are the mean and variance of the short rate in the limit as time goes to infinity? Prove that the expected value of a CIR process is e-kt X(0) + (1 - e-kt). Use Bloomberg to get daily closing prices for the S&P 500 index for all of 2016. Assume that the closing price follows a geometric Brownian motion. Estimate and . Use a time basis of 1 year, so that the time between each day is 1/m, where m is the number of days for which you have data. Hint: Use the fact that the distribution of is known. Optional part: Repeat the same analysis using data from 2008. Do you see a difference in the estimated values