MTH5510: QRM – Exercises Set 1 Due date: August 31, 2020; 9:00am; Analysis the output of the Matlab code is mandatory. I am not interested just to the Matlab code. Please submit your solutions through Moodle. Please use LaTeX or MS Word for preparing your solution. Note: You might want to use Matlab for this exercise; adequately report and comment on your results (For a quick introduction to Matlab visit http://www.mathworks.com/access/helpdesk/ help/pdf_doc/matlab/getstart.pdf) Exercise I: This question refers to Example 1 from the lecture slides. 1. Take d = 1 and λ1 = 1. Suppose X1,t+∆ has mean zero and standard deviation 0.01, and St = 100. For each of the following distributions of X1,t+∆, simulate 10,000 realizations of L(t, t+ ∆) and plot the empirical distribution. Then compute the mean and standard deviation of L(t, t + ∆). Find the normal probability density function corresponding to this mean and standard deviation and plot it over the empirical distribution. • X1,t+∆ is a scaled Student’s t-distribution with 3 degrees of freedom • X1,t+∆ is a scaled Student’s t-distribution with 10 degrees of freedom • X1,t+∆ is a scaled Student’s t-distribution with 50 degrees of freedom • X1,t+∆ has a normal distribution By ”X1,t+∆ is a scaled Student’s t-distribution with degrees of freedom” we mean that X1,t+∆ has the Student’s t-distribution with ν degrees of freedom for some appropriate α ∈ R. You must find the appropriate value of α so that X1,t+∆ has the correct standard deviation. The Matlab functions randn and trnd will be useful for this question. Which of the resulting distributions of L(t, t+ ∆) above are normal distributions? How do you know? 2. for each of the distributions from the previous part, state the exact probability distribution of Lδ(t, t+ ∆). Exercise II: This question refers to Example 2 from the lecture slides. 1. Take St = 100, r = 0.05, and σt = 0.2. Suppose X1,t+∆ has a normal distribution with mean zero and standard deviation 0.01, X2,t+∆ has a normal distribution with mean zero and standard deviation 10−4, and X3,t+∆ has a normal distribution with mean zero and standard deviation 10−3. Further X2,t+∆ is independent from the other two risk factor changes, but X1,t+∆ and X3,t+∆ have correlation −0.5. (note: the Matlab mvnrnd may be useful for this question.) 1 Let T = 1,K = 100, and ∆ = 1/252. Simulate 10,000 realizations of L(t, t+∆) and plot the empirical distribution. Note that there is a positive probability for rt+∆ and σt+∆ to be negative with the distributions they are assigned. State whether you think this is a problem and why, and how you circumvent ti if necessary. 2. For the same distribution as the previous part, simulate 10,000 realizations of Lδ(t, t+ ∆) and plot the empirical distribution. Which of the three risk factors seems to contribute most to the Lδ(t, t+ ∆), and how did you decide this? Exercise III: Let X ∼ N (µ, σ2). Derive the formula: E[eX ] = eµ+ 1 2 σ2 . Exercise IV: Read ”An Academic Response to Basel II” and Chapter 1 of Quantitative Risk Man- agement. 2
