w_exam_one_solutions

# w_exam_one_solutions - Tuesday February 26th M339W/389W...

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Tuesday, February 26th M339W/389W Financial Mathematics for Actuarial Applications Spring 2013, University of Texas at Austin In-Term Exam I Instructor: Milica ˇ Cudina Notes : This is a closed book and closed notes exam. Time : 75 minutes TRUE/FALSE 1 (2) TRUE FALSE 2 (2) TRUE FALSE 3 (2) TRUE FALSE 4 (2) TRUE FALSE 5 (2) TRUE FALSE 6 (2) TRUE FALSE 7 (2) TRUE FALSE 8 (2) TRUE FALSE 9 (2) TRUE FALSE 10 (2) TRUE FALSE 11 (2) TRUE FALSE 12 (2) TRUE FALSE MULTIPLE CHOICE 1 (5) a b c d e 2 (5) a b c d e 3 (5) a b c d e 4 (5) a b c d e 5 (5) a b c d e 7 (5) a b c d e 6 (5) a b c d e FOR GRADER’S USE ONLY: T/F 1. 2. M.C. Σ

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2 Part I. TRUE/FALSE QUESTIONS Please note your answers on the front page. 1. You are trying to estimate the mean rate of return on a stock. Then, using more frequent observations of the stock price produces a more accurate estimate. Solution: FALSE 2. The symmetric simple random walk accumulates quadratic variation at rate one per unit of time. Solution: TRUE 3. Continuously compounded returns of stocks are multiplicative. Solution: FALSE See p. 353 in McDonald. 4. The product of any pair of lognormal random variables is also lognormally distributed. Solution: FALSE Let X be a lognormal random variable. Let us write X = e η where η is a normal random variable. Then, Y = 1 /X is also lognormal since it can be written as Y = e - η . However, XY = 1 – not a lognormal random variable. 5. A simple symmetric random walk has independent increments. Solution: TRUE 6. A simple symmetric random walk is an appropriate model for the evolution of stock prices. Solution: FALSE The simple symmetric random walk can have negative values, while the stock prices cannot become negative. 7. In our usual notation, the time - t forward price of a bond deliverable at T is F t,T [ P ( T, T + s )] = P ( t, T + s ) P ( t, T ) . Solution: TRUE 8. In our usual notation, let S (0) = 110 , S (1 / 2) = 118 and let S (1) = 114 be the observed prices of a share of stock. Then, the estimate of the mean continuously compounded annual rate of return of this stock ˆ α is less than 3%. Solution: FALSE ˆ α = ln (114 / 110) 0 . 0357 . 9. A caplet is a financial instrument used as protection against the increase in the interest rate for all repayment installments of a loan to be repaid over multiple periods.
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