W-W: 2 Course: M339W/M389W - Financial Math for Acutaries Page: 1 of 2 University of Texas at Austin Warm-up Worksheet #2 Prerequisite material. In preparation for the first in-term exam, please solve the following problems: Problem 2.1. Consider the following values of interest rates from an incomplete Black-Derman-Toy interest rate tree for the effective annual interest rates. r0 = 0.09, ru = 0.12, ruu = 0.15, rd = 0.08, rud = 0.13. (i) (2 points) Find the volatility σ1 of the interest rates at time−1. (ii) (3 points) Find the interest rate rdd missing from the tree. (iii) (5 points) Consider a 3−year caplet for the notional amount of $100 whose cap rate is given to be 11.5%. Calculate its price. Problem 2.2. Consider the following binomial interest-rate tree modeling the future evolution of annual continuously compounded interest rates. The period-length is one year. 0.06 0.055 0.05 0.05 0.045 0.04 The risk-neutral probability is given to be equal to 1/2. What is the price of a zero-coupon bond redeemable in three years for $1,000? ˇ Instructor: Milica Cudina W-W: 2 Course: M339W/M389W - Financial Math for Acutaries Page: 2 of 2 Problem 2.3. The current stock price is $90. According to your coworker, the time−1 price of this stock will be uniformly distributed between $80 and $120. What is the expected rate of return on this stock for this one year period and the model she is suggesting? Problem 2.4. The product of independent lognormal random variables is also lognormally distributed. True or false? Problem 2.5. Consider the following values of interest rates from an incomplete Black-Derman-Toy interest rate tree for the effective annual interest rates. r0 = 0.10, ru = 0.15, ruu = 0.19, ruuu = 0.27 rd = 0.13, rud = 0.15, ruud = 0.20. Calculate the yield volatility in year one of a zero-coupon bond with maturity two years from today. ˇ Instructor: Milica Cudina

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