PSTAT 170 QUIZ 4 INTRODUCTION TO MATHEMATICAL FINANCE MAY 127, 2021 HAL W. PEDERSEN The quiz is a take home assignment due at 8:00pm Pacific on June 8, 2021. Provide clear answers to the questions. Please Submit each question on its own page. This quiz consists of 4 questions. Good luck! Problem 1. A stock price index at time t has the distribution according to the random variable St = 1500× exp ( µt+ σ √ tZ ) where Z has a standard normal distribution and time is measured in years. Parameter values are µ = 0.10 and σ = 0.30. The stock price index also pays a continuous dividend at the rate δ = 0.03. You invest $5,000 in the stock index at time 0 and your dividends are immediately reinvested in the stock index over the life of your investment. (1) What is the probability that value of your investment at the end of one-year is less than or equal to $4,000? (2) What is the probability that the one-year total return on your investment is less than or equal to -10%? (i.e. what is the probability that you lose at least 10% on your intial investment over the year?) (3) Compute the 0.01 percentile of your one-year investment total return? [i.e. de- termine the value Y0.01 such that 0.01 = P ( TR[0,1] ≤ Y0.01 ) ] Problem 2. A stock price index at time t has the distribution according to the random variable St = 50× exp ([ µ− 0.5σ2 ] t+ σ √ tZ ) where Z has a standard normal distribution and time is measured in years with µ = 0.07. What is the smallest value of σ so that the probability that the stock price declines by at least 20% over the next week is at least 2%? (One year is assumed to be 365 days.) 1 2 PSTAT 170 Problem 3. A stock price index pays a continuous dividend at annual rate δ = 0.02 and the stock price index follows the lognormal model with ln ( St/S0 ) ∼ N ( µt, σ2t ) where S0 = 25, µ = 0.07 and σ = 0.29. The risk-free rate of interest is r = 0.045. You have purchased a European put option on the stock index with strike price K = 24 that expires at time 1. You are to perform a Monte Carlo simulation to price this European put option. You are given the following table of standard normal random draws. Draw Number Standard Normal Draw 1 −1.50 2 −0.80 3 −0.75 4 1.05 5 −2.18 6 0.47 7 1.65 8 −1.65 (1) Based on this sequence of standard normal random draws, what is the Monte Carlo estimate for the price this European put option at time 1? (2) What is the standard deviation of the Monte Carlo estimate? PSTAT 170 3 Problem 4. A stock price index follows the lognormal stock price model ln ( St/S0 ) ∼ N ( µt, σ2t ) . You are given the following observations on the stock index price. Observation Observation Time Stock Index Price Observed 1 0 900 2 1/12 951 3 2/12 983 4 3/12 1070 5 4/12 956 6 5/12 827 7 6/12 880 8 7/12 1020 9 8/12 1095 10 9/12 1232 Estimate µ and σ using this data.




































































































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