PSTAT 170 - 11:00AM ELECTION, QUIZ 2 INTRODUCTION TO MATHEMATICAL FINANCE APRIL 29, 2021 HAL W. PEDERSEN The quiz is a take home assignment due at 8:00pm Pacific on May 6, 2021. (Due date revised to 8:00pm Pacific on May 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. The S&R index spot price is $2100 and the continuously compounded risk-free rate is 5%. You observe a 9-month forward price of $2155.854. (a) (2 points) What dividend yield is implied by this forward price? (b) (4 points) Suppose you believe the dividend yield over the next 9 months will be 0.5%. What arbitrage would you undertake? (c) (4 points) Suppose you believe the dividend yield will be 3% over the next 9 months. What arbitrage would you undertake? Problem 2. Suppose call and put prices are given by Strike Price 45 50 60 Call Price 12 10 5 Put Price 3 5 10 (a) (3 points) Show that you can not effect arbitrage only based on put options. (Apply the tests for different strike prices on pages 281-282 of the textbook.) (b) (2 points) Find the convexity violations. (c) (5 points) What spread (i.e. option position) would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage. Problem 3. Let S = $100, K = $95, r = 8%, T = 0.5, and δ = 0. Let u = 1.1, d = 0.9, and n = 2. Construct the binomial tree for pricing and hedging a European put option. At each node provide the premium, ∆ and B. 1 2 PSTAT 170 Problem 4. You are interested in the hedging of an American put option written on a stock index with a continuous dividend payment. You are using a binomial tree for analyzing the American put option and are given the following information. • Current price of the stock index is S = 30. • The American put option expires in 4 periods. • The continuously compounded interest rate is r = 0.06 per period. • The continuous dividend yield rate is δ = 0.04 per period. • The up and down stock index price movement per period is governed by u = 1.25 and d = 0.75. • The strike price is K = 28. (a) (5 points) Determine the optimal exercise strategy for the American put option. (This means that you indicate at which nodes in the tree you would exercise the put option.) (b) (5 points) What is the replicating portfolio (i.e. the values of (∆, B)) at each of the three nodes at time 2? (The three nodes at time 2 are (2, 0), (2, 1) and (2, 2) which are sometimes identified as corresponding to the movements dd, du and uu respectively.)














































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