PSTAT 170 Spring 2016 - Assignment 4 Problem 1. A European binary (or Digital ) option pays $5 if the stock ends above $57 after 3 months and nothing otherwise. The following 3-period binomial tree represents the monthly stock price movements: 71.46 67.42 63.60 S(0) = 60 64.72 61.06 57.60 58.61 55.30 53.08 Assuming cont. compounded interest rate of r = 3% and no dividends, find the replicating portfolios for each date if the stock prices moves according to S(0) = 60 S(1) = 57.60 S(2) = 61.06 S(3) = 64.72. Verify that your replicating strategy is self-financing at each step. Problem 2. A customized Asian floating strike Put option in a 2-period binomial tree pays out (S0 + S1 + S2 ) S2 ) max(0, 3 where Si is the stock price at the end of the i-th period. If S0 = 40, r = 0.045, = 0.15, = 0 and each period is 2 months long, find the no-arbitrage price of this Asian option today. Problem 3. Consider a binomial model with = 0.2, = 0.05 and interest rate r of 5% a year, both compounded continuously. Using T = 1 maturity of one year, initial stock price S(0) = 100 and N = 4 periods, find the premium of the American Call C A (K) for K = 95. At which states is early exercise rational? Suppose the stock moves are U p/U p/Down/Down. Compute the replicating portfolio along that scenario. Problem 4 & 5. Using the provided R script as a starting point, implement the binomial tree option pricing algorithm for European options. 100 , and interest rate r of 5% a (1) (6pt) Consider a binomial model with u = 1.01, d = 101 year, compounded continuously. Using T = 1 maturity of one year, initial stock price S(0) = 100 and N = 15 periods, plot the premium of the European Put P E (K) as a function of strike K, with K = 85, 85.5, 86, . . . , 110. (2) (3pt) Also compute the corresponding Delta. Hand-in annotated plots of Put premium and Delta as a function of K (be sure to label axes, etc.). (3) (3pt) In a brief paragraph summarize your findings. Discuss the asymptotes, slopes and convexity of the plots. Relate to the properties we discussed in Chapter 9. (4) (8pt) Modify the binomial tree pricing algorithm to compute prices of an American Put P A with maturity T and strike K. Re-compute P A (K) as a function of strike K, with K = 85, 85.5, 86, . . . , 110. Comment on the results. In particular, compare to the values of P E (K) in part 1. Hand-in your code, a plot of the difference P A (K) P E (K) and a summary of your findings. 1