Pstat 170 Winter 2024 – Asn 4 Problem 1 A European binary (or Digital ) option pays $3 if the stock ends above $63 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 = 4% and no dividends, find the replicating portfolios for each date if the stock prices moves according to S(0) = 60 → S(1) = 57.6 → S(2) = 61.06 → S(3) = 64.72. Verify that the replicating strategy is self-financing at steps n = 1, n = 2 and compute the terminal value of the replicating portfolio at n = 3. Problem 2 , δ = 0 and interest rate r =3% a year, Consider a binomial model with u = 1.04, d = 100 104 compounded continuously. Using T = 1 maturity of one year, initial stock price S(0) = 40 and N = 4 periods: a) find the premium of the European Call C(K) for K = 36, 37, 38, 39, 40, 41, 42, 43, 44, 45. You’re encouraged to use a computer to do this faster. Create a plot of K 7→ C(K). b) Write down a mathematical formula for the function K 7→ C(K). Hint: this is a piecewise function. Try computing e.g. the Call premium when K = 40 ϵ when ϵ = 0.1, 0.01, . . . to see the pattern. c) Compute the prices of the European Put with K = 38, 40, 42 and verify that PutCall parity holds. Problem 3 Consider a binomial model with σ = 0.24, δ = 0.06 and interest rate r=5% annual, both compounded continuously. Using T = 1 maturity of one year, initial stock price S0 = 100 and N = 4 periods, consider the American Call C Am with strike K = 95. 1. In which scenarios is early exercise rational? 2. Find the premium of this Call today C0Am . 3. Suppose the stock moves are U p/U p/Down/Down. Compute the replicating portfolio and the exercise strategy along that scenario. Problem 4 Using the posted R script as a starting point, implement the binomial tree option pricing algorithm for European options. 1. Consider a binomial model with σ = 0.18, and interest rate r of 2% a year, compounded continuously. Using T = 1/2 maturity of half a year, initial stock price S(0) = 100 and N = 20 periods, plot the premium of the European Put P E (K) as a function of strike K, with K = 85, 85.5, 88, . . . , 133. 2. What is the smallest slope of K 7→ P E (K)? What is the largest slope of K 7→ P E (K)? 3. Is the function K 7→ P E (K) concave or convex? How do the above questions relate to the material in Chapter 9? Problem 5 Use the same setting as Problem 4. Modify the binomial tree pricing algorithm to compute prices of an American Put P A (K) with maturity T and strike K. 1. Compute P A (K) as a function of strike K, with K = 85, 85.5, 86, . . . , 133. Hand-in the plot of K 7→ P A (K). 2. Compute and plot the difference between the American and European Put premia P A (K) − P E (K). For what strike is this difference the largest? When is it the smallest?