math finance problems
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?
Collepals.com Plagiarism Free Papers
Are you looking for custom essay writing service or even dissertation writing services? Just request for our write my paper service, and we'll match you with the best essay writer in your subject! With an exceptional team of professional academic experts in a wide range of subjects, we can guarantee you an unrivaled quality of custom-written papers.
Get ZERO PLAGIARISM, HUMAN WRITTEN ESSAYS
Why Hire Collepals.com writers to do your paper?
Quality- We are experienced and have access to ample research materials.
We write plagiarism Free Content
Confidential- We never share or sell your personal information to third parties.
Support-Chat with us today! We are always waiting to answer all your questions.
