Pstat 170 Fall 2022 – Asn 4
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Pstat 170 Fall 2022 – Asn 4
Problem 1
A European binary (or Digital) option pays $5 if the stock ends above $60 after 3 months and nothing otherwise. The following 3-period binomial tree represents the monthly stock price movements:
71.46
67.42
63.60 64.72
S(0) = 60 61.06
57.60 58.61
55.30
53.08
Assuming cont. compounded interest rate of r = 3% and no dividends, find the repli- cating portfolios for each date if the stock prices moves according to
S(0) = 60 → S(1) = 57.6 → S(2) = 61.06 → S(3) = 58.61.
What is the terminal value of your portfolio at n = 3?
Verify that your replicating strategy is self-financing at steps n = 1,n = 2.
Problem 2
Consider a binomial model with σ = 0.25, δ = 0.06 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, consider the American Call CAm with strike K = 96.
1. At which states is early exercise rational?
2. Find the premium of this Call today t = 0.
3. Suppose the stock moves are Down/Up/Up/Down. Compute the replicating port- folio and the exercise strategy along that scenario.
Problem 3
Using the provided R script as a starting point, implement the binomial tree option pricing algorithm for European options.
1. Consider a binomial model with u = 1.01,d = 
, and interest rate r of 3% a 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 PE (K) as a function of strike K, with K = 85, 85.5, 86, . . . , 110.
2. 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. 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.
Problem 4
Use the same setting as Problem 3. Modify the binomial tree pricing algorithm to compute prices of an American Put PA with maturity T and strike K .
Re-compute PA (K) as a function of strike K, with K = 85, 85.5, 86, . . . , 110. Com- ment on the results. In particular, compare to the values of PE (K) in Problem 3. Hand-in
your code, a plot of the difference PA (K) − PE (K) and a summary of your findings.
Problem 5
Let h = T/N be the length of one time-step in the binomial tree model. Set
u = exp (σ ^h) and d = exp (−σ ^h) .
Fix T = 1, σ = 0.25, S(0) = K = 100 and interest rate of r = 5% yearly, compounded continuously. There are no dividends.
1. Compute the price C(0,S0 ) of a European Call with the above parameters using N = 4, 8, 15, 30, 60, 90, 120 (i.e. varying the number of steps, while keeping the maturity fixed and using the particular scaling of u and d above. Note that as N grows, h shrinks.)
2. Also compute the Black-Scholes price of this Call CBS (0,S0 ). Comment on the answer in relation to what you obtained in part 1.
3. Repeat part 2 for σ = 0.2.0.3, 0.5. What happens to C(0,S0 ) as a function of σ?
2022-11-16