only need to do question 3to 7 and provide the step and graph
FNCE 445 Problem Set #6 Consider one-period binomial trees (Q1 and Q2): 1. Let S = 100,K = 105, r = 8%, T = 0.5 and δ = 0. Let u = 1.3 and d = 0.8. (a) What are ∆, B, and the premium for a European call? (b) What are ∆, B, and the premium for a European put? 2. Let S = 100,K = 95, r = 8%, T = 0.5 and δ = 0. Let u = 1.3 and d = 0.8. (a) What are ∆, B, and the premium for a European put? (b) Suppose you observe a put price of $8. What is the arbitrage? (c) Suppose you observe a put price of $6. What is the arbitrage? Consider multi-period binomial tree models in which n refers the number of periods (Q3 - Q7): 3. Let S = 100,K = 95, r = 0.08, T = 1 and δ = 0. Let u = 1.3, d = 0.8, and n = 2. Construct the binomial tree for a call option. At each node provide the premium, ∆, and B. 4. Repeat the option-price calculation in the previous question for stock prices of 80, 90 and 110. What happens to the initial option ∆ as the stock price increase? 5. Let K = 95, σ = 0.3, r = 0.08, T = 1 and δ = 0. Let u = 1.3, d = 0.8, and n = 2. Construct the binomial tree for a put option for S = 80, 110, and 130. At each node provide the premium, ∆, and B. What happens to the initial put ∆ as the stock price increase? 1 6. Try the supplementary exercise (posted at D2L) and refer the solution. Then, repeat the problem assuming that the stock pays a continuous dividend 8% per year. Calculate the prices of the American and Eu- ropean puts and calls. Which options are early-exercised? 7. Suppose that the exchange rate is $0.92/Euro. The dollar-denominated interest rate is 4% and the euro-denominated interest rate is 3%. u = 1.2, d = 0.9, T = 0.75, n = 3, and K = $1.00. a. What is the price of a 9-month European put? b. What is the price of a 9-month American put? 8. Consider a one-period binomial tree when the underlying stock pays the continuous dividend with yield δ. The interest rate is r. a. Suppose u < e(r−δ)h.="" show="" that="" there="" is="" an="" arbitrage="" opportunity.="" b.="" suppose="" d=""> e(r−δ)h. Show that there is an arbitrage opportunity. In answering the following problems, refer the standard normal density table to compute N(d1) and N(d2). It is helpful if you verify your solution by using the spreadsheets on the CD-ROM from the textbook. 9. Suppose S = 100,K = 95, σ = 30%, r = 0.08, δ = 0.03 and T = 0.75 a. Compute the BS price of a call. b. Compute the BS price of a call for which S = 100e−0.03×0.75,K = 95e−0.08×0.75, σ = 30%, r = 0, δ = 0 and T = 0.75. How does your answer compare to that for (a)? c. What is the 9-month forward price for the stock? d. (optional) Compute that price of a 95-strike 9-month call option on a futures contract. e. (optional) What is the relationship between your answer to (d) and the price you computed in (a) and (b)? 10. The exchange rate is U95/Euro, the yen-denominated interest rate is 1.5%, the euro-denominated interest rate is 3.5%, and the exchange rate volatility is 10%. 2 a. what is the price of a 90-strike yen-denominated euro put with 6 months to expiration? b. what is the price of a 1/90-strike euro-denominated yen call with 6 months to expiration? c. What is the relation between your answer in (a) and (b)? 11. Consider a bull spread where you buy a 40-strike put and sell a 45- strike put. Suppose σ = 0.30, r = 0.08, δ = 0, and T = 0.5. Use the excel spreadsheets when you compute gamma and vega. Use the normal table when you compute delta. a. Suppose S = 40. What are delta, gamma, and vega? b. Suppose S = 45. What are delta, gamma, and vega? c. Consider a bull spread when you buy a 40-strike call and a 45- strike call. Repeat (a) and (b). You can refer the answer key of Exercise 11.14 of the text book. Compare the results with the result of (a) and (b) above. Which greeks are identical? Why are they identical? 12. Suppose S = 40, σ = 0.3, r = 0.08, and δ = 0. Suppose you sell a 45-strike call with 91 days to expiration. (a) What is delta? (b) If the option is on 100 shares, what investment is required for a delta-hedged portfolio? What is your overnight profit if the stock tomorrow is 39? What if the stock price is 40.50? 3