代写MTH202 Introduction to Financial Mathematics Problem Set调试R语言程序

MTH202

Introduction to Financial Mathematics

Problem Set

1. A 74 stock pays a 0.75 dividend every six months, with the first dividend coming one month from today. The continuously compounded risk free rate is 0.05.

(a) What is the price of a prepaid forward contract that expires one year from today?

(b) What is the price of a forward contract that expires on the same date?

2. Suppose that the stock is 74, and the continuously compounded interest rate is 0.05.

(a) What is the 9-month forward price, assuming dividends are zero?

(b) If the 9-month forward price is 77, what is the annualized forward premium?

(c) If the 9-month forward price is 75.309, and the stock pays constant quarterly dividends, with the first dividend coming three months from today, and the last one coming immediately before the expiration of the forward contract, what is quarterly dividend payments the stock makes?

3. XYZ stock trades at 66, the risk free rate is 5%, and XYZ stock pays constant quarterly dividends of 0.45. Assuming that the first dividend is coming three months from today, and the last one coming immediately before the expiration of the forward contract. You can trade XYZ single stock futures with an expiration of nine month.

(a)Suppose you observe a 9-month forward price of 67.50, What arbitrage would you undertake?

(b)Suppose you observe a 9-month forward price of 66.75, What arbitrage would you undertake?

4. XYZ stock trades at 54 per share. XYZ stock does not pay any dividends. The cost of an European call option with strike price 50 and expiration date in three months is 8 per share. The risk free annual interest rate continuously compounded is 4%.

(i) Find the no-arbitrage price of a European put option with the same strike price and expi-ration time.

(ii) Suppose that the price of an European put option with the same strike price and expiration time is 3, find an arbitrage strategy and its profit per share.

5. Suppose that the current price of XYZ stock is 31. XYZ stock does not give any dividends. The risk free annual effective interest rate is 10%. The price of a three-month 30-strike European call option is 3. The price of a three-month 30-strike European put option is 2.25. Find an arbitrage opportunity and its profit per share.

6. Heather has a short position in 1200 shares of stock XYZ. She is supposed to return this stock one year from now. To insure her position, she buys a 65-strike call. The premium of a 65-strike call is 14.31722. The current price of one share of XYZ stock is 75. The risk free annual effective rate of interest is 5%.

(i)Make a table with Heathers profit when the spot price at expiration is 40,50, 60,70, 80,90.

(ii)Assuming that she does not buy the call, make a table with her profit when the spot price at expiration is 40,50, 60,70, 80,90.

(iii)Draw the graphs of the profit versus the spot price at expiration for the strategies in (i) and in (ii).

7. Suppose call and put prices are given by

Strike                 50        55

Call Premium       9         10

Put Premium        7         6

Which no-arbitrage property is violated? What spread position would you use to effect arbi-trage? Demonstrate that the spread position is an arbitrage.

8. You are given the following:

– The current price to buy one share of XYZ stock is 500.

– The stock does not pay dividends.

– The annual risk-free interest rate, compounded continuously, is 6%.

– A European call option on one share of XYZ stock with a strike price of K that expire in one year costs 66.59.

– A European put option on one share of XYZ stock with a strike price of K that expires in one year costs 18.64.

Using put-call parity, calculate the strike price, K.

9. Consider a European put option on a stock index without dividends, with 6 months to expiration and a strike price of 1000. Suppose that the continuously compounded risk-free rate is 4% , that the put costs 74.20 today. Calculate the price that the index the index must be in 6 months so that being long in the put would produce the same profit as being short in the put.

10. A stock is currently selling for 100. The risk-free interest rate is 4% annual effective. A 1-year forward contract on the stock is available with a forward price of 106.

Is there an arbitrage opportunity? If it is, determine the amount of guaranteed profit.

(Hint:long stock + short forward)

11. For a stock currently selling for 100 that has quarterly dividends of 1 with the next dividend due in 3 months, the prepaid forward price for a 1-year forward is 96.15. Show that the current risk free interest rate as a nominal interest rate compounded quarterly is 6.1863%.

12. An investor enters into a position by shorting a stock and purchasing a call option on the stock with an exercise price of 50 and two years to expiration. The current stock price is 52, the call premium is 8, and the risk-free interest rate is 2% annual effective. Determine he range of spot prices at expiration of the stock in order for the investor to lose more than 4 on this position.

13. An investor uses 40-strike and 50-strike call options to enter into a 1-year bull spread on a stock. The 40-strike call costs 9 and the 50-strike call costs 3. The risk-free interest rate is 3% annual effective. Determine spot price of the stock in 1-year in order for the investor’s profit on the position to be 0. (Hint : Bull spread: A position in which you buy a call and sell another identical call with a higher strike price.)

14. For a European put option on a stock (assume non-dividend payment), a lower bound for the price is KE−rT − S0. Suppose that S0 = 37, K = 40, r = 5% per annum (compounded continuously), and T = 0.5 years.

(a) Calculate the lower bound for this case.

(b) Consider the situation where the European put prices is 1.5. Is it theoretically possible to make a risk free profit? if yes,how?

15. A call option with expiry 1 year and strike price 1000 quoted at 201. The share price is quoted today at 1025. Assuming the risk free rate of interest is 5% and neglecting the dividend payments, use Put-Call Parity to value a put option with the same expiry date and strike price. By using arbitrage arguments,explain why this is the correct price.

16. The share of a company are priced at 200 and a European call option on the shares with exercise of 160 is priced at 20. Neglect the cost of borrowing. Show how to make a risk-free point.

17. On a specific date the S&P index of shares is at 4000 and the price of a contract to buy the index 1 year in the future is 4250. If the risk free rate is 5% (compounded continuously), is it theoretically possible to make a risk free profit? If yes, how(Neglect dividend payment).

18. A stock currently sells for 55, An investor uses 70-strike and 50-strike call options to enter into a 1-year bear spread on the stock. The 50-strike call costs 18 and the 70-strike call costs 4. The risk-free interest rate is 3% compounded continuously. Determine the investor’s maximum loss on the position. (Hint:Bear Spread: The opposite of a bull spread is called a bear spread; That is selling low strike call and buying high strike call.)

19. Assume today’s stock price is 35, the option’s strike price is 45, the continuously compounded risk-free interest rate is 6%, the time to expiration is nine months, the continuously compounded dividend yield is 2%, and the stock’s annualized standard deviation is 30%.

(a) Construct the three-period binomial tree for the stock price.

(b) Construct the value of an American put option. Is the option early exercised? If so, at which node(s) do you exercise early?

20. Assume today’s stock price is 15, the option’s strike price is 15, the continuously compounded risk-free interest rate is 6%, the time to expiration is 10 months, and the continuously compounded dividend yield is 2%. u = 1.32, d = 0.79. Construct a two-period binomial tree for a call option. At each node, provide the premium, ∆ and B.

21. Let S0 = 28, K = 22, σ = 0.30, continuously compounded interest rate r = 0.06, and continuous dividend yield δ= 0.04. T = 8 months, and number of binomial periods n = 2.

(a) What are u and d?

(b) Construct the binomial tree for the stock.

(c) Calculate the price of a European and American call.

(d) Is there a difference in the premium of the European and American call? Why(or why not)?