期权复习题11.docx

1、期权复习题11 期权复习题1. Explanationspeculators wish to take a position in the market.Either they are betting that a price will go up or they are betting that it will go down. They use derivatives to get extra leverageHedgers are interested in reducing a risk that they already face. Arbitrage involves lockin

2、g in a risk-less profit by entering simultaneously into transactions in two or more markets. A call option gives the holder the right to buy an asset by a certain date for a certain price. Put option: A put option gives the holder the right to sell an asset by a certain date for a certain price. Fut

3、ures contract: It is an agreement to buy or sell an asset for a certain price at a certain time in the future.In-the-money option：it would lead to a positive cash flow to the holder if it were exercised immediately.risk-neutral valuation: Firstly, assume that the expected return from the stock price

4、 is the risk-free rate r, then calculate the expected payoff from the option, at last, discounting the expected payoff at the risk-free rate Butterfly spreads: A butterfly spread involves positions in options with three different strike prices: buying two call options with strike prices X1 and X3, a

5、nd selling two call options with a strike price X2, X1 X2 X3Factors affecting stock option pricing: stock price, strike price, risk-free interest rate, volatility, time to maturity, and dividends. Bull spreads: A bull spread can be created using two call options with the same maturity and different

6、strike prices. The investor buys the call option with the lower strike price and shorts the call option with the higher strike price. Bull spreads can also be created by buying a put with a low strike price and selling a put with a high strike price.Bear spreads: A bear spread can be created by sell

7、ing a call with one lower strike price and buying a call with another higher strike price 2. What is the difference between a long forward position and a short forward position?Solution: When a trader enters into a long forward contract ,she is agreeing to buy the underlying asset for a certain pric

8、e at a certain time in the future. When a trader enters into a short forward contract ,she is agreeing to sell the underlying asset for a certain price at a certain time in the future.3. “when a futures contract is traded on the floor of the exchange, it may be the case that the open interest increa

9、ses by one, stays the same, or decreases by one .” Explain this statement.If both sides of the transaction are entering into a new contract, the open interest increases by one. If both sides of the transaction are closing out existing position, the open interest decreases by one. If one party is ent

10、ering into a new contract while the other party is closing out an existing position, the open interest stays the same.4.The price of gold is currently $500 per ounce. The forward price for delivery in one year is $700. An arbitrageur can borrow money at 10% per annum. What should the arbitrageur do?

11、 Assume that the cost of storing gold is zero.5.Give two reasons why the early exercise of American options on a non-dividend-paying stock is not optimal.6 Three call options on a stock have the same expiration date and strike prices of $45, $50, and $55. The market prices are $7, $5, and $2, respec

12、tively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy. 7.What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility

13、 is 35% per annum, and the time to maturity is six months? If this option is a European call option, what is the price?8. What is the price of a European put option on the S&P500 that is six months from maturity, the current value of the index is 500, the exercise price is 500, the risk-free interes

14、t rate is 10% per annum, the volatility of the index is 30% per annum., Continuous dividend yields is 2% per annum?9.Consider one year American put option on a non-dividend-paying stock when the stock price is $300, the strike price is $300, the risk-free interest rate is 10% per annum, and the vola

15、tility is 40% per annum. Divide the year into three 4-month time intervals and use the tree approach to estimate the value of the option. 10. Consider the price of a stock, S, which following the processwhere is a standard Brownian motion. For the first three years, ; for the next three years, . If

16、the initial value of stock price is $10, what is the expect value of the stock price at the end of year 6?11. What is the difference between forward contract when the forward price is 40 and a call option with a strike price of 40? (2) Suppose that a U.S. company knows that it is due to receive 10,0

17、00 from one of its British exporter in 30 days. It is faced with a significant foreign exchange risk. If you are management, how to hedging the foreign exchange risk by using forward contract or call option respectively? the foreign exchange rates of 30-day forward on pound is 1.60the strike exchang

18、e rate of a 30-day call option on pound is 1.60.12 Explain carefully the meaning of the terms convenience yield and cost of carry. What is the relationship between the futuers price F, the spot price S, the convenience yield,y, and the cost of carry,c?The convenience yield,y,（便利收益）for a commodity is

19、 a measure of the benefits realized from ownership（所有权）of the physical commodity（具体商品、实物商品） that are not realized by the holders of a futures contract.The cost of carry, c , is the storage cost plus the interest that is paid to finance the asset less the income earned on the asset.relationship betwe

20、en and the futuers price, the spot price, the convenience yield and the cost of carry is13. A one-year-long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.(a)What are the forw

21、ard price and the initial value of the forward contract?(b)Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract?14 .Explain the differences between forward contract and futures contract?15 .Expl

22、ain the differences between exchanged traded and Over-the- counter ?Exchange Tradedstandard productstrading floor or computer tradingvirtually no credit riskOver-the-Counternon-standard productstelephone marketsome credit risk16. Calculate the value of a six-month at-the-money European call option o

23、n a stock index when the index is at 500, the risk-free interest rate is 10% per annum, the volatility of the index is 20% per annum, and the dividend yield on the index is 3% per annum.17 If a stock price, S, follows geometric Brownian motion1) What is the process followed by the variable ? Show th

24、at also follows geometric Brownian motion.2)The expected value of ST is . What is the expected value of ?3) The varaince of ST is .What is the variance of ?18 Show that the probability that a European call option will be exercised in a risk-neutral world is, . Using risk-neutral valuation to value t

25、he complicated digtial option whose payoff at maturity is 19 Suppose that a portfolio is delta neutral and has a gamma of = -3,000, The delta and gamma of a particular traded call option are T = 0.62 and = 1.50, respectively. What position in the traded call option and in the underlying asset would

26、make the portfolio both gamma neutral and delta neutral? = The portfolio can be made gamma neutral by including a long position ofwT = - / T = 3000/ 1.5 = 2,000 traded call options in the portfolio. However, the delta of the portfolio will then change from zero to = 2,000 T = 2,000 0.62 = 1,240 = A

27、quantity, 1,240, of the underlying asset must be sold from the portfolio to keep it delta neutral.20. The stock price process assumed satisfiesSuppose that f is the price of a call option or other derivative contingent on S. Using no arbitrage opportunity to derive the Black-Scholes Differential Equ

28、ation21. Give the definitions of delta, gamma, vega, theta, and rho of the derivative。Give the proof : 22. A stock price is currently $40, It is known that at the end of six months it will be either $36 or $44. The risk-free interest rate is 10%. Suppose that is the stock price at the end of six mon

29、ths. What is the value of a derivative that pays off at this time ?23. Suppose stock prices follow the processes Show that the probability that a European call option will be exercised in a risk-neutral world is, . Since 24. Suppose two stock prices follow the processes We assume that prices of the

30、two stocks are independent and that no dividends are paid. Consider a derivative that pays off $Q at time T if the price of stock A is below and the price of stock B is below.Using risk-neutral valuation to value the derivative. The final payooff: In risk-netural world, Where Since Where , =25. Comp

31、anies A and B have been offered the following rates per annum on $20 million five-year loan:Company A wants to borrow floating-rate loan, Company B wants to borrow fixed-rate loan. Design a swap that will net bank, acting as intermediary, 0.04% per annum in total and appear to be equally attractive to both companiesSolution:Difference between the interest rates in fl rate markets b=0.7%Difference between the interest rates in fixed rate markets a=1.2%The swap arrangement appears to improve the position of both A and B 0.23% per annum