期权期货-复习

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1、 复习题Explanation 名词解释speculators 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 locking in a

2、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. Futures （f

3、orward）contract: It is an agreement to buy or sell an asset for a certain price at a certain time in the future.short selling: The investors broker borrows the shares from another clients account and sells them in the usual way. To close out the position the investor must purchase the shares. The br

4、oker then replaces them in the account of the client from whom they were borrowed.In-the-money option/At the money/ Out out of the moneyTime value, intrinsic value, option value,risk-neutral valuation: Firstly, assume that the expected return from the stock price is the risk-free rate r, then calcul

5、ate the expected payoff from the option, at last, discounting the expected payoff at the risk-free rate Factors affecting stock option pricing: stock price, strike price, risk-free interest rate, volatility, time to maturity, and dividends. Long position of forward:A callable bond（可提前赎回债券）: It conta

6、ins provisions(条款) that allow the issuing firm(发行公司) to buy back the bond at a predetermined price at certain times in the future.risk-neutral valuation: Firstly, assume that the expected return from the stock price is the risk-free rate r, then calculate the expected payoff from the option, at last

7、, discounting the expected payoff at the risk-free rate Swaps： Swaps are private agreements between two companies to exchange cash flows in the future according to a prearranged formula. 新型期权的常用产品Asian option, barrier option, lookback option, compound option, forward start option, as you like option

8、(choose option), convitable bond(可转换债券Lookback options : the payoffs from lookback options depend on the maximum or minimum stock price reached during the life of the optionOut options(敲出期权)：Option dies if stock price hits barrier before option maturity In options(敲入期权)：A knock-in option ( 敲入期权 ) is

9、 an option that comes into existence only if the underlying asset price reaches a certain barrier before option maturity .Barrier options are options where the payoff depends on whether the underlying assets price reaches a certain level before option maturity.Forward start options ：are options that

10、 are paid for now but will start at some time in the future. The strike price is usually equal to the price of the asset at the time the option starts ,ie, the option is at the money. Asian options： the payoffs from asian options depend on the average price of the underlying asset during at least so

11、me part of the life of the option. 价差组合期权bottom vertical strangle: a bottom vertical strangle can be created by buy a put with lower strike prices and buy a call with higher strike prices. Bull spreads: A bull spread can be created using two call options with the same maturity and different strike p

12、rices. 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 selling a ca

13、ll with one lower strike price and buying a call with another higher strike price Butterfly spreads: A butterfly spread involves positions in options with three different strike prices: buying two call options with strike prices X1 and X3, and selling two call options with a strike price X2, X1 X2 X

14、3 2 .Explain the differences between forward contract and futures contract?Private contract between 2 partiesExchange tradedNon-standard contractStandard contractUsually 1 specified delivery dateRange of delivery datesSettled at maturitySettled dailyDelivery or final cashsettlement usually occursCon

15、tract usually closed outprior to maturityFORWARDSFUTURES3. 签订一份期货合约对open interest(开放权益)和volum of trade（交易量）的影响如果交易双方签订一种新合约，那么未平仓合约数增长1个。如果交易双方就同一种合约进行平仓，那么未平仓合约数减少一种。如果一方签订一种新合约，而另一方同步将已有合约平仓，那么未平仓合约数不变。交易量都是增长。4 解：公司A的比较优势在英镑而需要美元借款，公司B相反，因此存在互换的基本。英镑上的利差为0.4，美元上的利差为0.8，因此互换的总获利为0.4。已知银行获利0.1，则两个公

16、司各获利0.15。因此A事实上以6.85的利率借美元，而B事实上以10.45的利率借英镑。在银行承当所有市场风险的状况下，互换安排如下图：美元6.85美元6.2英镑11英镑11美元6.2英镑10.45公司A银 行公司B1）互换收益的分派在公司A、公司B和银行间的比例分别为 35%、35%、30%。如何设计 ？2）若分派比例为50%、25%、25%，如何设计 ？美元6.86美元6.2英镑11英镑11美元6.2英镑10.46公司A银 行公司B美元6.8美元6.2英镑11英镑11美元6.2英镑10.5公司A银 行公司B5. 6 基于同一股票的看跌期权有相似的到期日.执行价格为$70、$65和$60，

17、市场价格分为$5、$3和$2. 如何构造蝶式差价期权.请用一种表格阐明这种方略带来的赚钱性.股票价格在什么范畴时，蝶式差价期权将导致损失？ 7 基于同一股票的有相似的到期日敲定价为 $70的看涨期权价格为 $4. 敲定价$65 的看涨期权的价格为 $6。解释如何构造bull/bear spread option.请用一种表格阐明这种方略带来的赚钱性.股票价格在什么范畴时，价差期权将导致损失？ 8. Oil: An Arbitrage Opportunity?Suppose that:- The spot price of gold is US$95- The quoted 1-year fut

18、ures price of gold is US$125- The 1-year US$ interest rate is 5% per annumIs there an arbitrage opportunity？Suppose that:- The spot price of gold is US$95- The quoted 1-year futures price of gold is US$80- The 1-year US$ interest rate is 5% per annumIs there an arbitrage opportunity?8 远期/期货价格公式及其价值公

19、式B-S公式的使用 1).What is the price of a European call 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 is 35% per annum, and the time to maturity is six months? 2). Suppose the current value of the ind

20、ex is 500, continuous dividend yields of index is 4% per annum, the risk-free interest rate is 6% per annum . if the price of three-month European index call option with exercise price 490is $20, What is the price of a three-month European index put option with exercise price 490? by put-call parity

21、 3) What is the price of a European futures put option：current futures price is $19, the strike price is $20, the risk-free interest rate is 12% per annum, the volatility is 20% per annum, and the time to maturity is five months? （保存2位小数）Solution: In this case F=19,X=20, r=0.12, =0.20, T-t=0.42, The

22、 price of the European put is 4) 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 forward price and the initial value of the forward contract?(b

23、) 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?The forward price, ， The initial value of the forward contract is zero. (a) The delivery price K in the contract is $44.21. The value of th

24、e forward contract after six months is given: The forward price, 5) 计算基于无红利支付股票的欧式看跌期权价格，其中执行价格为50，现价为50，有效期3个月期，无风险年收益率为10%，波动率为每年30%。若在两个月后预期支付的红利为1.50，则计算会有何变化解 欧式看跌期权价格是 若在两个月后预期支付的红利为1.50，在本题中我们在使用BS公式前必须从股票价格中减去红利的贴现值，因此应当是 其她变量不变 在本题中 欧式看跌期权价格是 6) 求无红利支付股票的欧式看涨期权的价格。其中股票价格为52，执行价格为50，无风险年收益率为

25、12%，年波动率为30%，到期日为3个月。在本题中 欧式看涨期权的价格是 7) 求无红利支付股票的欧式看跌期权的价格。其中股票价格为69，执行价格为70，无风险年收益率为5%，年波动率为35%，到期日为6个月。在本题中 欧式看跌期权价格为 9. 试证明标的资产无红利发放时的欧式看涨期权和看跌期权满足的平价公式 10. 试证明标的资产有红利发放时的欧式看涨期权和看跌期权满足的平价公式 11. 1）证明在风险中性环境下，到期的欧式看涨期权被执行的概率为 ,2） 使用风险中性定价原理，假设股票1的价格和股票2的价格分别服从几何布朗运动，且独立，给到期损益为如下形式的欧式衍生品定价： Solution

26、: Since Since Where , 12. Use two-step tree to value an American 2-year put option on a non-dividend-paying stock, current stock price is 50, the strike price is $52, and the volatility of stock price is 30% per annum, the risk-free interest rate is 5% per annum. （保存2位小数）In this case, S=50, X = 52,

27、= 0.3, t =1, r=0.05 , the parameters necessary to construct the tree are , 91.11067.40.93502507.4337.0414.96 27.4424.56 13 If a stock price, S, follows geometric Brownian motion1) What is the process followed by the variable ? Show that also follows geometric Brownian motion.2)The expected value of

28、ST is . What is the expected value of ?3) The varaince of ST is .What is the variance of ?4) Using risk-neutral valuation to value the derivative, whose payoff at maturity is1)We now use Itos lemma to derive the process followed by ,Define , So that also follows geometric Brownian motion. 2) . , 3)

29、Since and varaince of ST is .Similarly, by We get the varaince of is 14 In a risk-neutral world, suppose stock prices follow geometric Brownian motion1) What is the process followed by the variable by Itos lemma? Show that also follows geometric Brownian motion.2) The expected value of is . What is

30、the expected value of ?4) Using risk-neutral valuation to value the derivative, whose payoff at maturity is 13. Show that the probability that a European call option will be exercised in a risk-neutral world is, . Using risk-neutral valuation to value the complicated digtial option whose payoff at m

31、aturity is Solution: Since 14. 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 EquationSolution: Since f is a function of S and t, let：f = f (S, t)，it follows from Ito lemma that The appropriate portfolio is Short: one derivative Long: shares Value of the portfolio at time : The change of in the value of the portflio in time dt is given: Choosing to make the portflio in time dt is iskless，thus