3以股指外汇利率为标的的期权

1、第十五章 股指、外汇、期货与利率为标的的期权郑振龙 陈蓉厦门大学金融系课程网站： Email: 引言股价指数期权、外汇期权和期货期权定价原理相同，都可以看成是支付连续红利资产的期权。利率期权则具有高度的复杂性。本章将分析股价指数期权、外汇期权和期货期权的定价原理，并对利率期权进行初步的介绍。 14:02Copyright 2012 Zheng, Zhenlong & Chen, Rong, XMU2欧式股价指数期权、外汇期权和期货期权的定价股价指数期权、外汇期权和期货期权都可以被视为支付连续红利的资产，因而欧式的股价指数期权、外汇期权和期货期权都可以在支付连续收益的欧式期权定价模型中得到应用。

2、14:02Copyright 2012 Zheng, Zhenlong & Chen, Rong, XMU3默顿模型根据默顿模型，标的股票支付连续红利的欧式看涨期权和看跌期权的价值分别为当q0时，默顿模型就转化为基本的B-S-M模型。 14:02Copyright 2012 Zheng, Zhenlong & Chen, Rong, XMU4外汇期权默顿模型中的S是外汇汇率，q是外汇的连续复利，则是外汇汇率的波动率。因此外汇的欧式看涨期权的价值为 外汇的欧式看跌期权的价值为14:02Copyright 2012 Zheng, Zhenlong & Chen, Rong, XMU5期货期权当无收

3、益标的资产服从几何布朗运动时，其期货价格F同样服从几何布朗运动欧式期货看涨期权和欧式期货看跌期权的价值分别为 14:02Copyright 2012 Zheng, Zhenlong & Chen, Rong, XMU6标的资产支付连续红利的期权价格的敏感性14:02Copyright 2012 Zheng, Zhenlong & Chen, Rong, XMU7标的资产支付连续红利的期权价格的敏感性14:02Copyright 2012 Zheng, Zhenlong & Chen, Rong, XMU8利率期权利率期权的分析和定价要困难得多，这是因为：利率期权的标的资产利率的随机过程比股票价

4、格或是汇率的变化要复杂得多，几何布朗运动难以较好地捕捉利率的随机运动规律。特定时刻的利率不是一个数值，而是整条利率期限结构，所以我们用以描述利率随机运动规律的模型必须能捕捉整条利率曲线的特征。整条利率期限结构上不同到期时刻的利率的波动率都是互不相同的；最后，在利率期权中，利率本身影响期权的到期回报，同时又要充当回报的贴现率，这进一步加大了利率期权的复杂性。 14:02Copyright 2012 Zheng, Zhenlong & Chen, Rong, XMU9利率期权的种类利率期权的种类交易所交易的利率期权场外交易的利率期权（案例15.4）内嵌的利率期权14:02Copyright 201

5、2 Zheng, Zhenlong & Chen, Rong, XMU10Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Options on Stock Indices and CurrenciesChapter 1511Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Index OptionsThe most popular underlyi

6、ng indices in the U.S. are The S&P 100 Index (OEX and XEO)The S&P 500 Index (SPX)The Dow Jones Index times 0.01 (DJX)The Nasdaq 100 Index (NDX)Contracts are on 100 times index; they are settled in cash; OEX is American; the XEO and all other options are European.12Fundamentals of Futures and Options

7、 Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Index Option ExampleConsider a call option on an index with a strike price of 560Suppose 1 contract is exercised when the index level is 580What is the payoff?13Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010

8、Using Index Options for Portfolio InsuranceSuppose the value of the index is S0 and the strike price is KIf a portfolio has a b of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars heldIf the b is not 1.0, the portfolio manager buys b put op

9、tions for each 100S0 dollars heldIn both cases, K is chosen to give the appropriate insurance level14Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Example 1 Portfolio has a beta of 1.0It is currently worth $500,000The index currently stands at 1000What trade

10、is necessary to provide insurance against the portfolio value falling below $450,000?15Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Example 2 Portfolio has a beta of 2.0It is currently worth $500,000 and index stands at 1000The risk-free rate is 12% per annu

11、mThe dividend yield on both the portfolio and the index is 4%How many put option contracts should be purchased for portfolio insurance? 16Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010If index rises to 1040, it provides a 40/1000 or 4% return in 3 monthsTotal

12、 return (incl. dividends)=5%Excess return over risk-free rate=2%Excess return for portfolio=4%Increase in Portfolio Value=4+31=6%Portfolio value=$530,000Calculating Relation Between Index Level and Portfolio Value in 3 months 17Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright Jo

13、hn C. Hull 2010Determining the Strike Price (Table 15.2, page 329)An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value18Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Currency OptionsCurrency options trade o

14、n the NASDAQ OMXThere also exists an active over-the-counter (OTC) marketCurrency options are used by corporations to buy insurance when they have an FX exposure19Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Range Forward ContractsHave the effect of ensuring

15、 that the exchange rate paid or received will lie within a certain rangeWhen currency is to be paid it involves selling a put with strike K1 and buying a call with strike K2When currency is to be received it involves buying a put with strike K1 and selling a call with strike K2Normally the price of

16、the put equals the price of the call 20Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Range Forward Contract continuedFigure 15.1, page 331 PayoffAsset PriceK1K2PayoffAsset PriceK1K2Short PositionLong Position21Fundamentals of Futures and Options Markets, 7th

17、Ed, Ch 15, Copyright John C. Hull 2010European Options on Stockswith Known Dividend YieldsWe get the same probability distribution for the stock price at time T in each of the following cases:1.The stock starts at price S0 and provides a dividend yield = q2.The stock starts at price S0eqT and provid

18、es no e22Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010European Options on StocksPaying Dividend YieldcontinuedWe can value European options by reducing the stock price to S0eqT and then behaving as though there is no dividend23Fundamentals of Futures and Opt

19、ions Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Extension of Chapter 10 Results(Equations 15.1 to 15.3)Lower Bound for calls:Lower Bound for putsPut Call Parity24Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Extension of Chapter 13 Results (Equations

20、15.4 and 15.5)25Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Valuing European Index OptionsWe can use the formula for an option on a stock paying a continuous dividend yieldSet S0 = current index levelSet q = average dividend yield expected during the life o

21、f the option26Using Forward/Futures Index Prices (equations 15.6 and 15.7, page 336)Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 201027Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Currency Options: The Foreign Interest R

22、ateWe denote the foreign interest rate by rfThe return measured in the domestic currency from investing in the foreign currency is rf times the value of the investmentThis shows that the foreign currency provides a yield at rate rf28Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyrig

23、ht John C. Hull 2010Valuing European Currency OptionsWe can use the formula for an option on a stock paying a continuous dividend yield : Set S0 = current exchange rate Set q = r29Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Formulas for European Currency Op

24、tions (Equations 15.8 and 15.9 page 337) 30Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Using Forward/Futures Exchange Rates(Equations 15.10 and 15.11, page 338) Using31Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010Th

25、e Binomial ModelS0u uS0d dS0 p(1 p )f = e-rDtpfu+(1 p)fd 32Fundamentals of Futures and Options Markets, 7th Ed, Ch 15, Copyright John C. Hull 2010The Binomial Modelcontinued33Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Futures OptionsChapter 1634Fundamental

26、s of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Mechanics of Call Futures OptionsWhen a call futures option is exercised the holder acquires A long position in the futuresA cash amount equal to the excess of the futures price at previous settlement over the strike price35

27、Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Mechanics of Put Futures OptionWhen a put futures option is exercised the holder acquiresA short position in the futuresA cash amount equal to the excess of the strike price over the futures price at previous sett

28、lement36Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010The PayoffsIf the futures position is closed out immediately:Payoff from call = F KPayoff from put = K Fwhere F is futures price at time of exercise37Fundamentals of Futures and Options Markets, 7th Ed, Ch

29、 16, Copyright John C. Hull 2010Potential Advantages of FuturesOptions over Spot OptionsFutures contract may be easier to trade than underlying assetExercise of the option does not lead to delivery of the underlying asset Futures options and futures usually trade in adjacent pits at exchangeFutures

30、options may entail lower transactions costs 38Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Put-Call Parity for European Futures Options (Equation 16.1, page 347)Consider the following two portfolios:1. European call plus Ke-rT of cash 2. European put plus lo

31、ng futures plus cash equal to F0e-rT They must be worth the same at time T so thatc+Ke-rT=p+F0 e-rT39Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Other RelationsF0 e-rT K C P (F0 K)e-rTp (F0 K)e-rT40Fundamentals of Futures and Options Markets, 7th Ed, Ch 16,

32、 Copyright John C. Hull 2010Futures Price = $33Option Price = $4Futures Price = $28Option Price = $0Futures price = $30Option Price=?Binomial Tree ExampleA 1-month call option on futures has a strike price of 29. 41Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 20

33、10Consider the Portfolio:long D futuresshort 1 call optionPortfolio is riskless when 3D 4 = 2D or D = 0.83D 4-2DSetting Up a Riskless Portfolio42Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Valuing the Portfolio( Risk-Free Rate is 6% )The riskless portfolio

34、is: long 0.8 futuresshort 1 call optionThe value of the portfolio in 1 month is 1.6The value of the portfolio today is 1.6e 0.06/12 = 1.59243Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Valuing the OptionThe portfolio that is long 0.8 futuresshort 1 option i

35、s worth 1.592The value of the futures is zeroThe value of the option must therefore be 1.59244Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Generalization of Binomial Tree Example (Figure 16.2, page 349)A derivative lasts for time T and is dependent on a futu

36、resF0d dF0u uF0 45Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Generalization(continued)Consider the portfolio that is long D futures and short 1 derivativeThe portfolio is riskless when F0u D - F0 D uF0d D- F0D d46Fundamentals of Futures and Options Markets

37、, 7th Ed, Ch 16, Copyright John C. Hull 2010Generalization(continued)Value of the portfolio at time T is F0u D F0D uValue of portfolio today is Hence = F0u D F0D ue-rT47Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Generalization(continued)Substituting for D

38、we obtain = p u + (1 p )d erT where 48Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010Growth Rates For Futures PricesA futures contract requires no initial investmentIn a risk-neutral world the expected return should be zeroThe expected growth rate of the futur

39、es price is therefore zeroThe futures price can therefore be treated like a stock paying a dividend yield of rThis is consistent with the results we have presented so far (put-call parity, bounds, binomial trees) 49Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 20

40、10Valuing European Futures OptionsWe can use the formula for an option on a stock paying a continuous yieldSet S0 = current futures price (F0)Set q = domestic risk-free rate (r )Setting q = r ensures that the expected growth of F in a risk-neutral world is zero50Fundamentals of Futures and Options M

41、arkets, 7th Ed, Ch 16, Copyright John C. Hull 2010Blacks Model (Equations 16.7 and 16.8, page 351)The formulas for European options on futures are known as Blacks model51Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright John C. Hull 2010How Blacks Model is Used in PracticeEuropea

42、n futures options and spot options are equivalent when future contract matures at the same time as the otion.This enables Blacks model to be used to value a European option on the spot price of an asset52Using Blacks Model Instead of Black-Scholes (Example 16.5, page 352)Consider a 6-month European call option on spot gold6-month futures price is 620, 6-month risk-free rate is 5%, strike price is 600, and volatility of futures price is 20%Value of option is given by Blacks model with