ch13期權(quán)期貨與衍生證券（第五版）

1、Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.1,Options onStock Indices, Currencies, and FuturesChapter 13,Options, Futures, and Other Derivatives, 5th edition © 2002 by Jo

2、hn C. Hull,13.2,European Options on StocksProviding a Dividend Yield,We 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

3、 a dividend yield = q2.The stock starts at price S0e–q T and provides no income,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.3,European Options on StocksProviding Dividend Yie

4、ldcontinued,We can value European options by reducing the stock price to S0e–q T and then behaving as though there is no dividend,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.4,E

5、xtension of Chapter 8 Results(Equations 13.1 to 13.3),Lower Bound for calls:,Lower Bound for puts,Put Call Parity,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.5,Extension of Chapt

6、er 12 Results (Equations 13.4 and 14.5),Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.6,The Binomial Model,,,S0u ?u,S0d ?d,S0 ?,p,(1 – p ),f=e-rT[pfu+(1-p)fd ],Options, Futures,

7、and Other Derivatives, 5th edition © 2002 by John C. Hull,13.7,The Binomial Modelcontinued,In a risk-neutral world the stock price grows at r-q rather than at r when there is a dividend yield at rate qThe probabi

8、lity, p, of an up movement must therefore satisfypS0u+(1-p)S0d=S0e (r-q)Tso that,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.8,Index Options,Option contracts are on 100 time

9、s the indexThe most popular underlying indices are the Dow Jones Industrial (European) DJXthe S&P 100 (American) OEXthe S&P 500 (European) SPXContracts are settled in cash,Options, Futures, and Other Deriva

10、tives, 5th edition © 2002 by John C. Hull,13.9,Index Option Example,Consider 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?,Opt

11、ions, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.10,Using Index Options for Portfolio Insurance,Suppose the value of the index is S0 and the strike price is KIf a portfolio has a b of 1.

12、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 options for each 100S0 dollars heldIn both cases, K

13、 is chosen to give the appropriate insurance level,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.11,Example 1,Portfolio has a beta of 1.0It is currently worth $500,000The index c

14、urrently stands at 1000What trade is necessary to provide insurance against the portfolio value falling below $450,000?,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.12,Example 2,P

15、ortfolio has a beta of 2.0It is currently worth $500,000 and index stands at 1000The risk-free rate is 12% per annumThe dividend yield on both the portfolio and the index is 4%How many put option contracts should be

16、purchased for portfolio insurance?,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.13,If index rises to 1040, it provides a 40/1000 or 4% return in 3 monthsTotal return (incl dividen

17、ds)=5%Excess return over risk-free rate=2%Excess return for portfolio=4%Increase in Portfolio Value=4+3-1=6%Portfolio value=$530,000,Calculating Relation Between Index Level and Portfolio Value in 3 months,Options, F

18、utures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.14,Determining the Strike Price (Table 13.2, page 274),An option with a strike price of 960 will provide protection against a 10% decline in the

19、portfolio value,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.15,Valuing European Index Options,We can use the formula for an option on a stock paying a dividend yieldSet S0 =

20、 current index levelSet q = average dividend yield expected during the life of the option,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.16,Currency Options,Currency options trad

21、e on the Philadelphia Exchange (PHLX)There also exists an active over-the-counter (OTC) marketCurrency options are used by corporations to buy insurance when they have an FX exposure,Options, Futures, and Other Derivat

22、ives, 5th edition © 2002 by John C. Hull,13.17,The Foreign Interest Rate,We denote the foreign interest rate by rfWhen a U.S. company buys one unit of the foreign currency it has an investment of S0 dollarsThe re

23、turn from investing at the foreign rate is rf S0 dollarsThis shows that the foreign currency provides a “dividend yield” at rate rf,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.18

24、,Valuing European Currency Options,A foreign currency is an asset that provides a “dividend yield” equal to rfWe can use the formula for an option on a stock paying a dividend yield : Set S0 = current exchange rate

25、 Set q = r?,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.19,Formulas for European Currency Options (Equations 13.9 and 13.10, page 277),,Options, Futures, and Other Derivatives,

26、5th edition © 2002 by John C. Hull,13.20,Alternative Formulas(Equations 13.11 and 13.12, page 278),Using,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.21,Mechanics of Call Fu

27、tures Options,When a call futures option is exercised the holder acquires 1. A long position in the futures 2. A cash amount equal to the excess of the futures price over the strike price,Options, Futures, and

28、 Other Derivatives, 5th edition © 2002 by John C. Hull,13.22,Mechanics of Put Futures Option,When a put futures option is exercised the holder acquires 1. A short position in the futures 2. A cash amount equa

29、l to the excess of the strike price over the futures price,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.23,The Payoffs,If the futures position is closed out immediately:Payof

30、f from call = F0 – KPayoff from put = K – F0where F0 is futures price at time of exercise,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.24,Put-Call Parity for Futures Options (E

31、quation 13.13, page 284),Consider the following two portfolios:1. European call plus Ke-rT of cash 2. European put plus long futures plus cash equal to F0e-rT They must be worth the same at time T so thatc+Ke-

32、rT=p+F0 e-rT,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.25,,,Futures Price = $33Option Price = $4,Futures Price = $28Option Price = $0,Futures price = $30Option Price=?,Binomi

33、al Tree Example,A 1-month call option on futures has a strike price of 29.,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.26,Consider the Portfolio:long D futuresshort 1 ca

34、ll optionPortfolio is riskless when 3D – 4 = -2D or D = 0.8,Setting Up a Riskless Portfolio,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. H

35、ull,13.27,Valuing the Portfolio( Risk-Free Rate is 6% ),The riskless portfolio is: long 0.8 futuresshort 1 call optionThe value of the portfolio in 1 month is -1.6The value of the portfolio today i

36、s -1.6e – 0.06/12 = -1.592,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.28,Valuing the Option,The portfolio that is long 0.8 futuresshort 1 option is worth -1.5

37、92The value of the futures is zeroThe value of the option must therefore be 1.592,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.29,Generalization of Binomial Tree Example (Figur

38、e 13.3, page 285),A derivative lasts for time T and is dependent on a futures price,,,F0u ?u,F0d ?d,F0 ?,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.30,Generalization(contin

39、ued),Consider the portfolio that is long D futures and short 1 derivativeThe portfolio is riskless when,,F0u D - F0 D – ?u,F0d D- F0D – ?d,,,Options, Futures, and Other Derivatives, 5th edition

40、 © 2002 by John C. Hull,13.31,Generalization(continued),Value of the portfolio at time T is F0u D –F0D – ?uValue of portfolio today is – ?Hence ? = – [F0u D –F0D – ?u]e-rT,Options, Futures, and Other D

41、erivatives, 5th edition © 2002 by John C. Hull,13.32,Generalization(continued),Substituting for D we obtain? = [ p ?u + (1 – p )?d ]e–rT where,Options, Futures, and Other Derivatives, 5th edition © 2002

42、 by John C. Hull,13.33,Valuing European Futures Options,We can use the formula for an option on a stock paying a dividend yieldSet S0 = current futures price (F0)Set q = domestic risk-free rate (r )Setting q

43、= r ensures that the expected growth of F in a risk-neutral world is zero,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.34,Growth Rates For Futures Prices,A futures contract requ

44、ires no initial investmentIn a risk-neutral world the expected return should be zeroThe expected growth rate of the futures price is therefore zeroThe futures price can therefore be treated like a stock paying a div

45、idend yield of r,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.35,Black’s Formula (Equations 13.17 and 13.18, page 287),The formulas for European options on futures are known as

46、Black’s formulas,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.36,Futures Option Prices vs Spot Option Prices,If futures prices are higher than spot prices (normal market), an Ameri

47、can call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spotWhen futures prices are lower than spot prices (inverted market) the re

48、verse is true,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,13.37,Summary of Key Results,We can treat stock indices, currencies, and futures like a stock paying a dividend yield of q