1008.-期權及其衍生市場chapter

1、2022/9/111Options onStock Indices, Currencies, and FuturesChapter 122022/9/112European Options on StocksPaying Continuous DividendsWe 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 continuous dividend y

2、ield = q2.The stock starts at price S0eq T and provides no e2022/9/113European Options on StocksPaying Continuous DividendscontinuedWe can value European options by reducing the stock price to S0eq T and then behaving as though there is no dividend2022/9/114Extension of Chapter 7 Results(Equations 1

3、2.1 to 12.3)Lower Bound for calls:Lower Bound for putsPut Call Parity2022/9/115Extension of Chapter 11 Results (Equations 12.4 and 12.5)2022/9/116The Binomial Model(Risk-neutral world)S0u uS0d dS0 p(1 p )f=e-rTpfu+(1-p)fd 2022/9/117The Binomial ModelcontinuedIn a risk-neutral world the stock price g

4、rows at r-q rather than at r when there is a dividend yield at rate qThe probability, p, of an up movement must therefore satisfypS0u+(1-p)S0d=S0e (r-q)Tso that2022/9/118Index OptionsOption contracts are on 100 the indexThe most popular underlying indices are the S&P 100 (American) OEXthe S&P 500 (E

5、uropean) SPXContracts are settled in cash2022/9/119Index 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?2022/9/1110Using Index Options for Portfolio InsuranceSuppose the value of the index is S0

6、 and the strike price is XIf 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 options for each 100S0 dollars heldIn both cases, X is chosen to give the appropr

7、iate insurance level2022/9/1111Example 1Portfolio has a beta of 1.0It is currently worth $5 millionThe index currently stands at 1000What trade is necessary to provide insurance against the portfolio value falling below $4.8 million?2022/9/1112Example 2Portfolio has a beta of 2.0It is currently wort

8、h $1 million and index stands at 1000The risk-free rate is 12% per annum(每3個月計一次復利）The dividend yield on both the portfolio and the index is 4%(每3個月計一次復利）How many put option contracts should be purchased for portfolio insurance? 2022/9/1113If index rises to 1040, it provides a 40/1000 or 4% return i

9、n 3 monthsTotal return (incl. dividends)=5%Excess return over risk-free rate=2%Excess return for portfolio=4%Increase in Portfolio Value=4+3-1=6%Portfolio value=$1.06 millionCalculating Relation Between Index Level and Portfolio Value in 3 months 2022/9/1114Determining the Strike Price An option wit

10、h a strike price of 960 will provide protection against a 10% decline in the portfolio value2022/9/1115Valuing 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

11、f the option2022/9/1116Currency OptionsCurrency options trade 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 exposure2022/9/1117The Foreign Interest RateWe denote the foreign

12、interest rate by rfWhen a U.S. company buys one unit of the foreign currency it has an investment of S0 dollarsThe return from investing at the foreign rate is dollarsThis shows that the foreign currency provides a “dividend yield” at rate rf2022/9/1118Valuing European Currency OptionsA foreign curr

13、ency is an asset that provides a continuous “dividend yield” equal to rfWe can use the formula for an option on a stock paying a continuous dividend yield : Set S0 = current exchange rate Set q = r2022/9/1119Formulas for European Currency Options 2022/9/1120Alternative Formulas Using2022/9/1121Mecha

14、nics of Call Futures OptionsMost of Futures options are American.The maturity date is usually on, or a few days before, the earliest delivery date of the underlying futures contract.When a call futures option is exercised the holder acquires 1. A long position in the futures 2. A cash amount equal t

15、o the excess of the futures price over the strike price 2022/9/1122Mechanics of Put Futures OptionWhen a put futures option is exercised the holder acquires 1. A short position in the futures 2. A cash amount equal to the excess of the strike price over the futures price2022/9/1123The PayoffsIf the

16、futures position is closed out immediately:Payoff from call = F0-XPayoff from put = X-F0where F0 is futures price at time of exercise2022/9/1124Why Futures Option instead of Spot Option?Futures is more liquid and easier to get the price informationCan be settled in cashLower transaction cost 2022/9/

17、1125Put-Call Parity for Futures OptionConsider the following two portfolios:1. European call plus Xe-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+Xe-rT=p+F0 e-rT2022/9/1126Futures Price = $33Option Price = $4Futures Price = $28

18、Option Price = $0Futures price = $30Option Price=?Binomial Tree ExampleA 1-month call option on futures has a strike price of 29. 2022/9/1127Consider the Portfolio:long D futuresshort 1 call optionPortfolio is riskless when 3D 4 = -2D or D = 0.83D 4-2DSetting Up a Riskless Portfolio2022/9/1128Valuin

19、g 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 is -1.6e 0.06/12 = -1.5922022/9/1129Valuing the OptionThe portfolio that is long 0.8 futuresshort 1 option is worth -1.59

20、2The value of the futures is zeroThe value of the option must therefore be 1.5922022/9/1130Generalization of Binomial Tree Example A derivative lasts for time T & is dependent on a futuresF0u uF0d dF0 2022/9/1131Generalization(continued)Consider the portfolio that is long D futures and short 1 deriv

21、ativeThe portfolio is riskless when F0u D - F0 D uF0d D- F0D d2022/9/1132Generalization(continued)Value of the portfolio at time T is F0u D F0D uValue of portfolio today is Hence = F0u D F0D ue-rT2022/9/1133Generalization(continued)Substituting for D we obtain = p u + (1 p )d erT where 2022/9/1134Gr

22、owth 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 futures price is therefore zeroThe futures price can therefore be treated like a stock paying a dividend yield of r2022/9/1135Future

23、s price vs. expected future spot pticeIn a risk-neutral world, the expected growth rate of the futures price is zero,so Because FT=ST, so2022/9/1136Valuing European Futures OptionsWe can use the formula for an option on a stock paying a continuous dividend yieldSet S0 = current futures price (F0)Set

24、 q = domestic risk-free rate (r )Setting q = r ensures that the expected growth of F in a risk-neutral world is zero2022/9/1137Blacks Formula The formulas for European options on futures are known as Blacks formulas2022/9/1138European Futures Option Prices vs Spot Option PricesIf the European futures option matures at the same time as the futures contract, then the two options are in theory equivalent.If the European call future option matures before the futures contract, it is