财务工程Ch2

1財務工程第二講次股票選擇權的特性(PropertiesofStockOptions)§1.影響選擇權價格的因素(FactorsAffectingOptionPrices)‧影響選擇權價格的因素有六個：1.Currentstockprice(S)2.Strikeprice(X)3.Timetoexpiration(T)4.Volatilityofstockprice()5.Risk-freerate(Fr)6.Dividends(d)‧其影響方向如下表：EuropeanAmericancallputCallputS+－+－X－+－+T+？++++++Fr+－+－d－+－+1.T↑產生兩效果：(1)T↑volatility↑,任何股價情況均可能產生(2)T↑presentvalueofstrikingprice↓,C↑;歐式P？但對美式put有提早履約的機會∴P↑2.r↑產生兩效果：(1)r↑rTTSeSE)(↑(2)r↑presentvalueoffuturecashflow↓twoeffectsmakeP↓；1steffectmakesC↑2ndeffectmakesC↑twoeffectsmakeC↑2TimevaluedecaycurveEffectofvolatilityincrease/decreaseonlongstraddle305月買權權利金交叉比對表(台股選擇權報價)資料日期：2004/05/07履約價/指數5750以下5800585059005950600060506100615062006250540038642947351956661366171075980885755003073463874304755205676146627107595600236271309348389432476521568615662570017620623927431135039143347752256958001271511792092412763133523934354795900881071291541812112442783153543946000597390110132157184214246280317610038486176931121351591872162486200233140506378951151371621896300141925324152658098118140640081115202634435467831006500469121621283545567066002357913172229374767001234571014182430680011123468111519690000112235691270000001112345771000000011123472000000000112273000000000011105月賣權權利金交叉比對表(台股選擇權報價)資料日期：2004/05/074履約價/指數5750以下5800585059005950600060506100615062006250540030231813107543215500514032251914118643560080655342332620151296570012010083685544352721161258001711451231038570574636292359002322011731481251058772594738600030226723420417615012810790746161003813423042692362061781531301109262004674243833443062712392081811551336300557512468425385346308274241211183640065160455851346942738634731127624365007476996526055595144714283883493136600845796748700652606560515472430390670094489584579774870065360656151647368001,04399494489584679774970165460756269001,1431,0931,04399494489584679774970165470001,2431,1931,1431,0931,04399494489584679874971001,3421,2931,2431,1931,1431,0931,04399494489584672001,4421,3921,3421,2921,2421,1931,1431,0931,04399494473001,5421,4921,4421,3921,3421,2921,2421,1931,1431,0931,043資料來源：元富證券，‧以數學式表示這些效果：0)(0)()()()2(0)(0)(221198502)(0)(0)()(0)(0)(2222221TXNXeTrCXNTSCtXNXeXNTStCtXNeXCXNSC),p.in,(RubinsteTdNXeCdNXeTrCdrNdNTXeTCdNeXCdNSCrTrTrTrTErTErTErTEE§2.選擇權部位的利潤圖(ProfitDiagrams)與向量表示(VectorNotationPrices)1.BuyCall:01syntheticlong:0110112.SellCall:01syntheticshort:0110113.BuyPut:10longcall:1110014.ShortPut:10sellingput:0111106★四種基本選擇權部位的利潤圖§3.選擇權價格(權利金)的上下界(UpperandlowerboundsforoptionPrices)選擇權之價格有其界限範圍，若選擇權之價格高於上界或低於下界，則會有獲利性的套利機會。茲分述如下：‧符號EC=歐式買權價格AC=美式買權價格EP=歐式賣權價格AP=美式賣權價格00利潤利潤-c＋cXXSTSTA.longacallB.shortacall00利潤利潤-p＋pXXSTSTC.longaputD.shortaput71.上界(UpperBounds))(tTrEAEAEXePXPandXPSCandSC2.不支付股利股票買權之下界(LowerBoundforCallsonnon-dividend-payingStocks))(tTrEXeSCand0EC[證明]考慮下列兩投資組合：PortfolioA:OneEuropeanCallOptions+Cash(=$)(tTrXe)PortfolioB:OneShareThevalueofportfolioAattimeT:XSTXST1Call0XST$)(tTrXeXXTotalXTS∵attimeT,portfolioAisworthMax),(XST∵attimeT,portfolioBisworthTSMaxTTSXS),(,i.e.,ValueofportfolioAValueofportfolioBAssumethatallinvestmentvehiclesaredefault-free.Thus,attimet,PresentvalueofportfolioAPresentvalueofportfolioBThatis,)0,max(0,,)()()(tTrEEtTrEtTrEXeSCCandXeSCorSXeC)0,(rTXeSMaxCS)(tTrXe)(tTrXeCC=SC=Max(S-X,0)X83.不支付股利股票歐式賣權之下界(LowerBoundforEuropeanputsonnon-dividend-payingstocks)SXePtTrE)(and0EP[證明]考慮下列兩投資組合：PortfolioC:1Europeanputoption+1sharePortfolioD:Cash(=)(tTrXe)ThevalueofportfolioCattimeT:XSTXST1putTSX01shareTotalTSTSXTSAttimeT,portfolioCisworthMax),(TSXAttimeT,portfolioDisworthXXXSMaxT),(i.e.,ValueofportfolioCValueofportfolioDAssumethatallinvestmentvehiclesaredefault-free.Thus,attimet,PresentvalueofportfolioCPresentvalueofportfolioDThatis,)0,(0,)()()(SXeMaxPPandSXePXeSPtTrEEttrEtTrESS)(tTrXe)(tTrXePPXX9§4.提早履約：不支付股利股票之買權(EarlyExercise:callsonanon-dividend-payingstock)[結論]：ItisneveroptimaltoexerciseanAmericancalloptiononanon-dividend-payingstockearly.(對於美式的買權(無支付股利之股票），提早行使決不是最適化。)[證明]1.以較直覺的方法說明如下：設現在S=50,X=40，投資者擁有此種美式買權。他也許想目前的股價被低估(underprice)，機會難得，立即行使call權利，以持有股票，並準備投資一個月後才出售股票，則這不是最佳策略。因為：（一）若在選擇權一個月後到期時，才行使權利，則可節省在這一個月的融資成本。（二）說不定在一個月後，股價TS可能低於$40，則此時，根本不必行使而能以更低的價格買到股票。準此，投資人在Americancall到期以前行使權利，得不到好處。又若投資者認為目前的股價過高(overprices)，那他可能懷疑是否提早行使call的權利？在此情況下，投資人最好以賣出call而非行使call。因賣出call之利益大於$10(因有timevalue+intrinsicvalue)。而若行使call之權利，只有得到intrinsicvalue,$10。2.正式證明(Formalargument)：考慮下列兩投資組合：PortfolioE:1Americancall+Cash(=)(tTrXe)PortfolioF:1share若call在時間τ)0(T時，被提早行使權利，則PortfolioE之價值為：SXeXSTr)(∴PortfolioE之價值PortfolioF之價值(1)在時間T時，ValueofportfolioE=Max),(XSTValueofportfolioF=TS),(XSMaxSTT∴PortfolioEisalwaysworthasmuchas,andissometimesworthmorethan,PortfolioF.(2)由(1)(2)知，若option提早立即行使權利，valueofportfolioEvalueofportfolioF，但若直到到期日才行使AmericanCall,則valueofportfolioEvalueofportfolioF.∴不發放股利之AmericanCall絕不會在到期