财务工程Ch2.docx

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1、编号：时间：2021年x月x日书山有路勤为径，学海无涯苦作舟页码：第13页 共13页財務工程第二講次 股票選擇權的特性 (Properties of Stock Options)1. 影響選擇權價格的因素(Factors Affecting Option Prices) 影響選擇權價格的因素有六個：1. Current stock price (S)2. Strike price (X)3. Time to expiration (T)4. Volatility of stock price ()5. Risk-free rate ()6. Dividends (d) 其影響方向如下表：Eur

2、opeanAmericancallputCallputS+X+T+？+d+1. T 產生兩效果：(1) Tvolatility, 任何股價情況均可能產生(2) T present value of striking price, C; 歐式P？但對美式put有提早履約的機會 P2. r產生兩效果：(1) r(2) rpresent value of future cash flowtwo effects make P；1st effect makes C2nd effect makes C two effects make CTime value decay curveEffect of vo

3、latility increase/decrease on long straddle05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5800585059005950600060506100615062006250 5400386429473519566613661710759808857 5500307346387430475520567614662710759 5600236271309348389432476521568615662 5700176206239274311350391433477522569 5800127151179

4、209241276313352393435479 590088107129154181211244278315354394 6000597390110132157184214246280317 61003848617693112135159187216248 620023314050637895115137162189 6300141925324152658098118140 64008111520263443546783100 65004691216212835455670 660023579131722293747 67001234571014182430 6800111234681115

5、19 6900001122356912 700000011123457 710000000111234 720000000001122 730000000000111 05月賣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5800585059005950600060506100615062006250 54003023181310754321 5500514032251914118643 560080655342332620151296 5700120100836855443527211612 58001711451231038570574636

6、2923 59002322011731481251058772594738 6000302267234204176150128107907461 610038134230426923620617815313011092 6200467424383344306271239208181155133 6300557512468425385346308274241211183 6400651604558513469427386347311276243 6500747699652605559514471428388349313 6600845796748700652606560515472430390

7、6700944895845797748700653606561516473 68001,043994944895846797749701654607562 69001,1431,0931,043994944895846797749701654 70001,2431,1931,1431,0931,043994944895846798749 71001,3421,2931,2431,1931,1431,0931,043994944895846 72001,4421,3921,3421,2921,2421,1931,1431,0931,043994944 73001,5421,4921,4421,3

8、921,3421,2921,2421,1931,1431,0931,043 資料來源：元富證券，以數學式表示這些效果： 2. 選擇權部位的利潤圖(Profit Diagrams)與向量表示(Vector Notation Prices) 1. Buy Call: synthetic long: 2. Sell Call: synthetic short: 3. Buy Put: long call: 4. Short Put: selling put: 四種基本選擇權部位的利潤圖00 利潤 利潤-ccXXSTSTA. long a callB. short a call 00 利潤 利潤-pp

9、XXSTSTC. long a putD. short a put3. 選擇權價格(權利金)的上下界(Upper and lower bounds for option Prices)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= 歐式買權價格 = 美式買權價格= 歐式賣權價格 = 美式賣權價格1. 上界(Upper Bounds) 2. 不支付股利股票買權之下界(Lower Bound for Calls on non-dividend-paying Stocks) and 證明 考慮下列兩投資組合：Portfolio A: One E

10、uropean Call Options + Cash (= $)Portfolio B: One ShareThe value of portfolio A at time T:1 Call0$XXTotalX at time T, portfolio A is worth Max at time T, portfolio B is worth he value of portfolio acesricesMax, i.e., Value of portfolio AValue of portfolio B Assume that all investment vehicles are de

11、fault-free. Thus, at time t, Present value of portfolio APresent value of portfolio B That is, CC= SC = Max(S-X, 0)XS3. 不支付股利股票歐式賣權之下界(Lower Bound for European puts on non- dividend-paying stocks) and 證明 考慮下列兩投資組合：Portfolio C: 1 European put option + 1 sharePortfolio D: Cash (=)The value of portfoli

12、o C at time T:1 put01 shareTotalXAt time T, portfolio C is worth MaxAt time T, portfolio D is worth X i.e., Value of portfolio CValue of portfolio DAssume that all investment vehicles are default-free. Thus, at time t,Present value of portfolio CPresent value of portfolio DThat is, PP X S SX 4. 提早履約

13、：不支付股利股票之買權(Early Exercise: calls on a non-dividend- paying stock)結論：It is never optimal to exercise an American call option on a non-dividend-paying stock early. (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(under price)，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

14、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American call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time value + intrinsic value)。而若行使call之權利，只有得到intrinsic value, $10。2. 正式

15、證明(Formal argument)：考慮下列兩投資組合：Portfolio E: 1 American call +Cash (=)Portfolio F: 1 share若call在時間時，被提早行使權利，則Portfolio E之價值為： Portfolio E之價值 Portfolio F之價值 (1)在時間T時，Value of portfolio E = MaxValue of portfolio F = Portfolio E is always worth as much as, and is sometimes worth more than, Portfolio F. (

16、2)由(1) (2)知，若option提早立即行使權利，value of portfolio E value of portfolio F . 不發放股利之American Call絕不會在到期日以前行使權利。 即：3. Mathematical proof： 若提早行使權利， It is never optimal to exercise an American call early.5. 提早履約：不支付股利股票之賣權(Early Exercise: puts on a non-dividend- paying stock)結論：It can be optimal to exercise

17、an American put option on a non-dividend-paying stock early. (對於美式的賣權(無支付股利之股票)，提早行使可能是最適化。)證明 考慮下列兩投資組合： Portfolio G: 1 American put + one share Portfolio H: Cash (=) 若put在 t value of H (3) 若put持有至到期日，則： Value of G = max Value of H = X Value of G Value of H (4)由(3) (4)知，提早行使可能使G更好，尤其在 deep-in-the m

18、oney時更適用。2. Mathematical Proof： 若early exercise is optimal value of put option = 美式put有時，其價值等於內在價值，而故歐式put必定有時其價值低於內在價值6. 買賣權等式關係(put-call parity)對於具有相同履約價、相同到期日的歐式買權與賣權，下列恆等式必成立；否則，會產生套利機會。此恆等式稱為put-call parity. 證明 考慮下列兩投資組合： Portfolio A: 1 European C + Cash Portfolio C: 1 European put + one share

19、at t = T (maturity) value of A = value of C = max 選擇權均為歐式，無法提早行使權利。兩Portfolios之現值相同。即： Q.E.D.*Relationship between American call and put prices 考慮下列兩投資組合： Portfolio I: European call + $ X Portfolio J: American Put + 1 share (有相同X , T)在t = T（即若American put不提早行使權利） value of J = max value of I = max若Am

20、erican Put在 t = 時提早行使權利： value of J = X value of I = 由(1) (2)得： 7. 股利的影響(Effect of Dividends)1. Lower Bound for call and puts 考慮下列兩投資組合： Portfolio A: 1 European call + cash ( =D + X) Portfolio B: 1 shareUsing similar argument, 考慮下列兩投資組合： Portfolio C: 1 European put +1 share Portfolio D: Cash (=D +)U

21、sing similar argument , 2. Early exerciseWhen dividends are expected, we no longer assert American calls will not be exercised early.3. put-call parity 考慮下列兩投資組合： Portfolio I: 1 European Call + Cash (= D + X ) Portfolio J: 1 American put + 1 shareRegardless of what happens, value of Portfolio I value of Portfolio J 但對於 non-dividend-paying stocks, 吾人已證明： 第 13 页 共 13 页