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Ch 3 Simple Arbitrage Relationships for Options

Ch 3 Simple Arbitrage Relationships for Options.

hayden-gordon
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Ch 3 Simple Arbitrage Relationships for Options

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  1. Ch 3 Simple Arbitrage Relationships for Options Learning objective : 1. What are option’s prices (premiums) bounds when market is N-A-O ? 2. Are there any price relationships between put and call ? 一、 price bounds for call 二、price bounds for put 三、 put – call parity 四、dividend’s effect

  2. Notations : S(t); 標的(以下均設股票) 在t 時點之股價 C(t); 美式買權(AC)在t時點之價格(權利金) c(t); 歐式買權(EC)在t時點之價格(權利金) P(t); 美式賣權(AP)在t時點之價格(權利金) p(t); 歐式賣權(EP)在t時點之價格(權利金) K ; 履約價格 B(t,T); T時點確定1元之t點現值,也是將T確定貨幣轉換 為t點貨幣之無限貼現因子。只要t →T間之無險 利率> 0，則0 < B(t,T) <1

  3. CF(t); t 時點之 cash flow（inflow） T ; 各種 options 契約之到期日 Assumptions 1. 除第四節外，假設現股無任何股利發放。 2. 全文之AC，EC，AP，EP 契約除特別提及，否則均 設標的股票、履約價、到期日等條件都相同，只區分 買權或賣權，可期前履約（美式）或否（歐式）。

  4. Notions : 履約價權利價主研究課題 1.F(t,T)如何參考S(t)決定？ 2. V(t)? ( less important ! why? ) F(t,T) Fd 1.F (t,T)的決定因素, 和S(t)的關係 2.F (t,T)和F（t,T)或options prices 的關係 在考慮保證金帳戶淨收入後始終為0 名目=S(T)=F (T,T) 實質=S(T)+進場後保 證金總收入 =進場期貨價格 = F (t,T) Ft 各種premium如何決定？和S(t)或其他影響因素關係？ K (在集中交易市場是契約制式化規定) opt. premiums

  5. 一、price bounds of calls 〈Result-C-1〉relationship of C(t) and c(t) C(t) ≧ c(t) proof : [t,T] 之間，任何 EC 持有人可行為者，AC 持有人皆可行為，且後者權力更廣。 〈Result-C-2〉C(t) , c(t) are nonnegative c(t) ≧ 0 proof : 1. opt.持有人是limited liability.其未來給付最 差是0，不可能還須支出，故任何opt.權力 價值≧0 。 另證→2. if C(t)<0 , long EC now , CF(t)= -c(t) > 0

  6. S(T)>K , exercise call , CF(T)= S(T) - K > 0 S(T)≦K , EC is worthless , CF(T) = 0 if N-A-O , then c(t) ≧ 0 At T , if Note : • 〈R-C-1〉＆〈R-C-2〉 C(t) ≧ 0 • 〈R-C-1〉＆〈R-C-2〉still hold even CF occurs during (t,T)

  7. 〈Result-C-3〉upper bound of C(t) & c(t) C(t) ≦ S(t) proof : if C(t) > S(t) 現在(t)，買股賣AC，CF(t)= C(t) – S(t) > 0 ， 在[t,T]間任一時點 ，若AC被執行，則交付 持股，取得K>0。若AC未被執行，則抱股 CF ＝0，此為A-0 if N-A-O , C(t) ≦ S(t)

  8. Note : 1. 由〈R-C-1〉和〈R-C-3〉 c(t) ≦ S(t) 2. 訂金(如房屋買賣時)制度，雷同買權的交 易。房價是？訂金是？T？EC or AC？和 call的交易差別在？係可以想像C(t) > S(t) 的房地產交易嗎？ 你可以作何study? 承包工程保證金

  9. 〈Result-C-4〉lower bound of C(t) & c(t) c(t) ≧ max { 0,s(t)-K‧B(t,T) } proof : < R-C- 2 > 證明於下

  10. date t strategy空現股 無險投資(T到期)KB(t,T) 買Call CF(t) S(t) -KB(t,T) -c(t) > 0 date T strategy 還卷 回收無險投資 CF(T) -S(T) K if S(T)>K,則執行EC CF(t)=S(T)-K S(T)≦K , EC is worthless CF(T)=0

  11. 另證 portfolio A :持有EC，無險投資KB(t,T) portfolio B : 持股 就 portfolio A 而言，在 T if S(t) > K , 則以無險投資之本息K交付執行EC，取得現貨 此時value of portfolio A at T =S(T)=max{ S(T),K } if S(T)≦K，則EC無價值 此時value of portfolio A at T = K =max{ S(T),K } 就portfolio B 而言，在 T value of portfolio B = S(T)≦ max{ S(T),K } = value of portfolio A at T → present value of portfolio A ≧ present value of portfolio B i.e. c(t) + KB(t,T) ≧S(t) or c(t) ≧S(t) – KB(t,T) but c(t) ≧ 0 →c(t) ≧max{0,S(t) - KB(t,T)}

  12. Note : 1. by 〈R-C-1〉&〈R-C-4〉 → C(t) ≧ max { 0,S(t) – KB(t,T)} 2. 此一下限規範在現股(t,T)兼有股利發放(即 CF )時，需加 以修訂。 例：元大35，標的股：微星科技 K＝111.75元 執行比例1：1 4/8(一)收盤價 C＝37.6 S＝134 T 今年11月4日半年 rf (＝央行發行之180元NCD)＝ 2.5 ％ B( 0,180天) = (1+0.025×1/2)-1 ≒ 0.9877 S – KB( 0,180天) ＝134元 – 111.75元 × 0.9877 ＝ 134 – 110.37 ＝ 23.63 例： 台指權 K＝6200 T＝4月18日 4/8 收盤價 c=60 S=6190.83

  13. 〈Result-C-5〉歐式、美式買權等價 if (1) underlying stock has no dividend during (t,T) (2) riskless interest rate > 0 （i.e B(t,T) < 1） AC will not be exercised befor maturity , which implies C(t) = c(t) proof : 代表AC之執行價值 持有AC，其價值始終優於執行價值 故AC始終不會被期前執行，契約等同EC

  14. Note : 1. 此一結果只適用現股不發放股利，若現股有股 利，則前述C(t) ≧S(t) – KB(t,T)未必成立，故最 後結論未必成立。 2. 縱然現股無股利，就P(t)和p(t)而言，沒有此相同 價值之結果 結論：1. 許多研究發現：台灣卷商發行的認購權證訂價偏 高。市場(主要是卷商)的一種看法是：台灣權證 多屬於美式，投資人可隨時要求履約，故價應訂 高。對於這種論調，你有何見解？

  15. 2. 如果現在call在價內（i.e. S(t) > K），且預期未來股價 很可能跌至K以下（i.e. call成價外），難道還不執行？ Ans：是的 若執行 當然執行AC後，可以選擇馬上出脫持股兌現執行 利得S(t) – K > 0，但不執行，將AC出售是更好策 略，後者得C(t)。依下限定理C(t) > S(t) – K，賣AC 優於執行AC。 3. 美式options訂價比歐式難，但就call而言，比一定理 implies某些狀況(現股無股利)AC和EC訂價完全相同。　　 抱股, 承擔股價可能降至K以下的風險，又無股利 付K, 損失資金的利息收入

  16. C(t),c(t) c(t)=S(t)-B(t,T) × K S(t)=C(t) OB 區 有A-O 〈R-C-3〉 PRICE BOUND of AC＆EC OB 區 有A-O 〈R-C-4〉 〈R-C-2〉 45° 45° S(t) 0 KB(t,T)

  17. 〈補充〉買權價格的其他性質 Ct(‧)，ct(‧)在t時，買權之價格 例 令 〈性質一〉 §.

  18. 〈性質二〉 intuition 1 : intuition 2 : 時點 間呢？ 就歐式買權而言，前述intuition 1解釋適用？intuition 2 ? 在S 極高( call is deep-in ) or S 極低( call is deep-out ) Ct(T2)可能等於Ct(T1)，EC亦同

  19. >< >< 說明：中（主隊）-日（客隊）棒賽 得分 S 得分 K option’s time value是投資人為期待好or更好 結果之付出 觀眾為期待獲勝所付出之關心或 時間 。 ct or Ct max{ 0,S(t) – KB(t,T) } time value S(t) 0 KB(t,T) 〈性質三〉 就AC言，K＝S 時，AC之 time value最大；就EC言，KB(t,T)=S時，EC之time value最大。

  20. 〈性質四〉 time value 愈小。（time value decay） §. Exercise Prices & Call prices 〈性質五〉 （proof）(一)之證明 portfolio A : long EC(K1)，short EC(K2) portfolio B : 面額X2 – X1之riskless asset， matures at T

  21. payoff at T

  22. A在到期日，無論S(T)為何，其payoff ≧0 price of A now = ct(K1) – ct(K2) ≧ 0 Q.E.D (一) 之另證 if ct(K2) > ct(K1)，則買入EC(K1)，賣出EC(K2) CF (t=now) = ct(K2) – ct(K1)> 0 at T. if S(T) < K1，EC(K1)不執行，EC(K2)不會被執行CF(T) = 0 K1≦S(T)<K2，EC(K1)執行EC(K1) , EC(K2) …CF(T) = S(T) - K1≧0 S(T)≦K2，執行EC(K1) , EC(K2)也會被執行CF(T) = S(T) - K1+K2 - S(T) i.e. A-O存在，故N-A-O，ct(K2)≦ct(K1)

  23. （二）之證明 前面之portfolio A，改成long AC(K1)，short AC(K2) 相同推論at T，payoff of portfolio A≧ 0 in any case . 此時 被執行 則執行

  24. 〈性質六〉 （proof） (一)之證明 由前頁portfolio A，B之未來(T)之payoff 比較知 payoff of portf. B ≧ payoff of portf. A S(T) …(1) price of B ≧price of A t≦T …(2) i.e. (X2-X1) B(t,T)≧ct(K1)-ct(K2) t≦T Q.E.D

  25. (二)之證明 就AC而言可隨時執行，若某一 AC(K2)被執行，同時執行AC(K1)，則在 得K2 - K1，再投資無險資產，期末(T)得(K2 – K1)B-1(t,T)≧K2 – K1，前面(一)之證明不適用。 將B改成面額是(K2 -K1)B-1(t,T)，則(一)之推論(1)(2)適用 N-A-O 證明

  26. 補充：Early exercising of American Options 一、call 執行買權，則 pay striking K : later exercising can save interest than earlier exercising . late early…(1) stock price rises after, early and late exercising. can earn the same capital gain . late early …(2) stock price goes down after, early exercising burden capital loss, but delay your option of exercising decision can choose not exercising and protect yourself from loss .late early. …(3) get stock :

  27. dividend : early exercising ( before holder-of-record date = ex-dividend date ) can get dividend but not for later exercising ( after-ex-divi. date ) date early….(4) exercising decision criterion at t . 〈性質一〉 (4) (1) (3) >< 早執行股利之現值 早執行所付K之利息損失現值 早執行損失之等待價值 exercising not exercising 則 at t 股利愈大或riskless interest rate愈小，愈可能提早執行

  28. 〈引理一〉if there is no dividend , never exercising at T time 0 ex-dividend maturity 〈引理二〉有股利之AC，唯一可能提早執行之時點是每一 ex-dividend date 之前一瞬間 〈引理三〉

  29. 二、put 執行put receive striking K : early exercising can earn interest than late . early late…(1) if stock price goes down, early and late exercising can escape form downside loss . early late …(2) if stock price goes up , early exercising looses capital gain, but delay your option of exercising decision can choose not exercising and selling at spot price late early. …(3) give up stock :

  30. dividend : early exercising (ex : before dividend date ) will lose dividend , but not for later exercising ( after- ex - dividend date ) late early….(4) exercising decision criterion at t . 〈性質二〉 >< 早執行股利損失之價值 早執行利息收入之現值 exercising not exercising at t the larger the interest rate or the smaller the dividend . the better for early exercising AP 〈引理一〉if there is no-dividend , it is still possible for exercising of AP . when K(1-B) > time value(t)

  31. 二、price bounds of puts 〈Result-P-1〉relationship of P(t) and p(t) P(t) ≧ p(t) 〈Result-P-2〉P(t) and p(t) are nonnegative proof : 同〈R-C-2〉proof 之 1 另證：參考〈R-C-2〉proof 之 2，證明 p(t)≧0 （考慮 if p(t) < 0 ，可如何進行套利？） Note：前述兩結果在 underlying 有 CF 仍成立 〈Result-P-3〉upper bound of P(t) P(t) ≦ K proof：賣權是現貨空頭的保險工具，股價愈低，此保險工 具愈有價值，最小股價是0，此時賣權可實現其最 大價值，即K.

  32. 另證：if P(t) > K，則賣AP，並無險投資 K CF(t) = P(t) – K > 0

  33. 〈Result-P-4〉upper bound of p(t) p(t) ≦ KB(t,T) proof：if p(t) > KB(t,T) 現在(t)，賣EP，無險投資KB(t,T) CF(t) = p(t) – KB(t,T) > 0 T日，回收無險投資得K EP被執行 CF(T) = S(T) – K + K ≧ 0 EP未被執行 CF(T) = K > 0 if N-A-O , p(t) ≦ KB(t,T) if

  34. 〈Result-P-5〉lower bound of P(t) P(t) ≧max { 0 , K – S(t) } proof : 只須考慮當 K – S(t) > 0，P(t) ≧K – S(t)即可 if P(t) < K – S(t) 買AP，馬上執行CF(t) = K – S(t) > 0 That all ! if N-A-O , P(t) ≧ K – S(t) , 而P(t) ≧ 0 P(t) ≧ max { 0 , KB(t,T) – S(T) }

  35. 〈Result-P-6〉lower bound of p(t) p(t) ≧max { 0 , KB(t,T) – S(t) } proof : 只須證，if KB(t,T) – S(t) > 0 , then p(t) ≧ KB(t,T) – S(t) if p(t) < KB(t,T) – S(t) then，現在買入EP，無險借入KB(t,T)， 同時買入股票。 CF(t) = - P(t) + KB(t,T) – S(t) > 0 T日，有K負債，手中持股且擁有EP S(T)≧K，不執行EP，賣股償債.CF(T) = S(T) - K ≧0 S(T)<K，執行EP，交股取履約價償債. CF(T) = 0 if N-A-O，p(t) ≧ KB(t,T) – S(t)，又 p(t)≧0 p(t) ≧ max { 0 , KB(t,T) – S(t) } if

  36. Note：1 .只要無險利率為E，則 p(t) 下限 ≦ P(t) 下限 2. 若標的期中有 CF，前兩下界結論須加以修訂 p(t) , P(t) bounds of AP K 〈R-P-3〉 〈R-P-4〉 KB(t,T) bounds of EP 〈R-P-5〉 〈R-P-6〉 〈R-P-2〉 S(t) 0 KB(t,T) K

  37. 三、put – call parity (一) price relationship among EP、EC，underlying and riskfree asset 〈Result 12〉put – call parity — European Options p(t) = c(t) – S(t) + KB(t,T) proof : 考慮以下portfolio，持有EC，空現股，無險 投資KB(t,T)，其現在價值（所須成本） c(t) – S(t) + KB(t,T)

  38. S(T)≧K S(T) – K – S(T) + K = 0 = CF(T) S(T)<K 0 – S(T) + K = K – S(T) = CF(T) 前述投資組合在(0,T)間無cash flow，而在 T 日之 payoff = max { 0 , K – S(T) }，此 portfolio 之 CF 完全和 EP相同 c(t) – S(t) + KB(t,T) = p(t) 執行EC 還卷 投資回收 T日，if EC無價值

  39. 另證：A portfolio — 買入EC，無險投資KB(t,T) payoff of A at T = max { 0 , S(T) – K } + K = max { S(T) , K } B portfolio — 買入EP，買入股票 payoff of B at T = max { 0 , K – S(T) } + S(T) = max { S(T) , K } 而A、B兩投資組合，期前無現金流出入 兩投資組合之現在價值須相等 i.e. c(t) + KB(t,T) = p(t) + S(t)

  40. S(T) p(T) + S(T) = c(T) + K ≧ K portfolio insurance（why ?） p(T) S(T) 0

  41. implications and discussions of European • put – call parity — c(t) – p(t) – S(t) + KB(t,T) = 0 • price discovery among EC、EP , underlying and risk -free asset . • 2. mutually duplicate among EC、EP , underlying and • risk - free asset .（by linerar pricing principle 2 →1 ） • 3. independent of σ

  42. EC EP E . put – call parity underlying risk-free asset 4. 價平時（S＝K），c(t) > p(t) ∵ c(t) – p(t) = S(t) – KB(t,T) = K – KB(t,T) > 0

  43. （二）price relationship among AP、AC，underlying and riskfree asset 〈Result 13〉put – call parity — American Options P(t) + S(t) – KB(t,T) ≧ C(t) ≧ P(t) + S(t) – K proof：1. first in inequality： P(t) + S(t) – KB(t,T)≧C(t) by〈R 12〉 p(t) = c(t) – S(t) + KB(t,T) but P(t)≧p(t)（〈R-P-1〉）and C(t)=c(t) （〈R-C-5〉） p(t) ≧C(t) – S(t) + KB(t,T)

  44. 2. 2nd inequality：C(t) ≧ P(t) + S(t) – K if p(t) + S(t) – K > C(t) , 則 賣AP，空現股，無險投資K金額，買入AC CF(t) = P(t) + S(t) – K – C(t) > 0 for any ，若AP被執行，則 AP被執行損失 還卷 賣出無險資產 執行AC 還卷 T日

  45. Summary of Price Bound and Put – Call parity when underlying has no dividends

  46. Options are limited liability premium ≧ 0 for all options More optional in American type than in European type C(t) ≧ c(t) , P(t) ≧ p(t) Intuitions of above results !

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