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 pri ce relationships between put and call ? 一、 price bounds for call 二、 price bounds for put 二、 put – call parity 二、 dividend’s effect
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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
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
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)。
Notions :
履約價 權利價 主研究課題 )T,t(B )T,0(F)T,t(F)t(V Fd F(t,T)
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
在考慮保證金帳戶淨收入後始終為 0
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
1.F(t,T) 如何參考 S(t) 決定?
2. V(t)? ( less important ! why? )
opt.
K ( 在集中交易市場是契約制式化規定 )
premiums各種 premium 如何決定?和 S(t) 或其他影響因素關係?
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。
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 :
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〉
value(t) timeT)B(t,-1K )t~
,t(B)t~
(d (1)(4) (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
則 at t
exercising not exercising股利愈大或 riskless interest rate愈小,愈可能提早執
行
〈引理一〉 if there is no dividend , never exercising at T,tt
0 ex-dividend maturity
timeTt
~
〈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〉 for but , dividend-nonfor TC Tc 2t1t
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 :
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〉
value(t) time )t~
,t(B)t~
(d T)B(t,-1K ><
早執行利息收入之現值
早執行股利損失之價值
at texercising not exercisingthe 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)
二、 price bounds of puts
〈 Result-P-1 〉 relationship of P(t) and p(t) P(t) p(t)≧