看漲看跌平價定理

1、看漲看跌平價定理歐式看漲期權(quán)與看跌期權(quán)之間的平價關(guān)系無收益資產(chǎn)的歐式期權(quán)在標(biāo)的資產(chǎn)沒有收益的情況下，為了推導(dǎo)c和p之間的關(guān)系，我們考慮如下兩個組合：組合A： 一份歐式看漲期權(quán)加上金額為Xe-r(T-t)的現(xiàn)金組合B： 一份有效期和協(xié)議價格與看漲期權(quán)相同的歐式看跌期權(quán)加上一單位標(biāo)的資產(chǎn)在期權(quán)到期時，兩個組合的價值均為max(ST,X)。由于歐式期權(quán)不能提前執(zhí)行，因此兩 組合在時刻t必須具有相等的價值，即：c+Xe-r(T-t)=p+S (1.1)這就是無收益資產(chǎn)歐式看漲期權(quán)與看跌期權(quán)之間的平價關(guān)系(Parity)。它表明歐式看 漲期權(quán)的價值可根據(jù)相同協(xié)議價格和到期日的歐式看跌期權(quán)的價值推導(dǎo)出來，

2、反之亦然。如果式(1.1)不成立，則存在無風(fēng)險套利機會。套利活動將最終促使式(1.1)成立。2 .有收益資產(chǎn)歐式期權(quán)在標(biāo)的資產(chǎn)有收益的情況下，我們只要把前面的組合A中的現(xiàn)金改為D+Xe-r(T-t)， 我們就可推導(dǎo)有收益資產(chǎn)歐式看漲期權(quán)和看跌期權(quán)的平價關(guān)系：c+D+Xe-r(T-t)=p+S (1.2)美式看漲期權(quán)和看跌期權(quán)之間的關(guān)系1 .無收益資產(chǎn)美式期權(quán)由于Pp,從式(1.1)中我們可得：Pc+Xe-r(T-t)-S對于無收益資產(chǎn)看漲期權(quán)來說，由于c=C，因此：PC+Xe-r(T-t)-SC-PP+S由于c=C,因此，C+XP+S結(jié)合式(1.3)，我們可得：S-XC-PS-Xe-r(T-t

3、)(1.4)由于美式期權(quán)可能提前執(zhí)行，因此我們得不到美式看漲期權(quán)和看跌期權(quán)的精確平價關(guān)系， 但我們可以得出結(jié)論：無收益美式期權(quán)必須符合式(1.4)的不等式。有收益資產(chǎn)美式期權(quán)同樣，我們只要把組合A的現(xiàn)金改為D+X，就可得到有收益資產(chǎn)美式期權(quán)必須遵守的不等式：S-D-XC-PS-D-Xe-r (T-t)(1.5)1、定理Theorem 1(Put - call parity formula)(Call(K,T) - Put(K,T)erT + K = F0,T .If we use effective interest, the put - call parity formula becomes

4、:(Call(K,T) - Put(K,T)(1 + i)T + K = F0,TOften, F0,T = S0(1 + i)T . This forward price applies to assets which have neither cost nor benefit associated with owning them.In the absence of arbitrage, we have the following relation between call and put prices。Theorem 2(Put - call parity formula) For a

5、stock which does not pay any dividends,(Call(K,T) - Put(K,T)erT + K = S0erT2、證明Recall that the actions and payoffs corresponding to a call/put are:If ST KIf K STlong callno actionbuy the stockshort callno actionsell the stocklong putsell the stockno actionshort putbuy the stockno actionIf ST KIf K S

6、Tlong call0ST - Kshort call0-(ST - K)long putK - ST0short put-(K - ST )0Proof.Consider the portfolio consisting of buying onshare of stock and a strike put for one share; selling a K- strike call for one share;and borrowing S0 - Call(K,T) + Put(K,T). At time T, we have the following possibilities:If

7、 ST K, then the call is exercised and the put is not. We finish without stock and with a payoff for the call of K.In any case, the payoff of this portfolio is K. Hence, K should be equal to the return in an investment of S0 + Put(K,T) - Call(K,T) in a zero - coupon bond, i.e.K = (S0 + Put(K,T) - Cal

8、l(K,T)erT例子Example 1The current value of XYZ stock is 75.38 per share. XYZ stock does not pay any dividends. The premium of a nine - month 80 - strike call is 5.737192 per share.The premium of a nine- month 80 - strike put is 7.482695 pershare. Find the annual effective rate of interest.Solution: Th

9、e put - call parity formula states that (Call(K,T) - Put(K,T)(1 + i)T + K = S0(1 + i)T . So, (5.737192 - 7.482695)(1 + i)3/4 + 80 = 75.38(1 + i)T . 80 = (75.38 - (5.737192 - 7.482695)(1 + i)3/4 = (77.125503)(1 + i)3/4, and i = 5%.Example 2The current value of XYZ stock is 85 per share. XYZ stock does not pay any dividends. The premium of a six - month K - strike call is 3.329264 per share andthe premium of a oneSolution: The put - call parity formula states that (Call(K,T) - Put(K,T)(1 + i)T + K = S0(1 + i)T .So, (3.329264 - 10.384565)(1.065)0.5 + K = 85(1.0