看漲看跌平價定理

1、看漲看跌平價定理歐式看漲期權與看跌期權之間的平價關系無收益資產的歐式期權在標的資產沒有收益的情況下，為了推導c和p之間的關系，我們考慮如下兩個組合：組合A： 一份歐式看漲期權加上金額為Xe-r(T-t)的現(xiàn)金組合B： 一份有效期和協(xié)議價格與看漲期權相同的歐式看跌期權加上一單位標的資產在期權到期時，兩個組合的價值均為max(ST,X)。由于歐式期權不能提前執(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)這就是無收益資產歐式看漲期權與看跌期權之間的平價關系(Parity)。它表明歐式看 漲期權的價值可根據相同協(xié)議價格和到期日的歐式看跌期權的價值推導出來，

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

3、)(1.4)由于美式期權可能提前執(zhí)行，因此我們得不到美式看漲期權和看跌期權的精確平價關系， 但我們可以得出結論：無收益美式期權必須符合式(1.4)的不等式。有收益資產美式期權同樣，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美式期權必須遵守的不等式：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