看涨看跌平价定理

1、看涨看跌平价定理欧式看涨期权与看跌期权之间的平价关系无收益资产的欧式期权在标的资产没有收益的情况下，为了推导c和p之间的关系，我们考虑如下两个组合：组合A： 一份欧式看涨期权加上金额为Xe-r(T-t)的现金组合B： 一份有效期和协议价格与看涨期权相同的欧式看跌期权加上一单位标的资产在期权到期时，两个组合的价值均为max(ST,X)。由于欧式期权不能提前执行，因此两 组合在时刻t必须具有相等的价值，即：c+Xe-r(T-t)=p+S (1.1)这就是无收益资产欧式看涨期权与看跌期权之间的平价关系(Parity)。它表明欧式看 涨期权的价值可根据相同协议价格和到期日的欧式看跌期权的价值推导出来，

2、反之亦然。如果式(1.1)不成立，则存在无风险套利机会。套利活动将最终促使式(1.1)成立。2 .有收益资产欧式期权在标的资产有收益的情况下，我们只要把前面的组合A中的现金改为D+Xe-r(T-t)， 我们就可推导有收益资产欧式看涨期权和看跌期权的平价关系：c+D+Xe-r(T-t)=p+S (1.2)美式看涨期权和看跌期权之间的关系1 .无收益资产美式期权由于Pp,从式(1.1)中我们可得：Pc+Xe-r(T-t)-S对于无收益资产看涨期权来说，由于c=C，因此：PC+Xe-r(T-t)-SC-PP+S由于c=C,因此，C+XP+S结合式(1.3)，我们可得：S-XC-PS-Xe-r(T-t

3、)(1.4)由于美式期权可能提前执行，因此我们得不到美式看涨期权和看跌期权的精确平价关系， 但我们可以得出结论：无收益美式期权必须符合式(1.4)的不等式。有收益资产美式期权同样，我们只要把组合A的现金改为D+X，就可得到有收益资产美式期权必须遵守的不等式：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、证明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