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看涨看跌平价定理（Put–call parity）

　　看涨期权与看跌期权之间的平价关系。

1．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，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）

1．无收益资产美式期权

　　由于P>p,从式（1.1）中我们可得： 
　　P>c+Xe-r(T-t)-S
　　对于无收益资产看涨期权来说，由于c=C，因此： 
　　P>C+Xe-r(T-t)-S
　　C-P<S-Xe-r(T-t)（1.3）

　　为了推导出C和P的更严密的关系，我们考虑以下两个组合：
　　组合A：一份欧式看涨期权加上金额为X的现金
　　组合B：一份美式看跌期权加上一单位标的资产
　　如果美式期权没有提前执行，则在T时刻组合B的价值为max(ST,X)，而此时组合A的价值为max(ST,X)+ Xe-r(T-t)-X 。因此组合A的价值大于组合B。
　　如果美式期权在T-t 时刻提前执行，则在T-t 时刻，组合B的价值为X，而此时组合A的价值大于等于Xe-r(T-t) 。因此组合A的价值也大于组合B。
　　这就是说，无论美式组合是否提前执行，组合A的价值都高于组合B，因此在t时刻，组合A的价值也应高于组合B，即： 
　　c+X>P+S

　　由于c=C，因此， 
　　C+X>P+S
　　结合式（1.3），我们可得： 

  　　S-X<C-P<S-Xe-r(T-t)(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。

2．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-X<C-P<S-D-Xe-r（T-t） （1.5）

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:
(Call(K,T) − Put(K,T))(1 + i)T + K = F0,T

Often, 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。

(Put–call parity formula) For a stock which does not pay any
dividends,
(Call(K,T) − Put(K,T))erT + K = S0erT

Recall that the actions and payoffs corresponding to a call/put are:
                       If ST < K             If K < ST
     long call           no action           buy the stock
    short call          no action           sell the stock

long put         sell the stock        no action
short put       buy the stock        no action
                       

                       If ST < K          If K < ST 
long call                0                    ST − K
short call               0                −(ST − K)
long put            K − ST                   0
short put        −(K − ST )                0

Consider the portfolio consisting of buying one share of stock and a K–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:
1. If ST < K, then the put is exercised and the call is not. We finish without stock and with a payoff for the put of K.
2. If 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) − Call(K,T))erT

The 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 per share. Find the annual effective rate of interest.

Solution: The 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%.

The 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 and

the 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(1.065)0.5 and
K = (85 − 3.329264 + 10.384565)(1.065)0.5 = 95. year K–strike put is 10.384565 per share. The annual effective rate of interest is 6.5%. Find K.