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:
(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>Theorem 2
(Put–call parity formula) For a stock which does not pay any
dividends,
(Call(K,T) − Put(K,T))erT + K = S0erT
2、证明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
Proof. 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
3、例子Example 1 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%.
Example 2 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.
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