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.