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You are watching: Put call forward parity

Assume that:

F(0, T): the price created today for a forward contract expiring at time Tc0: the contact option price todayp0: the put option price todayBoth options expire once the forward contract expires: the moment until expiration is also T.The practice price that both alternatives is X.

Consider 2 portfolios. Portfolio A is composed of a long call and a lengthy position in a zero-coupon shortcut with face value of X - F(0, T). Investment portfolio B is composed of a long put and a lengthy forward.


As the two portfolios have specifically the very same payoff, your initial investments must be the very same as well. The is:


This equation is put-call parity for choices on front contracts.

As F(0, T) = S0(1 + r)T, us rearrange the equation as the follows:


Consider the complying with example:

T = 90 days, r = 5%, X = $95, S0 = $100, and the call price is $10. The placed price need to be c0 + X/(1 + r)T - S0 = 10 + 95/(1 + 0.05) (90/365) - 100 = $3.86.

Similarly, we have the right to compute the call price offered the price the the put.

Consider one more example. The options and a forward contract expire in 50 days. The risk-free price is 6%, and the practice price is 90. The forward price is 92, and also the call price is 5.5.

p0 = c0 + /(1 + r)T = 5.5 + (90 - 92)/1.06(50/365) = 3.52

Note the in this case X 0 (value of long call) + <-(X - ST)> (value of brief put) + (value of long bond) = ST - F(0, T), if ST (value of long call) + 0 (value of short put) + (value of lengthy bond) = ST - F(0, T), if ST > X.

As a front contract\"s payoff in ~ expiration is also ST - F(0, T), the portfolio\"s early stage value have to be equal to the initial value of the front contract (which is 0).

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Solving because that F(0, T), we attain the equation for the forward price in regards to the call, put, and also bond. Therefore, a man-made forward contract is a combination of a long call, a brief put and a zero-coupon link with confront value (X - F(0, T)). Note that we might either long or short this bond, depending upon whether the exercise price of these alternatives is reduced or greater than the front price.

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