# Put-call-forward parity for European options

There is a problem with the put-call parity arbitrage because it is difficult to short sell the underlying asset. But that problem can be solved by including forward in the equation in the place of the underlying asset as it is much easier to sell the forwards short.

F_{0}(T) = S_{0}*(1+r)^{-T} Or S_{0} = F_{0}(T)*(1+r)^{-T}

So, we will replace S0 in the put-call parity equation with F_{0}(T)*(1+r)^{-T} to get the put-call forward parity.

The equation for put-call-forward parity is given below:

c_{0} + X*(1+r)^{-T} = p_{0} + F_{0}(T)*(1+r)^{-T}

Or

c_{0} - p_{0} = [F_{0}(T) - X]*(1+r)^{-T}

Or

F_{0}(T) - X = (c_{0} - p_{0})*(1+r)^{T}

To create a synthetic protective put with the forward contract, we need to replace the underlying with the forward contract. We know that we can replicate the performance of underlying by going long into the forward contract and long into the risk-free bond that pays the forward price at the expiration.

Payoffs |
S |
S |
S |

Fiduciary Call |
|||

European call |
S |
0 |
0 |

Bond |
X |
X |
X |

Total |
S |
X |
X |

Protective Put with Forward Contract |
|||

European put |
0 |
0 |
X - S |

Forward |
S |
S |
S |

Bond |
F |
F |
F |

Total |
S |
S |
X |

__Check your concepts:__

(58.24) Which of the following is a correct version of put-call-forward parity equation?

(a) c_{0} + X(1+r)^{-T} = p_{0} + F_{0}(T)

(b) c_{0} + p_{0} = [X - F_{0}(T)](1+r)^{-T}

(c) F_{0}(T) - X = (c_{0} - p_{0})*(1+r)^{T}

__Solutions:__

(58.24) Correct Answer is C: F_{0}(T) - X = (c_{0} - p_{0})*(1+r)^{T} is the correct version of put-call-forward parity equation.

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