Assume that:

- F(0, T): the price established today for a forward contract expiring at time T
- c
_{0}: the call option price today - p
_{0}: the put option price today - Both options expire when the forward contract expires: the time until expiration is also T.
- The exercise price of both options is X.

Consider two portfolios. Portfolio A consists of a long call and a long position in a zero-coupon bond with face value of X - F(0, T). Portfolio B consists of a long put and a long forward.

As the two portfolios have exactly the same payoff, their initial investments should be the same as well. That is:

This equation is

As F(0, T) = S

Consider the following example:

T = 90 days, r = 5%, X = $95, S

Similarly, we can compute the call price given the price of the put.

Consider another example. The options and a forward contract expire in 50 days. The risk-free rate is 6%, and the exercise price is 90. The forward price is 92, and the call price is 5.5.

p

Note that in this case X < F(0, T), which means that we short the bond instead of buying the bond as in portfolio A above.

Continue with those assumptions at the beginning of this subject. Consider a portfolio consisting of a long call, short put and a long position in a zero-coupon bond with face value of X - F(0, T). At expiration the value of the portfolio is:

- 0 (value of long call) + [-(X - S
_{T})] (value of short put) + [X - F(0, T)] (value of long bond) = S_{T}- F(0, T), if S_{T}<= X. - [S
_{T}- X] (value of long call) + 0 (value of short put) + [X - F(0, T)] (value of long bond) = S_{T}- F(0, T), if S_{T}> X.

As a forward contract's payoff at expiration is also S

Solving for F(0, T), we obtain the equation for the forward price in terms of the call, put, and bond. Therefore, a synthetic forward contract is a combination of a long call, a short put and a zero-coupon bond with face value (X - F(0, T)). Note that we may either long or short this bond, depending on whether the exercise price of these options is lower or higher than the forward price.

rhardin: Is this really hard for anyone else or is it just me? Ugh. |

Tony1234: I'm right there with ya rhardin. I'm really happy derivatives only make up a small portion of the exam. |

edushyant: If you know put-call parity then forward put-call parity is not too difficult. |

Fabulous1: Just think of it like that: Instead of buying the underlying asset in the put call parity you take a long position in a forward contract. That has the same payout as the Bond of the in the put-call parity. Rearranging the terms gives you the put-call forward parity |

rcoyne: Substitution is our friend on this one. We know that put call parity says: Call+Strike/(1+r)^T=Put+Underlier. We also know that the underlier is priced at the PV of the Forward price: Underlier = F(0,T)/(1+r)^T. Substitute the PV of Forward price for Underlier in the put-call parity: Call+Strike/(1+r)^T = Put + Forward/(1+r)^T. Finally, since our two PV fractions have the same denominator, we can combine them on the same side of the equation as: Call + (Strike-Forward)/(1+r)^T = Put. |

mcbreatz: Now that is an explanation. |

nmech1984: coyne, you rock it. thanks a lot! |

urbanmonk: Clearly and simply put @rcoyne, many thanks! |