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We know that $$C-P = PV(F_{0,T}-K)$$

When we create a synthetic forward, we buy call and sell a put at the same strike price $K$. When we buy the call why do we assume the premium is positive? When we sell the put, why do we assume the premium is negative?

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  • $\begingroup$ You can accept one of the answers if you are satisfied by it :-) $\endgroup$
    – vonjd
    Jan 28 '13 at 16:15
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To see the connection between put-call parity and option price you should read this highly insightful paper by Espen Gaarder Haug & Nassim Nicholas Taleb:

Option traders use (very) sophisticated heuristics, never the Black– Scholes–Merton formula

It shows how you can heuristically derive option pricing formulas by adapting the tails and skewness by varying the standard deviation of a Gaussian - and then remove the risk parameter by using put-call parity.

Both authors are well known in the Quant-Community, they both have an academic as well as a practical background as traders.

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Both premiums are actually always positive by definition. The difference will be positive when the forward price exceeds the strike and vice versa.

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