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Let's say I have designed an option's portfolio. The portfolio includes long as well as short positions in European-style put and call contracts based on the same underlying asset with different strike prices.

Does it make sense to calculate the call-put parity for pairs of call and put with the identical strike prices? or is there is an approach to calculate the call-put parity for the option's portfolio as as a measure of the equivalence between call and put contracts?

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The put-call parity is just a no-arbitrage condition that exists for options at the same strike and expiration. If you want to know how your portfolio will change or how similar two options are you are going to want to look at the greeks. https://en.wikipedia.org/wiki/Greeks_(finance)

As an example, if you want long exposure to a stock but don't want to buy a call option you could sell a put option with a similar delta.

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