Put-call parity is a relationship between the prices of European call and put options with the same strike, expiration, and underlying. It is defined by a simple formula C + PV(K) = P + S, where PV(K) is present value of strike. This relationship should hold for all European options, otherwise there is an arbitrage opportunity. It does not hold for American options, although some relationship can still be defined.

## Put-Call Parity Formula

The put-call parity equation is:

\[C+PV(K)=P+S\]

… where:

- \(C\) = Call option price
- \(PV(K)\) = Present value of strike price (same strike for call and put)
- \(P\) = Put option price
- \(S\) = Underlying price

### Why Present Value of Strike?

The \(C\), \(P\), and \(S\) in the formula are clear – the current market price of the call option, the put option, and the underlying (e.g. current stock price). But why is there a present value of strike \(PV(X)\) and not just strike \(X\)?

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### Why Not Present Value of Underlying Price?

Why don’t we discount the underlying price in the same way as the strike price – why not \(PV(S)\), but just \(S\)?

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## Arbitrage

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## Example

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## Put-Call Parity for American Options

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