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Since American style options allow early exercise, put-call parity will not hold for American options (unless they are held to expiration). In practice, there is also a difference between calls and puts for European options as well. The full description is here: What causes the call and put volatility surface to differ?


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the call version pays $$ I_{S_T > K } S_T $$ the put version pays $$ -I_{S_T < K } S_T $$ Subtract to get a pay-off $$ S_T. $$ (ignoring the probability zero event of $S_T=K.$) So the prices subtract to give $S_0.$



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