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There are usually two ways to write proofs of equalities (like put-call parity) in quantitative finance. By replication, by constructing arbitrage. Both of these are actually the same, since the first one is done be making say, two portfolios, $A$ and $B$, and showing that they have the same outcome at time $t=T$. Then, by argument of LOOP (law of one ...

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I think it is simpler than looking at the distribution of returns. Since options can be perfectly hedged using synthetic positions, the distribution should not matter. Let's keep it in the put-call-parity universe. To make a synthetic call, you need to buy a put, buy a stock, and borrow the money at rate r. You're paying that back over time. To make a ...

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There are two ways (or shall i say at least two ways) to look at this. 1) The option price does not depend on the promise of the pay-offs alone, but also on the probability of those payoffs. As you alluded to, if you look at the probability distribution, you will see the unlimited payoffs is so unlikely as to be irrelevant. 2) The put call parity gives you ...

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