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Suppose we have a contract with payoff $P_Y$ in currency $Y$, where $P_Y$ on a variable in currency $Y$.

To calculate the value in $X$, we take the expected payout under $Y$-numeraire $E_Y(P_Y)$, discount using the discount rates for $Y$, then convert into $X$ by using the current spot rate. No quanto adjustment needed in this case.

Now suppose $P_Y$ is paid in currency $X$ instead, using the spot rate at expiry. This requires a quanto adjustment due to the correlation between the payoff and the FX forward rates.

These give different prices, because there's no quanto adjustment in the first case, while there is the second.

To me, this is counterintuitive. All that's needed to change from the first to the second is just a spot transaction at expiry. Why should they not give equal prices, and why would a quanto adjustment not be needed in the first case?

EDIT: Should be spot rate at inception, not expiry

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  • $\begingroup$ Are you sure you understand the payoffs? For the second one, you state: "Now suppose PY is paid in currency X instead, using the spot rate at expiry." But the phrase "using the spot rate at expiry" is unnecessary, since we are already in currency X. In your question, you have correctly stated the reason for the quanto adjustment: the correlation between the payoff and the FX rate. $\endgroup$ – dm63 Nov 18 '18 at 23:00
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in both situation consider it from the point of the seller of the option. And consider it from the hedging cost perspective. And for simplicity let's pretend that the payoff that you sold is a vanilla option. Assume that at trade date that X and Y trade 1-to-1.

in situation A: you sold the vanilla call and you start delta hedging. Everything happens in currency Y. At time of expiry you have fully replicated the vanilla payoff, and only think you have to do is to take your vanilla payoff and convert at spot to currency X. The only hedging cost you have is the delta hedging in currency Y. i.e. everything stays within the blackscholes framework.

in situation B: you sold the vanilla call but you know that you that you have to deliver the vanilla payout in currency X AND you have fixed the currency to 1-to-1. Now imagine that you the currency pair is strongly correlated with the equity movements (say it is stock of some exporting company). So let's say that if stock goes up, typically Y becomes really weak.

In this case if you did only the equity delta hedging. You end up with the payout max(S-K,0) and let's assume the stock went up. Your replication gave you the amount (S-K) in currency Y. Since the stock was correlated with the currency, the currency got weaker as per our assumptions. So you need to pay out (S-K) in currency X but you only have (S-K) in currency Y. but it does not trade 1-to-1 anymore, but much weaker. So you don't have sufficient money to pay your liabilities. To solve this, you have to adjust the amount of currencies at each delta hedging step.

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The OP states that option B pays off using the fx rate at expiry. That contract does not require a quanto adjustment. Perhaps the OP intended to say that the payoff of option B is using the fx rate at inception.

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