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Having two spot currency positions,

short EUR/USD
long  GBP/USD

We are looking for a way to diminish the risk of the spread going against us. The basic idea is to invert the positions, namely:

long  EUR/USD
short GBP/USD

However, this would not only eliminate the risk, but also any profits that might arise. We would like to be able to cancel the hedge position in exchange for paying a premium. How do we use options to execute this strategy?

Would creating the following positions achieve the desired effect?

put    short    EUR/USD
put    long     GBP/USD
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In that "portfolio" you've also got a short EURGBP position, because you're short EUR and long GBP and, both long and short USD some of which cancels out. So why not work out the amount you're short and cover that with a EURGBP call?

Otherwise, yes you can use options to offset your positions:

call    EUR/USD
put     GBP/USD
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In FX you always need to consider both sides. Probably question needs to be a bit ore clear

short EUR/USD => Short EUR or Long USD

So if you are buying Put on EUR it's a call on USD

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