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Let's assume a bank sells to a client a put of \$1,000,000 dollars on USDJPY at 110 in 6 months. The delta of this put is -0.6, spot is 112. So to hedge its position the bank has to short \$600,000 dollars.

Two questions :

  • Why couldn't the bank hedge in JPY as, if the client exercices, they will have to give him JPY instead of USD ?
  • Does it change something if this option's prenium is in EUR ? Does the bank stil have to hedge in USD ?
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    $\begingroup$ If you have a put on GOOG you hedge using GOOG stock, if you have a put on USD you hedge using USD. GOOG and USD are the underlying you are betting on. $\endgroup$ – Alex C Oct 20 '19 at 18:34
  • $\begingroup$ Yes, i know but as for those options there are two currencies, i was wondering if it was equivalent. So when i say 'could they hedge in JPY' it would have been the same for the put on GOOG if i've said 'could they hedge in USD' ? $\endgroup$ – TmSmth Oct 20 '19 at 19:16
  • $\begingroup$ On your second question, do you mean the option is EURUSD? Or it is USDJPY but the premium is paid in EUR? $\endgroup$ – AlRacoon Oct 22 '19 at 2:15
  • $\begingroup$ The prenium in EUR, thanks for your first answer, i'm waiting for the second answer to validate it ! $\endgroup$ – TmSmth Oct 22 '19 at 7:39
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When you delta hedge a currency option, you are hedging in both currencies. In your example, since you have a short position of $600,000 as your hedge, this would be against JPY; therefore you would be long 600,000 * 112 = 67.2MM JPY at the same time.

Regarding your second question, you would still hedge the delta using the underlying of the option--in this case, USDJPY. However, if you are a European bank that converted the USD premium into EUR, your functional currency, and wanted to hedge out your USD risk, you could trade EURUSD spot to turn USD exposure back to EUR. You can adjust this risk as your delta hedge changes.

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