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Please explain the concept of premium Adjusted Delta in FX market. In EURUSD, why delta changes if premium currency is changed from USD to EUR and how this new delta is related to the old one with premium paid in USD? How is this derived mathematically?

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The standard BS spot delta is a quantity in % of foreign currency (CCY1). The actual hedge quantity must be adjusted if the premium is paid in CCY1. FX options are special in that regards since for a stock option you wouldn't pay in shares typically.

Example: we are short EURUSD with 1MM EUR notional, the option premium received is 73669 EUR. Say delta is 60%, the hedge to our short is long 600M EUR. However, since we received our premium in EUR already, our actual hedge must be lower. Thus the premium adjusted delta is 600'000 - 73669 = 521'331 EUR.

Formally: $\Delta_{S, pa} = \Delta - \frac{v}{s} $

where $\Delta$ is our spot delta and the fraction is the premium $v$ in CCY1 using spot $s$ since BS returns the premium in CCY2 terms.

This is explained in this paper by Uwe Wystup and Dmitri Reiswich: https://mathfinance.com/wp-content/uploads/2017/06/CPQF_Arbeits20_neu2.pdf

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  • $\begingroup$ @Ussu: since many of your questions are about FX market conventions, you would do well to familiarize yourself with the work of Uwe Wystup (see above). He is an expert in the field. $\endgroup$ – Alex C Oct 29 '19 at 15:19

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