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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
    Commented Oct 29, 2019 at 15:19
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Delta: change in option price with respect to a change in spot price assuming all else equal. It is the number of shares needed to replicate one option. Specifically, when you are short an option, delta represents the number of shares needed to be bought to be delta-neutral.

Premium-adjusted delta: the delta of the position (portfolio) consisting of a short option position and the number of shares which can be bought using the premium (money) received from that short option.

Example: if an ATM call has 0.5 delta then a short position in that option will have a delta of -0.5. In order to be delta neutral, you need to long 0.5 shares. Let's say with the premium received from the short call you are only able to buy 0.1 shares. Then the premium-adjusted delta would be -0.5+0.1=-0.4 after buying the shares with the money funded from the short position.

An FX example and more info is given here: http://janroman.dhis.org/finance/FX/FX%20Volatility.pdf

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The above answers are correct in explaining how premium adjustment works.
However:

  • EURUSD is generally not Premium delta adjusted (the paper, albeit dated, from and Dmitri Reiswich and Uwe Wystup that was provided above actually confirms that)
  • The choice of premium currency therefore also has no impact on the delta level

If your vendor / pricer of choice displays a different delta in the case of EURUSD I would check the settings or restore defaults if possible.

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I would like to clarify the meaning of the premium adjusted delta. When you buy a forward, you will, for example, get eur and give usd. You can calculate the value of the two legs in eur resp. usd by applying usd eur at the usd leg, resp. eur usd at the eur leg and sum

What if you want to do the same for an option ? How much usd is required to be sold and how much eur is required to be bought ? Let's say the option delivers max(Eurusd - K; 0 ) in usd. It means 1 eur against K usd if it's a gain, 1 eur against the current eurusd else (0)

  1. You know that the amount of eur to buy ( eur is the foreign currency ) is the usual delta,

  2. How much usd do you have to sell ? As the sum of the legs is the option value v, you have v = eur to buy - usd to sell

If you say v is in eur then : v = delta - 'usd to sell, valued in eur'

So 'usd to sell, valued in eur' = delta - v

It's the premium adjusted delta !

For me it's pointless to delete v from delta just because you paid the premium in eur. Also, v is the current premium, not the one you actually paid at start.

Note that you can get this formula with d v / d usdeur. The proof is written in the good and synthetic article from Wystup.

Sum up : premium adjusted delta = delta - option value = ' usd to sell, valued in eur '

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