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# Delta Hedge

#### PL

##### Active Member
Dear all,

If a US bank enters in a EURGBP option how the Bank could hedge this position (Delta Hedging) in dollars?

Thank you

#### Aleksander Hansen

##### Well-Known Member
In a lot of ways.
Cross-currency swaps on GBPUSD and/or EURGBP. Or synthetically replicate the Greeks by taking positions in USD denominated assets. Most favorable outcome will be achieved if you find a cointegrated series that match, as well as the pure joint distribution (the copula, stripping away the marginals) so you don't have to rebalance all the time.

#### PL

##### Active Member
Hello Aleksander,
Thank you for your reply. Regarding the FX Swaps it was my first thought, but the time I tried to set up an example in my mind I found out that I didnt know what amount/Principal to hold for the FX Swaps....Suppose a dummy position on EURGBP with face 100 mil EUR and strike 0,9. How I should calculate the principal of the swaps? Could you please explain further the methodology? Additionally, could you also explain further the second solution?

I apologize if my questions are dummy, but I'm not a quant expert

Once again thanks a lot for your assistance and time you spent to reply

#### Aleksander Hansen

##### Well-Known Member
So, forget about the complications of swaps for a minute and let's think of it differently:

1) You have a portfolio [which happens to contain EURGBP options].

2) The question becomes, how do you make your [or generally any] portfolio delta-neutral?
Would you expect it to remain delta-neutral?

3) Is there any reason Greeks denominated in USD work differently from other denominations?

4) As a corollary, do you want to do anything about the currency exposure?

#### PL

##### Active Member
Hello again,
2. My portfolio...Lets suppose that we want to "neutralize" the portfolio on a daily base.
3. Basically, I do not know how to translate the Delta to USD...

#### WhizzKidd

##### Member
@David Harper CFA FRM

How does one approach a question like this:

The bank has a position on USDRUS options that has a delta of $2m and a gamma of$0.4m (per 1% move). The USDRUS exchange rate is 14.2. What position would you take to make the position delta neutral? After a short period of time, the USDRUS exchange rate moves to 13.916. Estimate the new delta. What additional trade is necessary to make the position delta neutral?

How does one hedge delta in the FX market?

I can only think of selling a call option, which will get rid of the gamma. This short call will possible reduce some of the delta, but how do you think about hedging the rest of the delta? Futures/forwards?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @WhizzKidd It's sometimes easier to help if we know the source (i.e., what is the source?) because some sources are more reliable than others. It appears we are given position delta without knowing the percentage greeks or actual position sizes (see https://www.bionicturtle.com/forum/threads/l1-t4-7-dynamic-delta-hedging.4839/ for introduction to distinction between position greeks and percentage greeks). If the goal is to neutralize both gamma and delta, we could (i) enter a short position in call options (i.e., with negative position gamma) and then (ii) short shares to neutralize the remaining positive position delta. So, typically you'd neutralize the gamma first and then the delta because shorting shares will neutralize the delta without impacting gamma; but options have both delta and gamma.

But the questions asks only "What position would you take to make the position delta neutral?" which can be achieved with a short currency futures contract, in this case with an approximate (the future has percentage delta of almost one) notional of $2.0 mm. We neutralize position delta on a dollar-for-dollar basis. The gamma will remain. Then, with respect to "After a short period of time, the USDRUS exchange rate moves to 13.916. Estimate the new delta.", I suppose the question is looking for: that's a 13.916/14.20-1 = -2.00% move exactly, such that the new delta =$0.40 * -2.0% = -$0.80 on the net position; or maybe it looking for the$2.0 mm - 2%*0.40 = $1.2 mm on the original exposure. I hope that helps! Last edited: #### WhizzKidd ##### Member @David Harper CFA FRM , thanks. Would it not be 2*0.4 instead of 2%*0.4, as it says that the gamma is$0.4m per 1% move?

Sorry, I see you did apply it as above.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@WhizzKidd yes, right, thanks! I guess since gamma is given as $0.4 m/.010 (ie,$0.4 per 1%), it is $0.4 m/Δ0.01 * Δ0.02 =$0.40 * 0.02/0.01 = $0.40 * 2, just as you say . Gamma is tricky because while delta is elegantly unit-less (delta =$Δc/$ΔS so dollars cancel), gamma's true units are un-intuitively denominated (1/$) since gamma is Δdelta/ΔS =Δ($Δc/$ΔS)/$ΔS = unit-wise Δ($/$^2) = Δ(1/$). Thanks!

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