FX Spot Delta market standard calculation (Trader View)

Question

I am just writing my thesis about FX instrument and hedging and one question popped up which I can't solve. Maybe it is silly but cant find anything about it how the delta of a fx spot is defined and I want to hedge it with an option in USD Deltas. The delta of an option is easy just the first derivative of the Garman-Kohlhagen option pricing formula.

I have a GBP/USD FX-SPot trade with T+2 settlement period and the deal is made today on the 8/13/2019. The spot date would be 8/15/2019 (physical exchange). I have the folling parameters:

$$\Delta_{USD_{T+2}} \approx Notional_{GBP} * pips $$

The question is how can I discount the delta to be the value of today.

How would I now discount the delta to today in terms of T+2 to T?

I would use the instantaneous fx spot rate: $$FX_{instantaneous_{GBP/USD}} = FX_{Spot}-(ON+TN)$$ but how can I use it in the approximation above?

If I would look into a USD/CHF FX-Spot trade the delta would look like:

$$\Delta_{USD_{T}} \approx Notional_{USD} * pips = \Delta_{CHF}/FX_{instantaneous_{USD/CHF}}$$

So my two questions:

1. How can I discount the USD $$\Delta$$ for GBP/USD FX-Spot?
2. Does the approximation makes sense for USD/CHF?

If not what approach should I use?

Answer 1

This will be too late for your thesis but you mix up a few concepts. If it does not help you anymore, it may help others.

Spot does not have delta (well it is delta one, as delta is defined as the change in value due to change in spot - hence, logically spot delta is 1, as you ask for change in spot for change in spot).

FX options are actually quoted in delta. Uwe Wystup and Dimitri Reswich explains this nicely. Many currencies switch from ATMS to AMTF after 1 year.

In terms of computing ATMD, ATMF, and ATMS; I simply refer to the aforementioned paper. Table 2 shows Spot vs forward delta.

What you refer to is the distinction between time to delivery vs time to expiry (I suppose). The later is usually from pricing date (when the structure is entered) to expiry (when spot fixing is observed). The former refers to the difference between delivery and premium date (what you say is T+2) but that is not generally applicable (there are T+1 currencies, deferred premia etc.).

Ignoring all this, Spot delta is really just forward delta, $N(d1)$, discounted $exp^{-ccy2*\tau}*FwdDelta$. This discount factor is using time to delivery (premium date to delivery date). Usually, premium date is spot date (which is in turn T+2 for many currencies).

Now that concludes the logical argument. Spot delivery is T+2. If option delta is premium date to delivery, you have zero time difference between premium and delivery in spot. Hence, why spot is delta 1 again.

ON and TN are not even forwards but swaps. It makes no sense to use these to (delta) hedge spot. You enter spot today, for delivery in T+2(in many cases).

Last but not least, delta hedging involves buying spot. If you would have to delta hedge spot again, it would defeat the purpose of hedging with spot in the first place.