Let's start with Delta. What we need from delta is a way of understanding how our P&L will change as the underlying FX rate moves, and therefore what position to take in the underlying to hedge our delta risk exposure. While it is generally pure to measure delta with respect to the forward rate, in FX options as in equities, it is very common to look at your spot delta. The reasons for this are that spot FX is very easy to trade and therefore to hedge with, and also that the forward points measuring the differential between spot and forward rates are relatively slow moving and therefore can be hedged separately and less frequently.

In order to convert the raw Black-Scholes delta into an equivalent spot FX position, we simply multiply the option delta by the notional in the base currency. This now tells us two things. Firstly, the P&L which would be generated by a move of 1 in the spot FX rate. Clearly a huge unrealistic move in FX. So we would scale that down to say a 0.01 move. That is one big figure, which tells us that a one big figure move would make or lose us $55,000. Secondly, it gives us our hedge ratio in terms of base currency spot position. So in the example we have here, a delta of 0.55 means that if we buy 10 million euros of the 1.10 call, we would be effectively long 5.5 million euros of spot FX and therefore would need to sell 5.5 million euros as our delta hedge. Does our delta depend on what currency our premium is paid in? Well, yes it does, at least our initial hedge because if the premium comes in the wrong currency, for example, we receive the premium in euros, but we actually want it in dollars, then that necessitates a spot FX trade to correct. If we're already doing a delta hedge on our option trade, then the premium conversion trade can be incorporated into our hedge requirements to give a modified initial delta hedge. If you don't convert your premium back into the currency in which you account for your P&L then you are running a spot FX risk. So now we'll do that using these numbers as an example. Here the client will be paying 186,000 euros for the option, but we want the premium in dollars. The next section shows the effect on the all in delta hedge. Here are the numbers for our delta. As you can see, the raw delta, if we were not including the conversion of premium, is long, 4.3 million euros. So we would need to sell 4.3 million euros as our hedge. The client is also paying us just under 200,000 euros, which we also want to sell. So once we incorporate that into our hedge requirements, we now need to sell 4.5 million euros. Note that this only needs to be done once on the initial delta hedge because the premium is only paid once.

From this point on any rebalancing of delta is done on changes in the raw delta.