fx vanilla option's forward delta in single currency

Question

According to Black formula , a vanila fx call option's pricing is

$$C(F,\tau) = D[N(d_+)F - N(d_-)K]$$ , where $\tau$ is the time to expiry, $D =e^{-r\tau}$ the discount factor, $F=S/D$ the outright forward rate, and $d_\pm =\frac{1}{\sigma\sqrt{\tau}}\left[\ln\frac{F}{K}\pm\frac12\sigma^2\tau\right]$.

If we look at the forward delta , it's $$\frac{\partial C}{\partial F}=DN(d_+)$$

Can I interprete that, with such an option shorted, if there's a outright forward rate deal at the same maturity, with $N(d_+)$ unit of currency 1, and $-N(d_-)F$ unit of currency 2, the delta will be fully hedged? Of course the ratio $\frac{N(d_-)}{N(d_+)}F$ is not at-the-money, but never mind that.

Answer 1

Actually the forward delta is the option's sensitivity to the PV of the forward contract with same maturity so it is $$ \frac{1}{D}\frac{\partial C}{\partial F} = N(d_{+}) $$ For an option on CCY1CCY2 with payoff in CCY2 the forward delta gives you the number of forwards on CCY1CCY2 required to hedge the option. The forwards can be struck at any pre-agreed rate since all forwards have the same sensitivity to $F$.