## FX Conventions

We list and derive the various FX conventions.

### Delta Conventions

The Black Scholes Delta, which is the derivative of the option price with respect to the spot price of the underlying, is given by:

$$\frac {\partial BS } {\partial S}= \Delta_{s, ua}=e^{-r_{f}\tau} \phi N{\left( \phi d_{1} \right )}$$

In the FX world, this delta is called the Unadjusted Spot delta, and in the hedging world, it represents the number of units of Currency 1 to buy in order to hedge a short call option position. An alternative representation of the Delta is the Premium Adjusted Delta, which adjusts the Delta for the Premium amount (option price). As the premium amount is in Currency 2, dividing it by the current spot rate (S) gives the equivalent amount in units of Currency 1. Thus:

$$\Delta_{s, pa}=\Delta_{s, ua}- \frac{1}{S} BS Price$$ $$\Delta_{s, pa}=e^{-r_{f}\tau} \phi N{\left( \phi d_{1} \right )} - \frac{1}{S} \left( S e^{-r_{f}\tau} \phi N{\left( \phi d_{1} \right )} - K e^{-r_{d}\tau} \phi N{\left( \phi d_{2} \right )} \right)$$ $$\Delta_{s, pa}=e^{-r_{f}\tau} \phi N{\left( \phi d_{1} \right )} - e^{-r_{f}\tau} \phi N{\left( \phi d_{1} \right )} + \frac{K}{S} e^{-r_{d}\tau} \phi N{\left( \phi d_{2} \right )}$$ $$\Delta_{s, pa}= \frac{K}{S} e^{-r_{d}\tau} \phi N{\left( \phi d_{2} \right )}$$

The delta can also be expressed in terms of the price (value) of the forward, instead of the spot. Let $$V_{f}$$ represent the current value of a $$\tau$$ maturity forward contract:

$$V_{f}=e^{-r_{d}\tau} \left( Current Forward Rate - Agreed Forward Rate \right)$$ $$V_{f}= e^{-r_{d}\tau} \left( S e^{\left(r_d-r_f \right)\tau} -k \right)$$ $$V_{f}= S e^{-r_f\tau} -k e^{-r_d \tau}$$ $$\frac{d V_{f}}{d S}= \frac{d }{d S} \left( S e^{-r_f\tau} -k e^{-r_d \tau} \right) = e^{-r_f\tau}$$ $$\Rightarrow \frac{d S}{d V_{f}}= e^{r_f\tau}$$

Now the Black Scholes Delta with respect to the forward rate can be easily computed using the chain rule:

$$\frac {\partial BS } {\partial V_f}= \frac {\partial BS } {\partial S} \frac{d S}{d V_{f}}$$ $$\Delta_{f, ua}= \Delta_{s, ua} e^{r_f\tau}$$ $$\Delta_{f, ua}= e^{-r_{f}\tau} \phi N{\left( \phi d_{1} \right )} e^{r_f\tau}$$ $$\Delta_{f, ua}= \phi N{\left( \phi d_{1} \right )}$$

$$\Delta_{f, pa}=\Delta_{s, pa} e^{r_f\tau}$$ $$\Delta_{f, pa}=\frac{K}{S} e^{-r_{d}\tau} \phi N{\left( \phi d_{2} \right )} e^{r_f\tau}$$ $$\Delta_{f, pa}=\frac{K}{S} e^{ \left( r_f-r_{d} \right)\tau} \phi N{\left( \phi d_{2} \right )}$$
$$\Delta_{s, ua}=e^{-r_{f}\tau} \phi N{\left( \phi d_{1} \right )}$$ $$\Delta_{s, pa}= \frac{K}{S} e^{-r_{d}\tau} \phi N{\left( \phi d_{2} \right )} =\frac{K}{F} e^{-r_{f}\tau} \phi N{\left( \phi d_{2} \right )}$$ $$\Delta_{f, ua}= \phi N{\left( \phi d_{1} \right )}$$ $$\Delta_{f, pa}=\frac{K}{S} e^{ \left( r_f-r_{d} \right)\tau} \phi N{\left( \phi d_{2} \right )} =\frac{K}{F} \phi N{\left( \phi d_{2} \right )}$$
Where, we have used $$F= S e^{\left(r_d-r_f \right)\tau}$$