## FX Conventions

We list and derive the various FX conventions.

### Strike given Volatility and Delta

Some calculations in FX require calculation of strike for given level of volatility and delta. As would be expected, the formula depends on the delta conventions, and we start with the Unadjusted Spot delta:

$$\Delta_{s, ua}=e^{-r_{f}\tau} \phi N{\left( \phi d_{1} \right )}$$ $$N{\left( \phi d_{1} \right )}=\phi e^{r_{f}\tau} \Delta_{s, ua}$$ $$d_{1}=\phi N^{-1} \left( \phi e^{r_{f}\tau} \Delta_{s, ua} \right )$$ $$\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{F}{K} \right ) + \frac{\sigma^{2}\tau}{2} }\right) =\phi N^{-1} \left( \phi e^{r_{f}\tau} \Delta_{s, ua} \right )$$ $$\ln{\left (\frac{F}{K} \right ) + \frac{\sigma^{2}\tau}{2} } =\phi {\sigma \sqrt{\tau}} N^{-1} \left( \phi e^{r_{f}\tau} \Delta_{s, ua} \right )$$ $$\ln{K}=ln{F}+\frac{\sigma^{2}\tau}{2}- \phi {\sigma \sqrt{\tau}} N^{-1} \left( \phi e^{r_{f}\tau} \Delta_{s, ua} \right )$$ $$K=F e^{ \frac{\sigma^{2}\tau}{2}- \phi {\sigma \sqrt{\tau}} N^{-1} \left( \phi e^{r_{f}\tau} \Delta_{s, ua} \right ) }$$

The derivation for the Unadjusted forward delta is similar, and is not repeated here:

$$\Delta_{f, ua}= \phi N{\left( \phi d_{1} \right )}$$ $$K=F e^{ \frac{\sigma^{2}\tau}{2}- \phi {\sigma \sqrt{\tau}} N^{-1} \left( \phi \Delta_{s, ua} \right ) }$$

For the premium adjusted spot delta, we can only derive an implicit formula for K:

$$\Delta_{s, pa}= \frac{K}{F} e^{-r_{f}\tau} \phi N{\left( \phi d_{2} \right )}$$ $$N{\left( \phi d_{2} \right )}=\phi \Delta_{s, pa}\frac{F}{K} e^{r_{f}\tau}$$ $$d_{2} =\phi N^{-1} \left( \phi \Delta_{s, pa}\frac{F}{K} e^{r_{f}\tau} \right)$$ $$\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{F}{K} \right ) - \frac{\sigma^{2}\tau}{2} }\right) =\phi N^{-1} \left( \phi \Delta_{s, pa}\frac{F}{K} e^{r_{f}\tau} \right)$$ $$\ln{\left (\frac{F}{K} \right ) - \frac{\sigma^{2}\tau}{2} } =\phi {\sigma \sqrt{\tau}} N^{-1} \left( \phi \Delta_{s, pa}\frac{F}{K} e^{r_{f}\tau} \right)$$ $$\ln{F} - \frac{\sigma^{2}\tau}{2} =\phi {\sigma \sqrt{\tau}} N^{-1} \left( \phi \Delta_{s, pa}\frac{F}{K} e^{r_{f}\tau} \right)+ \ln{K}$$

Which will then need to be solved using root finding procedure. We get a similar implicit expression for premium adjusted forward delta, which can be easily verified by repeating the steps in the preceding derivation:

$$\Delta_{s, pa}= \frac{K}{F} \phi N{\left( \phi d_{2} \right )}$$ $$\ln{F} - \frac{\sigma^{2}\tau}{2} =\phi {\sigma \sqrt{\tau}} N^{-1} \left( \phi \Delta_{s, pa}\frac{F}{K} \right)+ \ln{K}$$