## FX Conventions

We list and derive the various FX conventions.

### At the Money (ATM) Strikes

ATM strike is defined as the strike at which the delta of a call option is equal to the negative of the delta a put option. We therefore have four alternative definitions of ATM strikes corresponding to the four different delta conventions. Let's first derive a formula for the ATM strike using Unadjusted Spot Delta conventions:

$$e^{-r_{f}\tau} N{\left(d_{1} \right )}= e^{-r_{f}\tau} N{\left(-d_{1} \right )}$$ $$N{\left(d_{1} \right )}= N{\left(-d_{1} \right )}$$ $$d_{1}= -d_{1}$$ $$d_{1}= 0$$ $$\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{F}{K} \right ) + \frac{\sigma^{2}\tau}{2} }\right)=0$$ $$\ln{\left (\frac{F}{K} \right ) + \frac{\sigma^{2}\tau}{2} }=0$$ $$\ln{K} =ln{F} + \frac{\sigma^{2}\tau}{2}$$ $$K =F e^{\frac{\sigma^{2}\tau}{2} }$$ $$\Rightarrow K_{ATM} =F e^{\frac{\sigma^{2}\tau}{2} }$$

It is easily verified that the Unadjusted Forward Delta conventions give same formula for the ATM strike.

$$\frac{K}{S} e^{-r_{d}\tau} N{\left( d_{2} \right )}=\frac{K}{S} e^{-r_{d}\tau} N{\left( - d_{2} \right )}$$ $$N{\left(d_{2} \right )}= N{\left(-d_{2} \right )}$$ $$d_{2}= -d_{2}$$ $$d_{2}= 0$$ $$\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{F}{K} \right ) - \frac{\sigma^{2}\tau}{2} }\right)=0$$ $$\ln{\left (\frac{F}{K} \right ) - \frac{\sigma^{2}\tau}{2} }=0$$ $$\ln{K} =ln{F} - \frac{\sigma^{2}\tau}{2}$$ $$K =F e^{\frac{- \sigma^{2}\tau}{2} }$$ $$\Rightarrow K_{ATM} =F e^{-\frac{\sigma^{2}\tau}{2} }$$

Again, it is easily verified that the Premium adjusted Forward Delta conventions give the same formula for the ATM strike.