### 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.

For the Premium Adjusted Spot Delta conventions, we get:

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

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