Higher-order ATM asymptotics for the CGMY model via the characteristic function
Using only the characteristic function, we derive short-time at-the-money (ATM) call-price asymptotics for the exponential CGMY model with activity parameter $Y\in(1,2)$. The Lipton--Lewis formula expresses the normalized ATM call price, denoted $c(t,0)$, in terms of the characteristic exponent, which, upon rescaling at the rate $t^{-1/Y}$ from the $Y$-stable domain of attraction, yields $c(t,0) = d_{1} t^{1/Y} + d_{2} t + o(t)$ as $t\downarrow 0$. The first-order coefficient $d_{1}$ is the known stable limit from the domain of attraction of a symmetric $Y$-stable law, and $d_{2}$ is given by an explicit integral involving the characteristic exponent and the limiting stable exponent. We then extract closed-form higher-order coefficients by keeping the full Lipton--Lewis integrand intact and introducing a dynamic cutoff that partitions the domain into inner, core, and tail regions, establishing the expansion with controlled remainder. All coefficients are verified numerically against existing closed-form expressions where available.