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 $t^{-1\mathrm{/}Y}$ from the $Y$-stable domain of attraction, yields $c(t,0)=d_1{t}^{1\mathrm{/}Y}+d_{2}t+o(t)$ 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.