Higher-order ATM asymptotics for the CGMY model via the characteristic function

Battle Count: 123

The paper provides rigorous theoretical foundations for short-time ATM option pricing under CGMY models, which are widely used in practice for equity and FX derivatives. The explicit closed-form coefficients enable faster calibration and more accurate pricing of short-dated options. However, the results are primarily theoretical and require significant mathematical sophistication to implement. The bifurcation structure at Y=3/2 and Y=5/4 has practical implications for model selection and calibration. The work is more relevant to quantitative researchers and model developers than to direct trading strategy implementation.

The mathematical derivation involves complex analysis of characteristic functions, Fourier integrals, Laplace methods, and careful domain partitioning with dynamic cutoffs. Implementing the numerical verification requires high-precision quadrature with careful handling of singularities near w=0 and the Laplace concentration scale. The closed-form coefficients (involving gamma functions, trigonometric terms, and complex exponentials) are straightforward to evaluate, but the full five-term expansion with correct ordering depends on the value of Y relative to bifurcation points 3/2 and 5/4. The proof techniques (dominated convergence, Taylor-Lagrange remainders, super-exponential bounds) are mathematically demanding.

The paper provides fully explicit closed-form formulas for all coefficients (d1, d2, a2,1, a4,1, a1,2), detailed proofs with all steps, and comprehensive numerical verification tables (Tables 1-4) with specific parameter sets. The methodology is self-contained within the Fourier/characteristic function framework. However, no code repository is provided, and the numerical quadrature implementation details are not fully specified.