Gaussian Normalized Coordinates and Risk-Neutral CDF Deformations

Battle Count: 51

The paper provides deep theoretical foundations for understanding implied volatility surfaces, no-arbitrage conditions, and local volatility calibration. While not directly a trading strategy paper, its results on butterfly arbitrage detection (linear inequality in Bachelier, nonlinear factor in Black), CDF deformation bounds, and one-coordinate smile reconstruction are directly relevant to options desk risk management, volatility surface calibration, and arbitrage detection systems. The η-as-primitive-variable framework could inform parametric smile fitting used in trading systems.

The theoretical framework requires advanced knowledge of differential geometry, stochastic calculus, and option pricing theory. Practical implementation of the Bachelier linear butterfly criterion (Lq ≤ 0) is relatively straightforward. The Black butterfly factor and local volatility formulas require careful numerical differentiation. The one-coordinate ODE reconstruction (Section 5) involves solving a first-order nonlinear ODE. The Mills-ratio barrier checks are computationally simple but require careful handling of tail behavior.

The paper is purely theoretical with complete mathematical proofs. All derivations are self-contained and verifiable. No empirical data or code is required. However, the advanced differential geometry and stochastic calculus background makes independent verification non-trivial. A companion preprint (Sun [9]) is referenced for finite-strike monotonicity results.