Gatheral's Conjecture Revisited

Battle Count: 74

The paper is highly relevant to quantitative finance practitioners dealing with volatility derivatives, variance swaps, and model calibration. It provides a rigorous counterexample to Gatheral's conjecture, showing that the Heston model with perfect negative correlation produces integrated variance that is strictly smaller in convex order than its local-volatility projection. This has direct implications for pricing convex payoffs on realized variance, understanding convexity adjustments, and model selection. However, it is a theoretical result rather than a directly implementable trading strategy.

The mathematical content is extremely advanced, involving stochastic calculus, PDE theory, convex ordering, Chebyshev inequalities, Duhamel formulas, CIR process analysis, affine transforms, saddle-point methods, Malliavin-type arguments, and martingale problem theory. While the results are purely theoretical and do not require computational implementation, understanding and verifying the proofs requires deep expertise in mathematical finance and stochastic analysis. The paper spans 34 pages with extensive appendices.

Pure mathematical paper with complete, self-contained proofs. All theorems, lemmas, and propositions are fully proved with explicit constructions. The paper includes detailed appendices covering mimicking theorems, regularity of the projected surface, well-posedness of the local-volatility SDE, localized density estimates, and smooth approximation arguments. No computational experiments are needed to verify the results.