Martingale expansion for stochastic volatility

Battle Count: 104

Highly relevant to quantitative trading and derivatives pricing. The paper provides a unified, simplified framework for computing first-order corrections to Black-Scholes option prices under stochastic volatility models. The implied volatility expansion (Corollary 1, Remark 2) and skew formula (Remark 3) are directly applicable to option pricing, calibration, and hedging. The framework covers both small vol-of-vol and fast mean-reversion regimes, which are the two most common SV model classes in practice.

The theoretical framework is elegant but requires strong background in stochastic calculus, martingale theory, and asymptotic analysis. For practical implementation, one needs to: (1) identify the appropriate v_ε for the model, (2) compute the joint limit (X,Y) of (X_ε, Y_ε), (3) evaluate E[Y|X=x], and (4) apply the expansion formula. The Bergomi model example (Section 3) shows the computation is tractable for specific models. No code is provided.

The paper provides complete mathematical proofs (Section 4) for all main results (Theorem 1, Corollary 1). The methodology is fully analytical with no computational experiments. All assumptions are clearly stated. However, no numerical code or implementation examples are provided.