Martingale expansion for stochastic volatility

The martingale expansion provides a refined approximation to the marginal distributions of martingales beyond the normal approximation implied by the martingale central limit theorem. We develop a martingale expansion framework specifically suited to continuous stochastic volatility models. Our approach accommodates both small volatility-of-volatility and fast mean-reversion models, yielding first-order perturbation expansions under essentially minimal conditions.

💡 Research Summary

The paper develops a martingale expansion framework tailored to continuous stochastic volatility (SV) models, providing a refined approximation of the marginal distribution of the asset price beyond the normal approximation given by the martingale central limit theorem (MCLT). The authors consider a generic SV model with zero interest rates:

(dS_t = \sqrt{V_t},S_t,dB_t,\qquad B_t = \rho W_t + \sqrt{1-\rho^2},W_t^\perp,)

where ((W,W^\perp)) is a two‑dimensional Brownian motion, (\rho\in(-1,1)), and the variance process (V_t) depends on a small parameter (\varepsilon>0). As (\varepsilon\to0), the integrated variance (\int_0^T V_t^\varepsilon dt) converges in probability to a deterministic limit (v_\varepsilon). Under this scaling, the log‑price (\log S_T^\varepsilon) converges in law to a normal distribution with mean (-v_0/2) and variance (v_0).

The core contribution is a first‑order perturbation result that holds under essentially minimal assumptions, notably avoiding the heavy Malliavin calculus machinery used in earlier works. The authors introduce the standardized pair

(X_\varepsilon = \frac{1}{\sqrt{v_\varepsilon}}\int_0^T \sqrt{V_t^\varepsilon},dB_t,\qquad
Y_\varepsilon = \frac{1}{\varepsilon}\Bigl(\int_0^T V_t^\varepsilon dt - v_\varepsilon\Bigr),)

and assume that ((X_\varepsilon,Y_\varepsilon)) converges in law to ((X,Y)) with (X) standard normal and the conditional mean function (x\mapsto E