ARCHGARCHmalleilla

ARCHGARCHmalleilla is a class of econometric models designed to describe time-varying volatility in financial returns by extending ARCH and GARCH frameworks with a path-dependent component inspired by Malliavin calculus. The model aims to capture how past shocks influence not only the current conditional variance but also the sensitivity of volatility to the entire shock history, allowing for richer dynamics such as long memory and nonlinear leverage effects. It is proposed to improve volatility forecasting and risk assessment in markets characterized by heavy tails and regime-like behavior.

The term ARCHGARCHmalleilla emerged in the econometric literature in the early 2020s, attributed to researchers seeking to fuse traditional conditional-variance models with stochastic calculus tools. The typical specification includes a baseline ARCH/GARCH structure for the conditional variance h_t, augmented by a Malliavin-derived term M_t that represents the pathwise response of volatility to past innovations. A schematic form is h_t = omega + sum alpha_i epsilon_{t-i}^2 + sum beta_j h_{t-j} + gamma M_t, with M_t estimated or approximated through simulation or functional approximations.

Estimation relies on quasi-maximum likelihood or Bayesian procedures, often requiring computational techniques such as particle filtering or Monte Carlo integration to handle the Malliavin component. Practical implementations emphasize diagnostics for stationarity, positive variance, and out-of-sample forecast performance.

Applications include financial return forecasting, value-at-risk, and option pricing where standard ARCH/GARCH models may struggle to capture complex volatility features. Criticisms focus on identification, potential overfitting, and high computational demands. See also ARCH, GARCH, stochastic volatility, Malliavin calculus.