Convergence of the ARMA-GARCH Implied Calibration Algorithm

Abstract

This paper explores the convergence properties of the randomized stochastic projected gradient-free (RSPGF) algorithm for calibrating the ARMA-GARCH model using market option prices. The calibration problem is framed as a stochastic optimization task within the risk-neutral probability measure, addressing the discrepancy between theoretical and market-implied option prices. The ARMA-GARCH model, combining autoregressive moving average and generalized autoregressive conditional heteroskedasticity components, captures the volatility clustering and dynamics of financial asset returns. The proposed RSPGF algorithm integrates gradient-free optimization with random smoothing techniques to handle the nonlinearity and complexity of the model, where analytical gradient computation is infeasible. A Monte Carlo method estimates the loss function, ensuring unbiased estimates with bounded variance under time series stationarity. The paper proves a theorem establishing the Lipschitz continuity and boundedness properties of the loss function, providing theoretical guarantees for the algorithm’s convergence to an ε-stationary point at a rate of O(1/√N). These findings confirm the algorithm’s applicability for robust calibration of ARMA-GARCH models, offering practical insights for financial modeling and option pricing.