This paper introduces a new asymmetric stochastic volatility model designed to capture how both the sign and magnitude of past shocks influence future volatility. The proposed Leverage Propagation Stochastic Volatility (LPSV) model extends traditional formulations by allowing the feedback mechanism to evolve over time, offering a more persistent and realistic representation of leverage effects than standard asymmetric stochastic volatility models. Based on the intuition that the impact of negative shocks on volatility unfolds gradually, rather than instantaneously, the model encodes this ``leverage propagation'' directly in its structure. Under Gaussian assumptions, we establish stationarity conditions and derive closed-form expressions for variance, kurtosis, and a novel leverage propagation function that quantifies delayed transmission of asymmetry. A Monte Carlo study confirms the robustness of Bayesian inference via Markov chain Monte Carlo (MCMC), even under heavy-tailed shocks. In empirical applications, the LPSV model captures volatility clustering and asymmetric persistence more effectively than competing alternatives, using daily financial returns from the German DAX and U.S. S&P 500. Moreover, the model captures prolonged volatility responses to non-financial shocks -illustrated through PM2.5 air pollution data from Madrid during Saharan dust events, demonstrating its broader relevance for environmental volatility modelling. These findings highlight the versatility of the model to trace the dynamics of delayed volatility sensitive to sign in different domains where understanding the persistence of risk is crucial.