Returns of risky assets and their time-dependent volatility are often jointly modelled by stochastic volatility models (SVMs). Over the last few decades, several SVMs have been proposed to adequately capture the features of the relationship between the return and its volatility. The earliest SVM considers a hierarchical model, with current return as a function of the current latent volatility, which is further modelled as an auto-regressive process. In an attempt to make SVM more appropriate for complex realistic market behaviour, leverage parameter was introduced, which however led to violation of martingale difference property of the risky part of return (a necessary mean-zero condition that prevents arbitrage opportunities under mild regularity conditions). Subsequently, alternative SVMs had been developed and are currently in use. In this article, we propose mean-corrections for several generalizations of SVM with leverage that capture the complex market behaviour as well as satisfy martingale difference property of returns. We also establish a few theoretical results to characterize the key desirable features of these models, and present comparison with other popular competitors. Furthermore, three real-life examples (CITI bank stock price, euro–USD rate and S&P 500 index returns) have been used to demonstrate the performance of this new class of SVMs.