Advanced-type equilibria for a general class of zero-sum stochastic differential games have been studied in part by Escobedo-Trujillo et al. (J Optim Theory Appl 153:662–687, 2012), in which a comprehensive study of the so-named bias and overtaking equilibria was provided. On the other hand, a complete analysis of advanced optimality criteria in the context of optimal control theory such as bias, overtaking, sensitive discount, and Blackwell optimality was developed independently by Jasso-Fuentes and Hernandez-Lerma (Appl Math Optim 57:349–369, 2008; J Appl Probab 46:372–391, 2009; Stoch Anal Appl 27:363–385, 2009). In this work we try to fill out the gap between the aforementioned references. Namely, the aim is to analyze Blackwell-Nash equilibria for a general class of zero-sum stochastic differential games. Our approach is based on the use of dynamic programming, the Laurent series and the study of sensitive discount optimality.