This paper investigated the relationship between cryptocurrencies and emerging stock market indices using fractional integration and co-integration technique. Particularly, fractional integration is applied to examine stochastic properties of individual assets and fractional cointegration to analyse bivariate connectedness. Our findings unveil the absence of mean reversion in majority cases which indicates high persistence in series. Furthermore, bivariate analysis reveals disconnection between cryptocurrencies prices and stock indices. Surprisingly, a different picture emerges on using conditional volatility instead of prices. Like, conditional volatility-based estimation uncovers evidence of mean reversion in univariate analysis as expected. There is some evidence of cointegration on volatility grounds between cryptocurrencies and emerging stock market indices. Our findings implies that investment decision regarding digital currencies should be taken cautiously. As cryptocurrencies are extremely volatile with high degree of persistence which can make them counterproductive.