In this paper, we investigate the stochastic properties of six major cryptocurrencies and their bilateral linkages with six stock market indices using fractional integration techniques. From the univariate analysis, we observe that for Bitcoin and Ethereum, the unit root null hypothesis cannot be rejected; for Litecoin, Ripple and Stellar, the order of integration is found to be significantly higher than 1; for Tether, however, we find evidence in favour of mean reversion. For the stock market indices, the results are more homogeneous and the unit root cannot be rejected in any of the series, with the exception of VIX where mean reversion is obtained. Concerning bivariate results within the cryptocurrencies and testing for cointegration, we provide evidence of no cointegration between the six cryptocurrencies. Along the same lines, testing for cointegration between the cryptocurrencies and the stock market indices, we find evidence of no cointegration, which implies that the cryptocurrencies are decoupled from the mainstream financial and economic assets. The findings in this paper indicate the significant role of cryptocurrencies in investor portfolios since they serve as a diversification option for investors, confirming that cryptocurrency is a new investment asset class.