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In this paper, fractality and stationarity of average Sensex value of stock market have been investigated by enlightening the self-similarity characteristics and nature of frequency oscillation of the two crucial constraints, namely, the average Sensex values of BSE and NASDAQ. The time series of these parameters between 31st Oct 2007 and 1st Nov 2017, of Bombay Stock Exchange (BSE) and American stock exchange (NASDAQ) have been chosen for extracting the nature of scaling (fractality) and stationary behavior using statistical techniques. Methodologies like General Hurst Estimation (GHE) and Higuchi’s Fractal dimension (HFD) have been used to compute Hurst Exponent for the time series. According to experimental results, the said time series exhibits SRD - Short Range Dependent anti-persistent character. The stationarity behavior of the data-series has been determined using Continuous Wavelet Transform (CWT) method. Both the time series have been found to exhibit non-stationarity.