This paper deals with the problem of estimating the pointwise regularity of multifractional Brownian motion, assumed as a model of stock price dynamics. We (a) correct the shifting bias affecting a class of absolute moment-based estimators and (b) build a data-driven algorithm in order to dynamically check the local Gaussianity of the process. The estimation is therefore performed for three stock indices: the Dow Jones Industrial Average, the FTSE 100 and the Nikkei 225. Our findings show that, after the correction, the pointwise regularity fluctuates around 1/2 (the sole value consistent with the absence of arbitrage), but significant deviations are also observed.