In our recent paper “The Statistics of Bitcoin and Cryptocurrencies” (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2872158), we show the statistical properties of the most important cryptocurrencies, of which Bitcoin is the most prominent example. We characterize their exchange rates versus the US Dollar by fitting parametric distributions to them. It is shown that returns are clearly non-normal, with standard heavy-tailed distributions giving good descriptions of the data. The results are important for investment and risk management purposes.
The histogram of the Bitcoin/USD exchange rate returns. The normal as well as various heavy-tailed distributions are fitted to the data (from 2014 until 2016).
You can see that the returns are clearly not normal. Once you go to the class of heavy-tailed distributions, the fit becomes reasonably and you can, for simplicity, choose the t-distribution. This result resembles results from traditional fiat currency exchange rates.