  We propose a doubly subordinated Levy process, NDIG, to model the time series
properties of the cryptocurrency bitcoin. NDIG captures the skew and fat-tailed
properties of bitcoin prices and gives rise to an arbitrage free, option
pricing model. In this framework we derive two bitcoin volatility measures. The
first combines NDIG option pricing with the Cboe VIX model to compute an
implied volatility; the second uses the volatility of the unit time increment
of the NDIG model. Both are compared to a volatility based upon historical
standard deviation. With appropriate linear scaling, the NDIG process perfectly
captures observed, in-sample, volatility.
