Every quant faces the same dilemma the moment they build features from prices. A price series has memory, the level today tells you something about the level tomorrow, but it is non-stationary, so a model cannot learn from it. The textbook fix is to take returns, which are stationary, but a full first difference erases almost all of the memory: returns are very close to unpredictable from their own past. Fractional differentiation is the elegant middle ground. Instead of differencing once (an integer order of one) or not at all (order zero), it differences by a fractional amount, just enough to make the series stationary while keeping as much memory as possible. This lesson defines the fractional differencing operator, derives the weight series and its recursion, computes those weights and a differenced value by hand, and walks through the trade-off between the ADF stationarity test and the correlation with the original series that tells you how much memory survived. It builds directly on the stationarity lesson and feeds the features used in the rest of the module.
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