gs_quant.timeseries.statistics.exponential_std

exponential_std(x, beta=0.75)

Exponentially weighted standard deviation

Parameters:

• x (Series) – time series

• beta (float) – how much to weigh the previous price in the time series, thus controlling how much importance we place on the (more distant) past. Must be between 0 (inclusive) and 1 (exclusive)

Return type:

Series

Returns:

time series of standard deviation of the input series

Usage

Provides an unbiased estimator of exponentially weighted standard deviation of a series [\(X_0\), \(X_1\), \(X_2\), …]:

\(S_t = \sqrt{[EWMA(X_t^2) - EWMA(X_t)^2] * DF_t}\)

where \(EWMA(X_t)\) is the exponential moving average at \(t\) (see function exponential_moving_average()), \(DF_t\) is the debiasing factor (see Weighted sample variance for further details):

\(DF_t = \frac{(\sum_{i=0}^t w_i)^2} {(\sum_{i=0}^t w_i)^2 - \sum_{i=0}^t w_i^2}\)

where \(w_i\) is the weight assigned to \(i\) th observation:

\(w_i = (1-\beta)\beta^i\) for i<t; \(\beta^i\) for i=t

Examples

Generate price series and compute exponentially weighted standard deviation of returns

>>> prices = generate_series(100)
>>> exponential_std(returns(prices), 0.9)

See also

std() var() exponential_moving_average()