# high-low spread estimator (hlse)
def hlse(ohlc_df, frequency='daily'):
    """
    Computes the high-low spread estimator, an estimate of bid-offer spreads, a measure of liquidity risk.
    See Corwin & Schultz (2011) for details: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1106193

    Parameters
    ----------
    ohlc_df: DataFrame
        DataFrame with DatetimeIndex and Open, High, Low and Close (OHLC) prices from which to compute the high-low spread estimates.
    frequency: str, {'daily', 'weekly', 'monthly'}, default 'daily'
        daily: daily bid-offer spread estimate.
        weekly: weekly bid-offer spread estimate, resampled over a weekly frequency as the mean of daily estimates.
        monthly: monthly bid-offer spread estimate, resampled over a monthly frequency as the mean of daily estimates.

    Returns
    -------
    S: Series
        Datetimeindex and time series of high-low spread estimates.
    """

    # define vars: mid, 2 day high and 2 day low vars
    mid, high_2d, low_2d = (ohlc_df.high + ohlc_df.low)/2, ohlc_df.high.rolling(2).max(), ohlc_df.low.rolling(2).min()

    # compute adjustment for overnight price moves
    ohlc_df['gap_up'], ohlc_df['gap_down'] = ohlc_df.low - ohlc_df.close.shift(1), ohlc_df.high - ohlc_df.close.shift(1)
    # adjustment for gap up
    ohlc_df['high_adj'], ohlc_df['low_adj'] = np.where(ohlc_df.gap_up > 0, ohlc_df.high - ohlc_df.gap_up, ohlc_df.high), np.where(ohlc_df.gap_up > 0, ohlc_df.low - ohlc_df.gap_up, ohlc_df.low)
    # adjustment for gap down
    ohlc_df['high_adj'], ohlc_df['low_adj'] = np.where(ohlc_df.gap_down < 0, ohlc_df.high - ohlc_df.gap_down, ohlc_df.high), np.where(ohlc_df.gap_down < 0, ohlc_df.low - ohlc_df.gap_down, ohlc_df.low)

    # B beta
    B = (np.log(ohlc_df.high_adj/ohlc_df.low_adj))**2 + (np.log(ohlc_df.high_adj.shift(1)/ohlc_df.low_adj.shift(1)))**2
    # G gamma
    G = (np.log(high_2d/low_2d))**2
    # alpha
    alpha = ((np.sqrt(2 * B) - np.sqrt(B)) / (3 - 2 * np.sqrt(2))) - (np.sqrt(G/(3 - 2 * np.sqrt(2))))
    # replace negative values by 0
    alpha = pd.Series(np.where(alpha < 0, 0, alpha), index=alpha.index)
    # substitute alpha into equation 14 to get high-low spread estimate S
    S = (2 * (np.exp(alpha) - 1)) / (1 + np.exp(alpha))
    # resample using daily mean
    if frequency == 'weekly':
        S = S.resample('W').mean()
    if frequency == 'monthly':
        S = S.resample('M').mean()
    # drop NaNs
    S.dropna(inplace=True)

    return S