# Kyle's Lambda¶

A measure of market impact cost from Kyle (1985), which can be interpreted as the cost of demanding a certain amount of liquidity over a given time period.

## Definition¶

Following Hasbrouck (2009) and Goyenko, Holden, Trzcinka (2009), Kyle's Lambda for a given stock $i$ and day $t$, is calculated as the slope coefficient $\lambda_{i,t}$ in the regression:

ret_{i,t,n}= \delta_{i,t} + \lambda_{i,t} S_{i,t,n}+\epsilon_{i,t,n}

where for the $n$th five-minute period on date $t$ and stock $i$, $ret_{i,t,n}$ is the stock return and $S_{i,t,n}$ is the sum of the signed square-root dollar volume, that is,

S_{i,t,n}=\sum_k{sign}(dvol_{i,t,n,k}) \sqrt{dvol_{i,t,n,k}}

## Source Code¶

This example Python code is not optimized for speed and serves only demonstration purpose. It may contain errors.

It returns $\lambda \times 10^6$

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 # KylesLambda.py import numpy as np name = 'KylesLambda' description = """ A measure of market impact cost from Kyle (1985), which can be interpreted as the cost of demanding a certain amount of liquidity over a given time period. Result is Lambda*1E6. """ vars_needed = ['Price', 'Volume', 'Direction'] def estimate(data): price = data['Price'].to_numpy() volume = data['Volume'].to_numpy() direction = data['Direction'].to_numpy() sqrt_dollar_volume = np.sqrt(np.multiply(price, volume)) signed_sqrt_dollar_volume = np.abs( np.multiply(direction, sqrt_dollar_volume)) # Find the total signed sqrt dollar volume and return per 5 min. timestamps = np.array(data.index, dtype='datetime64') last_ts, last_price = timestamps[0], price[0] bracket_ssdv = 0 bracket = last_ts + np.timedelta64(5, 'm') rets, ssdvs, = [], [] for idx, ts in enumerate(timestamps): if ts <= bracket: bracket_ssdv += signed_sqrt_dollar_volume[idx] else: ret = np.log(price[idx-1]/last_price) if not np.isnan(ret) and not np.isnan(bracket_ssdv): rets.append(ret) ssdvs.append(bracket_ssdv) # Reset bracket bracket = ts + np.timedelta64(5, 'm') last_price = price[idx] bracket_ssdv = signed_sqrt_dollar_volume[idx] # Perform regression. x = np.vstack([np.ones(len(ssdvs)), np.array(ssdvs)]).T try: coef, _, _, _ = np.linalg.lstsq(x, np.array(rets), rcond=None) except np.linalg.LinAlgError: return None else: return None if np.isnan(coef[1]) else coef[1]*1E6 

Last update: May 26, 2020