# Kyle's LambdaÂ¶

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.

## DefinitionÂ¶

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,

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

## Source CodeÂ¶

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

It returns \(\lambda \times 10^6\)

```
# 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
```