Liquidity Measures¶
We have Limit Order Books of the stock GARAN for the trading days in 2017. Using bid and ask prices and volumes at different levels in the order book, we are interested in calculating the following liquidity measures for every minute interval, denoted by superscript k. Note that we are using a custom mean function defined as:¶
- $mean(\vec{\textbf{x}}) := \frac { \sum_{j=1}^{dim(\vec{\textbf{x}})} x_{j} } {dim(\vec{\textbf{x}})} $
i) Average $i^{th}$ level bid depth: $\text{$\bf ABD$}^k_i$¶
$= mean( ( \sum^i_{j=1} \text{bsize}^k_{j,1},...,\sum^i_{j=1}\text{bsize}^k_{j,|\tau_k|}) )$
ii) Average $i^{th}$ level bid TL depth: $\text{$\bf WABD$}^k_i$¶
$= mean( ( \sum^i_{j=1} \text{bid}^k_{j,1} \cdot \text{bsize}^k_{j,1},...,\sum^i_{j=1} \text{bid}^k_{j,|\tau_k|} \cdot \text{bsize}^k_{j,|\tau_k|}) )$
iii)Average $i^{th}$ level ask depth: $\text{$\bf AAD$}^k_i$¶
$= mean( ( \sum^i_{j=1} \text{asize}^k_{j,1},...,\sum^i_{j=1}\text{asize}^k_{j,|\tau_k|}) )$
iv) Average $i^{th}$ level ask TL depth: $\text{$\bf WAAD$}^k_i$¶
$= mean( ( \sum^i_{j=1} \text{ask}^k_{j,1} \cdot \text{asize}^k_{j,1},...,\sum^i_{j=1} \text{ask}^k_{j,|\tau_k|} \cdot \text{asize}^k_{j,|\tau_k|}) )$
v) Average $i^{th}$ level total depth: $\text{$\bf ATD$}^k_i = \text{$\bf ABD$}^k_i + \text{$\bf AAD$}^k_i $¶
vi) Average $i^{th}$ level total TL depth: $\text{$\bf WATD$}^k_i = \text{$\bf WABD$}^k_i + \text{$\bf WAAD$}^k_i $¶
vii) Average spread: $\text{$\bf AS$}^k$¶
$= mean( ( \text{ask}^k_{1,1} - \text{bid}^k_{1,1},...,\text{ask}^k_{1,|\tau_k|} - \text{bid}^k_{1,|\tau_k|}) )$
viii) Average (spread/mid_price): $\text{$\bf WAS$}^k_i$¶
$= mean( ( \frac {\text{ask}^k_{1,1} - \text{bid}^k_{1,1}}{ \frac {\text{ask}^k_{1,1} + \text{bid}^k_{1,1}} {2}},...,\frac {\text{ask}^k_{1,|\tau_k|} - \text{bid}^k_{1,|\tau_k|}}{ \frac {\text{ask}^k_{1,|\tau_k|} + \text{bid}^k_{1,|\tau_k|}} {2}}) )$
ix) Slopes: ask side slope, bid side slope and order book slope estimated by linear regression¶
, where
$i \in \{1,2,...,\text{LOB depth}=5\}$
$\tau_k= \{ \text{entry times in LOB which are inside the $k^{th}$ minute interval} \} $
$observable^{k}_{j,l}$ : $l^{th}$ value of the observable's values inside the $k^{th}$ minute interval at level $j$
We import the Limit Order Book data frame of GARAN from Jan 2, 2017, the first trading day of the year, as an example:¶
import pandas as pd
from LiqVars import *
date=filedates_str[0]
print(date)
df = get_df(date)
df
Calculating the measures¶
We would like to calculate the liquidity measures for the LOB information contained in each minute interval. We take 10:00 AM as our starting time.¶
In the end, we expect a single value for each measure for every minute.¶
Below we have all the entries that fall into the first minute interval [10:00:00.000 , 10:01:00.000) as an example.
We have set the parameters freq=1 and precision='m', meaning we calculate the measures every minute.
masks, indices = get_mask_to_avg(df, freq=1, precision='m')
df.iloc[masks[0][0]:masks[0][-1]]
Now we calculate the measures for Jan 2, 2017. Results are shown below, rounded with a relative error tolerance of 0.1% :¶
results = get_all(df,date)
get_significant_figures(results, 2, 0.001)
print([i[:5] for i in get_clocks_inbetween('10:00:00.000',get_clocks(df)[-1][-1], freq=1, precision='m') \
if not indices.count(i[:5])])
SLOPES¶
We start by calculating the slope of the map¶
$\hat{S}_{t_n}$ : Accumulated volume at $i^{th}$ level $\longrightarrow$ Average price per share
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; X_{\pm i}= \sum_{j=1}^i x_{\pm j}\longrightarrow
\hat{S}_{t_n}(X_{\pm i})= \frac { \sum_{j=1}^i p_{\pm j} x_{\pm j} } {X_{\pm i}} \;\; , \;\; x_{\pm j}$: Volume at $j^{th}$ level
for every time entry $t_n$ in the LOB, for ask side (+) and bid side (-). Here the best N ask prices are given by $p_{1}<...<p_{N}$
and the best N bid prices are given by $p_{-1}>...>p_{-N}$, which aligns with the order of levels in our LOB data frame.
Source: http://www.sciencedirect.com/science/article/pii/S0378426618301353
We take now the bid side as example and calculate $\hat{S}_{t_{3}}(X_{-i})$, where $t_{3}$ is the $3^{rd}$ row of our LOB:¶
bid_side_vols = get_vol(df,3,'bid')
bid_side_acc_vols = np.cumsum(bid_side_vols)
bid_side_prices = get_price(df,3,'bid')
print(bid_side_acc_vols,bid_side_prices)
bid_avg_price_per_share = [sum([i*k for i,k in zip(bid_side_prices[:m+1],bid_side_vols[:m+1])])/j \
for m,j in enumerate(bid_side_acc_vols)]
print(bid_avg_price_per_share)
We would like to apply linear regression on this map to get the slope but we observe a downwards slope on the bid side, whereas we would like to have an upwards one like we would get on the ask side.¶
def f(x,a,b): return a+b*x
popt_bid, _ = curve_fit(f, bid_side_acc_vols, bid_avg_price_per_share, p0=[1,1])
plt.plot(bid_side_acc_vols,bid_avg_price_per_share,'o')
plt.plot(bid_side_acc_vols,[f(i,*popt_bid) for i in bid_side_acc_vols])
print(popt_bid[-1])
To tackle this, we could reflect the graph at y-axis:¶
bid_side_acc_vols_reflected = -bid_side_acc_vols[::-1]
bid_avg_price_per_share_reflected = bid_avg_price_per_share[::-1]
popt_bid_reflected, _ = curve_fit(f, bid_side_acc_vols_reflected, bid_avg_price_per_share_reflected, p0=[1,1])
plt.plot(bid_side_acc_vols_reflected,bid_avg_price_per_share_reflected,'o')
plt.plot(bid_side_acc_vols_reflected,[f(i,*popt_bid_reflected) for i in bid_side_acc_vols_reflected])
print(popt_bid_reflected[-1])
For the limit order book slope we merge all the points from both ask and bid side and again apply linear regression:¶
ask_side_vols = get_vol(df,3,'ask')
ask_side_acc_vols = np.cumsum(ask_side_vols)
ask_side_prices = get_price(df,3,'ask')
ask_avg_price_per_share = [sum([i*k for i,k in zip(ask_side_prices[:m+1],ask_side_vols[:m+1])])/j \
for m,j in enumerate(ask_side_acc_vols)]
order_acc_vols = list(bid_side_acc_vols_reflected) + list(ask_side_acc_vols)
order_avg_price_per_share = list(bid_avg_price_per_share_reflected) + list(ask_avg_price_per_share)
popt_order, _ = curve_fit(f, order_acc_vols, order_avg_price_per_share, p0=[1,1])
plt.plot(order_acc_vols,order_avg_price_per_share,'o')
plt.plot(order_acc_vols,[f(i,*popt_order) for i in order_acc_vols])
print(popt_order[-1])
Finally, all together:¶
plot_index(df,3,date,dpi=300)
Mind that these slopes are averaged over every minute as described earlier to get the desired result:¶
get_significant_figures(results[results.columns[-3:]],2,0.001)