By: Dean Markwick's Blog -- Julia

Re-posted from: https://dm13450.github.io/2022/05/11/modelling-microstructure-noise-using-hawkes-processes.html

Microstructure noise is where the
price we observe isn’t the ‘true’ price of the underlying asset. The
observed price doesn’t diffuse as we assume in your typical
derivative pricing models, but instead, we see
some quirks in the underlying data. For example, there is an explosion of
realised variance as we use finer and finer time subsampling periods.

Last month I wrote about
calculating realised volatility
and now I’ll be taking it a step further. I’ll show you how this
microstructure noise manifests itself in futures trade data and how I
use a Hawkes process to come up with a price formation model that fits
the actual data.

The original work was all done in
Modeling microstructure noise with mutually exciting point processes. I’ll
be explaining the maths behind it, showing you how to fit the models
in Julia and hopefully educating you on this high-frequency finance
topic.

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A Bit of Background

My Ph.D. was all about using Hawkes processes to model when things
happen and how these things are self-exciting. You may have read my
work on Hawkes process before, either through my Julia package
HawkesProcess.jl,
or examples of me
using Hawkes processes to model terror attack data or just
how to calculate the deviance information criteria of a Hawkes process. The
real hardcore might have read my Ph.D. thesis.

But how can we use a Hawkes process to model the price of some
financial asset? There is a vast amount of research and work about how Hawkes
processes can be used in price formation processes for financial
instruments and high-frequency types of problems and this post will
act as a primer to anyone interested in using Hawkes processes for
such problems.

The High-Frequency Problem

At short timescales, a price moves randomly rather than with any
trend. Amazon stock might trade thousands of times in a minute but
that’s supply and demand changing rather than people thinking
Amazon’s future is going to change from one minute to the next. So we need
a different way of thinking about how prices move at short timescales
compared to longer timescales.

We can build nice mathematical models guessing how a price might move;
it might move like a random walk or maybe a random walk with jumps in
the price now and then. But, no matter what model we use, it must
match up with what is observed in the real world. One of
these observations is a phenomenon called ‘microstructure noise’ and we
only see it in high-frequency data.

Microstructure noise is a catch-all term for different things happening
in the market. This includes, bid-ask bounce, people buy and sell at
two different prices, so looks like the price is moving, but in
reality, just oscillating around the mid-price. The discreteness of
prices at these time scales also plays a part. There is a minimum
increment level that prices change buy and this can have real effects
on how prices move. Exchanges need to pay attention to their tick
sizes, as it can help or hinder liquidity if they are set
incorrectly. This then has a real effect that we can observe when we
calculate a realised volatility.

Realised volatility is measuring how much the price moved in a
period. These high-frequency effects are going to give the impression
of more movement than what the ‘true’ volatility is. So we end up
seeing our measurement of volatility explode as the time scale we use
gets smaller and smaller. Calculating the volatility using 1-second
intervals gives a larger value than if we used 1 minute
intervals. This means that our volatility estimation depends on the
time scale used, so what is the ‘real’ volatility?

We aren’t interested in working out what the real volatility
is. Instead, we want to build a model for a price that displays this
volatility scaling effect.

The Hawkes Process Model for a Price

How do we use Hawkes processes to build a model that will have this microstructure noise?

Lets call the price at time \(t\), \(X(t)\) and guess that it moves by
summing up the positive jumps \(N_1(t)\) and negative jumps \(N_2(t)\)
that also happen at time \(t\).

\[X(t) = N_1(t) – N_2(t)\]

When do these jumps occur? How are these jumps distributed? This is
where we use the Hawkes process.

A Hawkes process is a self-exciting point process. When something
happens it increases the probability of another event happening. This
is the self-exciting behaviour we want. Every time there is a positive
jump, there is an increase in the probability of a negative jump
happening and, likewise, when there is a negative jump there is a greater
probability of a positive jump.

Each jump causes the probability of the other jump happening like in
this picture

When someone buys and pushes the price higher by removing that
liquidity, it’s more likely that someone will now sell at the new
higher price introducing some downward pressure. This is mean
reversion where prices move higher and then outside forces
push it lower and vice versa.

With Hawkes processes there are three parameters:

• The background rate or when the jumps randomly occur. This is common to both positive and negative jumps.
• \(\kappa\) – the ‘force’ that pushes and increases the probability of the other jump happening.
• The kernel \(g(t)\) dictates how long the force lasts. This is an exponential decay with parameter \(\beta\).

We will fit a Hawkes process to a price series
to infer the 3 parameters that describe how the jumps behave. This
model will hopefully replicate the ‘microstructure noise’ effects we
see in practise.

Futures Trade Data

In the early days of my Ph.D., I answered an email that was advertising
for early grad students to do some prop trading. As part of the
interview, they gave me some data and asked me to write a simple
moving cross-over strategy. I failed miserably as I never
heard back from them. But I did get some nice data, which now that I’m
older and wiser, recognise as reported trades from a futures
exchange. This is the data we will be using today to calculate the
mode and luckily it’s similar to the data they use in the original
paper.

using CSV
using DataFrames, DataFramesMeta
using Plots
using Dates
using Statistics

All the usual packages when working with data in Julia.

rawData = CSV.read("fgbl__BNH14_clean.csv", DataFrame, header=false)
rename!(rawData, [:UnixTime, :Price, :Volume, :DateTime]);
first(rawData, 5)

To clean the data we convert the Unix timestamp to an actual DateTime object and pull out the hour and the date of the trade.

cleanData = @transform(rawData, DateTimeClean = DateTime.(:DateTime, dateformat"mm/dd/yyyyHH:MM:SS.sss"), 
                                DateTimeUnix = unix2datetime.(:UnixTime ./ 1000) )
cleanData = @transform(cleanData, Date = Date.(:DateTimeUnix),
                                  Hour = hour.(:DateTimeUnix));
first(cleanData[:,[:UnixTime, :DateTimeClean, :DateTimeUnix]], 5)

To get an idea of the data we are looking at I aggregate the total
number of trades and total volume of the trades over each day and plot
that as a time series.

dayData = groupby(cleanData, :Date)
dailyVolumes = @combine(dayData, TotalVolume = sum(:Volume),
                                  TotalTrades = length(:Volume),
                                  FirstTradeTime = minimum(:DateTimeUnix))

xticks = minimum(dailyVolumes.Date):Month(2):maximum(dailyVolumes.Date)
xticks_labels = Dates.format.(xticks, "yyyy-mm")

p1 = plot(dailyVolumes.Date, dailyVolumes.TotalVolume, seriestype=:scatter, label="Daily Volume", legend = :topleft, xticks = (xticks, xticks_labels))
p2 = plot(dailyVolumes.Date, dailyVolumes.TotalTrades, seriestype=:scatter, label= "Daily Number of Trades", legend = :topleft, xticks = (xticks, xticks_labels))
plot(p1, p2, fmt=:png)

It takes a while for the trading to take off in this future
contract. This is where it slowly becomes the front-month contract and
then is the most active.

Also, because trading doesn’t cross over the daylight saving dates, we don’t have to worry about timezones. Always a bonus!

What about if we look at what hour is the most active?

hourDataG = groupby(cleanData, [:Date, :Hour])
hourDataS = @combine(hourDataG, TotalHourVolume = sum(:Volume),
                                  TotalHourTrades = length(:Volume))
hourDataS = leftjoin(hourDataS, dailyVolumes, on=:Date)

hourDataS = @transform(hourDataS, FracVolume = :TotalHourVolume ./ :TotalVolume)

hourDataG = groupby(hourDataS, :Hour)
hourDataS = @combine(hourDataG, MeanFracVolume = mean(:FracVolume),
                                 MedianFracVolume = median(:FracVolume))
sort!(hourDataS, :Hour)
bar(hourDataS.Hour, hourDataS.MedianFracVolume * 100, title = "Fraction of Daily Volume", label=:none, fmt=:png)

Early in the morning (just after the exchange opens) and late afternoon (when the Americans start trading) is where there is the most activity.

For our analysis, we are going to be focused on the hours 14, 15, 16
to make sure that we have the most active period and this is the same
as what the original paper did, took a subset of the day.

How to Calculate the Volatility Signature

Let \(X(t)\) be the price of the future at time \(t\). The signature is the quadratic variation over a window of \([0, T]\), which is more commonly known as the realised volatility:

\[C(\tau) = \frac{1}{T} \Sigma _{n=0} ^{T/\tau} \mid X((n+1) \tau) – X(n \tau) \mid ^2 .\]

\(\tau\) is our sampling frequency, say every minute, etc.

To calculate the volatility across the trades we have to pay particular attention to the fact that these trades are irregularly spaced, so we need to fill forward the price for every \(t\) value.

function get_price(t::Number, prices, times)
    ind = min(searchsortedfirst(times, t), length(times))
    sp = ind == 0 ? 0 : prices[ind]
end

function get_price(t::Array{<:Number}, prices, times)
    res = Array{Float64}(undef, length(t))
    for i in eachindex(t)
        res[i] = get_price(t[i], prices, times)
    end
    res
end

Our get_price function here will return the last price before time \(t\).

To calculate the signature value we chose a \(\tau\) value, generate
the indexes between 0 and \(T\) using a \(\tau\) step size. Pull the
price at those times and calculate the quadratic variation. Again, we
add a method to calculate the signature for different \(\tau\)’s.

function signature(tau::Number, x, t, maxT)
    inds = collect(0:tau:maxT)
    prices = get_price(inds, x, t)
    
    rets = prices[2:end] .- prices[1:(end-1)]
    (1/maxT) * sum(abs.(rets) .^ 2)
end

function signature(tau::Array{<:Number}, x, t, maxT)
   res = Array{Float64}(undef, length(tau))
    for i in eachindex(res)
        res[i] = signature(tau[i], x, t, maxT)
    end
    res
end

Now let’s apply this to the data. We are only interested when the
future was actively trading and in the hours between 14:00 and 16:00.
We convert the times into seconds since 15:59:59 and calculate the
signature for all the dates, before taking the final average.

We are taking the log price of the last trade to represent our actual
\(X(t)\). We just look at 2014 dates too as that is when the future is
active.

cleanData2014 = @subset(cleanData, :Date .>= Date("2014-01-01"))

uniqueDates = unique(cleanData2014.Date)

eventList = Array{Array{Float64}}(undef, length(uniqueDates))
priceList = Array{Array{Float64}}(undef, length(uniqueDates))

signatureList = Array{Array{Float64}}(undef, length(uniqueDates))
avgSignature = zeros(length(1:1:200))

for (i, dt) in enumerate(uniqueDates)
   
    subData = @subset(cleanData2014, :Date .== dt, :Hour .<= 16, :Hour .>= 14)
    eventList[i] = getfield.(subData.DateTimeClean .- (DateTime(dt) + Hour(14) - Second(1)), :value) ./ 1e3
    priceList[i] = subData.Price
    
    signatureList[i] = signature(collect(1:1:200), log.(priceList[i]), eventList[i] .+ rand(length(eventList[i])), 3*60*60 + 1)
    avgSignature .+= signatureList[i]
end

avgSignature = avgSignature ./ length(eventList);

To plot the signature we take the average across all the dates and
then normalised by the \(\tau = 60\) value.

plot(avgSignature / avgSignature[60], seriestype=:scatter, 
    label = "Average Signature", xlabel = "Tau", ylabel = "Realised Volatility (normalised)", fmt=:png)

This is an interesting plot with big consequences in high-frequency finance.

This explosion in realised volatility at small timescales (\(\tau
\rightarrow 0\)) comes from microstructure noise. If prices evolved
as Brownian motion, the above plot would be flat for all timescales so
the above result contradicts lots of classical finance assumptions.

Practically, this is a pain if we are trying to measure the currently
volatility, it depends on the timescale we are looking at, there isn’t
one true volatility using the normal methods. Instead, we need to be
aware of these microstructure effects as we use a finer \(\tau\) over
which to calculate the volatility.

This is where the Hawkes model comes in. If we assume the price,
\(X(t)\) moves as stated in Equation (), can we produce a similar signature plot?

The Theoretical Signature Under a Hawkes Process

After doing some maths (you can read the paper for the full details), we arrive at the following equation for the theoretical signature.

If both \(N_1\) and \(N_2\) are realisations of Hawkes processes with
parameters \(\mu, \kappa\) and \(g(t) = \beta \exp (-\beta t)\) then their intensity can be written as

\[C(\tau) = \Lambda \left( k ^2 + (1 – k ^2) \frac{1 – e ^{-\gamma \tau}}{ \gamma \tau} \right),\]

where

\(\Lambda = \frac{2 \mu}{1 – \kappa}, k = \frac{1}{1 + \kappa}, \gamma = \beta (\kappa + 1)\).

These are from the paper and adjusted based on my parameterisation of the Hawkes process. This gives us our theo_signature function.

function theo_signature(tau, bg, kappa, kern)
    Lambda = 2*bg/(1-kappa)
    k = 1/(1 + kappa)
    gamma = kern*(kappa + 1)
    @. Lambda * (k^2 + (1-k^2) * (1 - exp(-gamma * tau)) / (gamma*tau))
end

Calibrating the Hawkes Process Model

We now move on to fitting the Hawkes process to the data. I’ll be using
a new method that takes a different approach to my package HawkesProcesses.jl.

We have a theoretical volatility signature from a Hawkes process
(theo_signature) and the above plot of what the actual signature
looks like it. Therefore, it is just a case of optimising over a loss
function to find the best fitting parameters. I’ll use root mean
square error as my loss function and simply use the Optim.jl package
to perform the minimisation.

signatureRMSE(x, sig) = sqrt(
    mean(
        (sig .- theo_signature(1:200, x[1], x[2], x[3])).^2
        )
)

using Optim
optRes = optimize(x->signatureRMSE(x, avgSignature/avgSignature[10]), rand(3))

paramEst = Optim.minimizer(optRes)

3-element Vector{Float64}:
 0.24402236592012655
 0.7417867072115396
 0.19569169443936185

These are the three parameters of the Hawkes process which appear
sensible.

plot(avgSignature, label="Observed", seriestype=:scatter)
plot!(avgSignature[10]*theo_signature(1:200, paramEst[1], paramEst[2], paramEst[3]), label="Theoretical", lw=3, xlabel = "Tau", ylabel = "Realised Variance", fmt=:png)

So a nice match-up between the theoretical signature and what we
observed. This gives some weight to the Hawkes model as a
representation of the price process.

Interpreting the Hawkes Parameters

Our \(\kappa\) value of 0.75 shows there is a large amount of
excitement with each price jump. a \(\beta\) value of 0.2 shows that this
mean reversion lasts for about 5 seconds. So if we see an uptick in
the price, we expect a downtick with a rough half-life of five
seconds. The opposite is also true, a downtick likely leads to an
uptick 5 seconds later.

Conclusion

Microstructure noise shows up when we start calculating volatility on
a high-frequency timescale. We have shown that it is a real effect
using some futures data and then built a Hawkes model to try and
reproduce this effect. We managed to get the right shape of the
volatility signature and found that was quite a bit of mean reversion
between the up and downticks that lasted around 5 seconds.

What’s next? In a future post, I will introduce another dimension and
show and there can also be correlation across assets under a similar
method to reproduce another high-frequency phenomenon. This will be
based on the same paper and show you how we can start looking at the
correlation between two assets and how that changes at high-frequency
time scales.

By: Dean Markwick's Blog -- Julia

Re-posted from: https://dm13450.github.io/2022/04/28/Volatility-methods.html

Volatility measures the scales of price changes and is an easy way to
describe how busy markets are. High volatility means there are periods
of large price changes and vice versa, low volatility means periods of
small changes. In this post, I’ll show you how to measure realised
volatility and demonstrate how it can be used. If you just want a live
view of crypto volatility, take a look at
cryptoliquiditymetrics where I have added in a new card with the volatility over the last 24 hours.

Enjoy these types of posts? Then you should sign up for my newsletter. It’s a short monthly recap of anything and everything I’ve found interesting recently plus
any posts I’ve written. So sign up and stay informed!

To start with we will be looking at daily data. Using my
CoinbasePro.jl package in a Julia we can get the last 300 days OHLC
prices.

I’m running Julia 1.7 and all the packages were updated using
Pkg.update() at the time of this post.

using CoinbasePro
using Dates
using Plots, StatsPlots
using DataFrames, DataFramesMeta
using RollingFunctions

From my CoinbasePro.jl
package, we can pull in the daily candles of Bitcoin. 86400 is the
frequency for daily data. Coinbase restrict you to just 300 data
points

dailydata = CoinbasePro.candles("BTC-USD", now()-Day(300), now(), 86400);
sort!(dailydata, :time)
dailydata = @transform(dailydata, :time = Date.(:time))
first(dailydata, 4)

Plotting this gives you the typical price path. Now realised
volatility is a measure of how varied this price was
over time. Was it stable or were there wild swings?

plot(dailydata.time, dailydata.close, label = :none, 
     ylabel = "Price", title = "Bitcoin Price", titleloc = :left, linewidth = 2)

To calculate this variation, we need to add in the log-returns.

dailydata = @transform(dailydata, :returns = [NaN; diff(log.(:close))]);
bar(dailydata.time[2:end], dailydata.returns[2:end], 
    label=:none, 
    ylabel = "Log Return", title = "Bitcoin Log Return", titleloc = :left)

We can start by looking at this from a distribution perspective. If we
assume the log-returns (\(r\)) are from a normal distribution, with
zero mean, the standard deviation of this distribution is the equivalent to the
volatility

\[r \sim N(0, \sigma ^2),\]

so \(\sigma\) is how we will refer to volatility. From this, you can
see how high volatility leads to wide variations in prices. Each
log-return sample has a wider range of values that it could be.

So by taking the running standard deviation of the log-returns we can
estimate the volatility and how it changes over time. Using the RollingFunctions.jl package this is a one-liner.

dailydata = @transform(dailydata, :volatility = runstd(:returns, 30))
plot(dailydata.time, dailydata.volatility, title = "Bitcoin Volatility", titleloc = :left, label=:none, linewidth = 2)

There was high volatility over June this year as the price of Bitcoin
crashed. It’s been fairly stable since then, hovering around 0.03
and 0.04. How does this compare though to the S&P 500 as a general
indicator of the stop market? We know Bitcoin is more volatile than
the stock market, but how much more?

I’ll load up the AlphaVantage.jl package to pull the daily prices of
the SPY ETF and repeat the calculation; adding the log-returns and
taking the rolling standard deviation.

using AlphaVantage, CSV

stockPrices = AlphaVantage.time_series_daily("SPY", datatype="csv", outputsize="full", parser = x -> CSV.File(IOBuffer(x.body))) |> DataFrame
sort!(stockPrices, :timestamp)
stockPrices = @subset(stockPrices, :timestamp .>= minimum(dailydata.time));

Again, add in the log-returns and calculate the rolling standard
deviation to estimate the volatility.

stockPrices = @transform(stockPrices, :returns = [NaN; diff(log.(:close))])
stockPrices = @transform(stockPrices, :volatility = runstd(:returns, 30));

volPlot = plot(dailydata.time, dailydata.volatility, label="BTC", 
               ylabel = "Volatility", title = "Volatility", titleloc = :left, linewidth = 2)
volPlot = plot!(volPlot, stockPrices.timestamp, stockPrices.volatility, label = "SPY", linewidth = 2)

As expected, Bitcoin volatility is much higher. Let’s take the log of
the volatility to look zoom in on the detail.

volPlot = plot(dailydata.time, log.(dailydata.volatility), 
               label="BTC", ylabel = "Log Volatility", title = "Log Volatility", 
               titleloc = :left, linewidth = 2)
volPlot = plot!(volPlot, stockPrices.timestamp, log.(stockPrices.volatility), label = "SPY", linewidth = 2)

Interestingly the SPY has had a resurgence in volatility as we move
towards the end of the year. One thing to point out though is the
slight difference in look back periods for the two products. Bitcoin
does not observe weekends or holidays, so 30 rows previously are
always 30 days, but for SPY this is the case as there are weekends and
trading holidays. In this illustrative example, it isn’t too much of an
issue, but if you were to take it further and perhaps look at the
correlation between the volatilities, this is something you would need
to account for.

A Higher Frequency Volatility

So far it has all been on daily observations, your classic dataset to
practise on. But I am always banging on about high-frequency finance,
so let’s look at more frequent data and understand how the volatility
looks at finer timescales.

This time we will pull the 5-minute candle bar data of both Bitcoin
and SPY and repeat the calculation.

1. Calculate the log-returns of the close to close bars
2. Calculate the rolling standard deviation by looking back 20 rows.

minuteData_spy = AlphaVantage.time_series_intraday_extended("SPY", "5min", "year1month1", parser = x -> CSV.File(IOBuffer(x.body))) |> DataFrame
minuteData_spy = @transform(minuteData_spy, :time = DateTime.(:time, dateformat"yyyy-mm-dd HH:MM:SS"))
minuteData_btc = CoinbasePro.candles("BTC-USD", maximum(minuteData_spy.time)-Day(1), maximum(minuteData_spy.time),300);

combData = leftjoin(minuteData_spy[!, [:time, :close]], minuteData_btc[!, [:time, :close]], on=[:time], makeunique=true)
rename!(combData, ["time", "spy", "btc"])
combData = combData[2:end, :]
dropmissing!(combData)
sort!(combData, :time)
first(combData, 3)

For 5 minute data, we will use a look-back period of 20 rows, which gives us 100 minutes, so a little under 2 hours.

combData = @transform(combData, :spy_returns = [NaN; diff(log.(:spy))],
                                :btc_returns = [NaN; diff(log.(:btc))])
combData = @transform(combData, :spy_vol = runstd(:spy_returns, 20),
                                :btc_vol = runstd(:btc_returns, 20))
combData = combData[2:end, :];

Plotting it all again!

vol_tks = minimum(combData.time):Hour(6):maximum(combData.time)
vol_tkslbl = Dates.format.(vol_tks, "e HH:MM")

returnPlot = plot(combData.time[2:end], cumsum(combData.btc_returns[2:end]), 
                  label="BTC", title = "Cumulative Returns", xticks = (vol_tks, vol_tkslbl),
                  linewidth = 2, legend=:topleft)
returnPlot = plot!(returnPlot, combData.time[2:end], cumsum(combData.spy_returns[2:end]), label="SPY",
                   xticks = (vol_tks, vol_tkslbl),
                   linewidth = 2)


volPlot = plot(combData.time, combData.btc_vol * sqrt(24 * 20), 
    label="BTC", xticks = (vol_tks, vol_tkslbl), titleloc = :left, linewidth = 2)
volPlot = plot!(combData.time, combData.spy_vol * sqrt(24 * 20), label="SPY", title = "Volatility",
               xticks = (vol_tks, vol_tkslbl), titleloc = :left, linewidth = 2)

plot(returnPlot, volPlot)

On the left-hand side, we have the cumulative return of the two
assets on the 30th of December, and on the right the corresponding
volatility. Bitcoin still has higher volatility whereas SPY has been
relatively stable with just some jumps.

Simplifying the Calculation

Rolling the standard deviation isn’t the efficient way of calculating the volatility and can also be simplified down to a more efficient calculation.

The standard deviation is defined as:

\[\sigma ^2 = \mathbb{E} (r^2) + \mathbb{E} (r) ^2\]

if we assume there is no trend in the returns so that the average is zero:

\[\mathbb{E} (r) = 0\]

then we get just the first term

\[\sigma ^2 = \frac{1}{N} \sum _{i=1} ^N r^2\]

which is simply proportional to the sum of squares. Hence why you will
hear that the realised variance is referred to as the sum of squares.

Once again, let’s pull the data and repeat the previous calculations
but this time adding another column that is the rolling summation of
the square of the returns.

minutedata = CoinbasePro.candles("BTC-USD", now()-Day(1) - Hour(1), now(), 5*60)
sort!(minutedata, :time)
minutedata = @transform(minutedata, :close_close_return = [NaN; diff(log.(:close))])
minutedata = minutedata[2:end, :]
first(minutedata, 4)

minutedata = @transform(minutedata,
                                    :new_vol_5 = running(sum, :close_close_return .^2, 20),
                                    :vol_5 = runstd(:close_close_return, 20))
minutedata = minutedata[2:end, :]
minutedata[1:5, [:time, :new_vol_5, :vol_5]]

vol_tks = minimum(minutedata.time):Hour(8):maximum(minutedata.time)
vol_tkslbl = Dates.format.(vol_tks, "e HH:MM")

ss_vol = plot(minutedata.time, sqrt.(288 * minutedata.new_vol_5), titleloc = :left, 
              title = "Sum of Squares", label=:none, xticks = (vol_tks, vol_tkslbl), linewidth = 2)
std_vol = plot(minutedata.time, sqrt.(288 * minutedata.vol_5), titleloc = :left, 
               title = "Standard Deviation", label=:none, xticks = (vol_tks, vol_tkslbl), linewidth = 2)
plot(ss_vol, std_vol)

Both methods show represent the relative changes equally. There are
some notable edge effects in the standard deviation method, but
overall, our assumptions look fine. The y-scales are different though
as there are some constant factor differences between the two methods.

Comparing Crypto Volatilities

Let’s see how the volatility changes across some different
currencies. We define a function that calculates the close to close
return and iterate through some different currencies.

function calc_vol(ccy)
    minutedata = CoinbasePro.candles(ccy, now()-Day(1) - Hour(1), now(), 5*60)
    sort!(minutedata, :time)
    minutedata = @transform(minutedata, :close_close_return = [NaN; diff(log.(:close))])
    minutedata = minutedata[2:end, :]
    minutedata = @transform(minutedata, :var = 288*running(sum, :close_close_return .^2, 20))
    minutedata
    minutedata[21:end, :]
end

Let’s choose the classics BTC and ETH, the meme that is SHIB and
finally EURUSD (the crypto version).

p = plot(legend=:topleft, ylabel = "Realised Volatility")
for ccy in ["BTC-USD", "ETH-USD", "USDC-EUR", "SHIB-USD"]
    voldata = calc_vol(ccy)
    vol_tks = minimum(voldata.time):Hour(4):maximum(voldata.time)
    vol_tkslbl = Dates.format.(vol_tks, "e HH:MM")
    plot!(voldata.time, sqrt.(voldata.var), label = ccy, 
          xticks = (vol_tks, vol_tkslbl), linewidth = 2)
end
p

SHIB has higher overall volatility. ETH and BTC have very comparable
volatilities moving together. EURUSD has the lowest overall (as we
would expect), but interesting to see how it moved higher just as the
cryptos did at about 9 am.

An Update to CryptoLiquidityMetrics

So I’ve taken everything we’ve learnt here and implemented it into
cryptoliquiditymetrics.com. It is a new panel (bottom right) and
calculated all through Javascript.

How does this help you?

Knowing the volatility helps you get an idea of how easy it is to trade
or what strategy to use. When volatility is high and the price is
moving about it might be better to be more aggressive and make sure your trade
happens. Whereas if it is a stable market without too much volatility
you could be more passive and just wait, trading slowly and picking
good prices.

Just another addition to my Javascript side project!

By: Dean Markwick's Blog -- Julia

Re-posted from: https://dm13450.github.io/2022/02/02/Order-Flow-Imbalance.html

I’ll show you how to calculate the ‘order flow imbalance’ and build a
high-frequency trading signal with the results. We’ll see if it is a
profitable strategy and how it might be useful as a market indicator.

A couple of months ago I attended the Oxford Math Finance seminar
where there was a presentation on order book dynamics to predict
short-term price direction. Nicholas Westray presented
Deep Order Flow Imbalance: Extracting Alpha at Multiple Horizons from the Limit Order Book. By
using deep learning they predict future price movements using common
neural network architectures such as the basic multi-layer perceptron
(MLP), Long Term Short Memory network (LSTM) and convolutional neural
networks (CNN). Combining all three networks types lets you extract
the strengths of each network:

• CNNs: Reduce frequency variations.
  • The variations between each input reduce to some common
    factors.
• LSTMs: Learn temporal structures
  • How are the different inputs correlated with each other such as
    autocorrelation structure.
• MPLs: Universal approximators.
  • An MLP can approximate any function.

A good reference for LSTM-CNN combinations is DeepLOB: Deep
Convolution Neural Networks for Limit Order Books
where they use this type of neural network to predict whether the
market is moving up or down.

The Deep Order Flow talk was an excellent overview of deep
learning concepts and described how there is a great overlap between
computer vision and the state of the order book. You can build an
“image” out of the different order book levels and pass this through
the neural networks. My main takeaway from the talk was the concept of
Order Flow Imbalance. This is a transformation that uses the order
book to build a feature to predict future returns.

I’ll show you how to calculate the order flow imbalance and see how
well it predicts future returns.

The Setup

I have a QuestDB database with the best bid and offer price and size
at those levels for BTCUSD from Coinbase over roughly 24 hours. To
read how I collected this data check out my previous post on
streaming data into QuestDB.

Julia easily connects to QuestDB using the LibPQ.jl package. I also
load in the basic data manipulation packages and some statistics
modules to calculate the necessary values.

using LibPQ
using DataFrames, DataFramesMeta
using Plots
using Statistics, StatsBase
using CategoricalArrays
using Dates
using RollingFunctions

conn = LibPQ.Connection("""
             dbname=qdb
             host=127.0.0.1
             password=quest
             port=8812
             user=admin""")

Order flow imbalance is about the changing state of the order book. I
need to pull out the full best bid best offer table. Each row in this
table represents when the best price or size at the best price changed.

bbo = execute(conn, 
    "SELECT *
     FROM coinbase_bbo") |> DataFrame
dropmissing!(bbo);

I add the mid-price in too, as we will need it later.

bbo = @transform(bbo, mid = (:ask .+ :bid) / 2);

It is a big dataframe, but thankfully I’ve got enough RAM.

Calculating Order Flow Imbalance

Order flow imbalance represents the changes in supply and demand. With
each row one of the price or size at the best bid or ask changes which
corresponds to change in the supply or demand, even at a high
frequency level, of Bitcoin.

• Best bid or size at the best bid increase -> increase in demand.
• Best bid or size at the best bid decreases -> decrease in demand.
• Best ask decreases or size at the best ask increases -> increase
  in supply.
• Best ask increases or size at the best ask decreases ->
  decrease in supply.

Mathematically we summarise these four effects at from time \(n-1\) to
\(n\) as:

\[e_n = I_{\{ P_n^B \geq P^B_{n-1} \}} q_n^B – I_\{ P_n^B \leq
P_{n-1}^B \} q_{n-1}^B – I_\{ P_n^A \leq P_{n-1}^A \}
q_n^A + I_\{ P_n^A \geq P_{n-1}^A \} q_{n-1}^A,\]

where \(P\) is the best price at the bid (\(P^B\)) or ask (\(P^A\)) and
\(q\) is the size at those prices.

Which might be a bit easier to read as Julia code:

e = Array{Float64}(undef, nrow(bbo))
fill!(e, 0)

for n in 2:nrow(bbo)
    
    e[n] = (bbo.bid[n] >= bbo.bid[n-1]) * bbo.bidsize[n] - 
    (bbo.bid[n] <= bbo.bid[n-1]) * bbo.bidsize[n-1] -
    (bbo.ask[n] <= bbo.ask[n-1]) * bbo.asksize[n] + 
    (bbo.ask[n] >= bbo.ask[n-1]) * bbo.asksize[n-1]
    
end

bbo[!, :e] = e;

To produce an Order Flow Imbalance (OFI) value, you need to aggregate
\(e\) over some time-bucket. As this is a high-frequency problem I’m
choosing 1 second. We also add in the open and close price of the
buckets and the return across this bucket.

bbo = @transform(bbo, timestampfloor = floor.(:timestamp, Second(1)))
bbo_g = groupby(bbo, :timestampfloor)
modeldata = @combine(bbo_g, ofi = sum(:e), OpenPrice = first(:mid), ClosePrice = last(:mid), NTicks = length(:e))
modeldata = @transform(modeldata, OpenCloseReturn = 1e4*(log.(:ClosePrice) .- log.(:OpenPrice)))
modeldata = modeldata[2:(end-1), :]
first(modeldata, 5)

Now we do the usual train/test split by selecting the first 70% of the
data.

trainInds = collect(1:Int(floor(nrow(modeldata)*0.7)))
trainData = modeldata[trainInds, :]
testData = modeldata[Not(trainInds), :];

We are going to fit a basic linear regression using the OFI value as
the single predictor.

using GLM

ofiModel = lm(@formula(OpenCloseReturn ~ ofi), trainData)

StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}}}}, Matrix{Float64}}

OpenCloseReturn ~ 1 + ofi

Coefficients:
─────────────────────────────────────────────────────────────────────────────
                  Coef.   Std. Error       t  Pr(>|t|)  Lower 95%   Upper 95%
─────────────────────────────────────────────────────────────────────────────
(Intercept)  -0.0181293  0.00231571    -7.83    <1e-14  -0.022668  -0.0135905
ofi           0.15439    0.000695685  221.92    <1e-99   0.153026   0.155753
─────────────────────────────────────────────────────────────────────────────

We see a positive coefficient of 0.15 which is very significant.

r2(ofiModel)

0.3972317963590547

A very high in-sample \(R^2\).

predsTrain = predict(ofiModel, trainData)
predsTest = predict(ofiModel, testData)

(mean(abs.(trainData.OpenCloseReturn .- predsTrain)),
    mean(abs.(testData.OpenCloseReturn .- predsTest)))

(0.3490577385082666, 0.35318460250890665)

Comparable mean absolute error (MAE) across both train and test sets.

sst = sum((testData.OpenCloseReturn .- mean(testData.OpenCloseReturn)) .^2)
ssr = sum((predsTest .- mean(testData.OpenCloseReturn)) .^2)
ssr/sst

0.4104873667550974

An even better \(R^2\) in the test data

extrema.([predsTest, testData.OpenCloseReturn])

2-element Vector{Tuple{Float64, Float64}}:
 (-5.400295917229609, 5.285718311926791)
 (-11.602503514716034, 11.46049286770534)

But doesn’t quite predict the largest or smallest values.

So overall:

• Comparable R2 and MAE values across the training and test sets.
• Positive coefficient indicates that values with high positive order flow imbalance will have a large positive return.

But, this all suffers from the cardinal sin of backtesting, we are using information from the future (the sum of the \(e\) values to form the OFI) to predict the past. By the time we know the OFI value, the close value has already happened! We need to be smarter if we want to make trading decisions based on this variable.

So whilst it doesn’t give us an actionable signal, we know that it can explain price moves, we know just have to reformulate our model and make sure there is no information leakage.

Building a Predictive Trading Signal

I now want to see if OFI can be used to predict future price
returns. First up, what do the OFI values look like and what
about if we take a rolling average?

Using the excellent RollingFunctions.jl package we can calculate
the five-minute rolling average and compare it to the raw values.

xticks = collect(minimum(trainData.timestampfloor):Hour(4):maximum(trainData.timestampfloor))
xtickslabels = Dates.format.(xticks, dateformat"HH:MM")

ofiPlot = plot(trainData.timestampfloor, trainData.ofi, label = :none, title="OFI", xticks = (xticks, xtickslabels), fmt=:png)
ofi5minPlot = plot(trainData.timestampfloor, runmean(trainData.ofi, 60*5), title="OFI: 5 Minute Average", label=:none, xticks = (xticks, xtickslabels))
plot(ofiPlot, ofi5minPlot, fmt=:png)

It’s very spiky, but taking the rolling average smooths it out. To
scale the OFI values to a known range, I’ll perform the Z-score
transform using the rolling five-minute window of both the mean and
variance. We will also use the close to close returns rather than the
open-close returns of the previous model and make sure it is lagged
correctly to prevent information leakage.

modeldata = @transform(modeldata, ofi_5min_avg = runmean(:ofi, 60*5),
                                  ofi_5min_var = runvar(:ofi, 60*5),
                                  CloseCloseReturn = 1e4*[diff(log.(:ClosePrice)); NaN])

modeldata = @transform(modeldata, ofi_norm = (:ofi .- :ofi_5min_avg) ./ sqrt.(:ofi_5min_var))

modeldata[!, :CloseCloseReturnLag] = [NaN; modeldata.CloseCloseReturn[1:(end-1)]]

modeldata[1:7, [:ofi, :ofi_5min_avg, :ofi_5min_var, :ofi_norm, :OpenPrice, :ClosePrice, :CloseCloseReturn]]

xticks = collect(minimum(modeldata.timestampfloor):Hour(4):maximum(modeldata.timestampfloor))
xtickslabels = Dates.format.(xticks, dateformat"HH:MM")

plot(modeldata.timestampfloor, modeldata.ofi_norm, label = "OFI Normalised", xticks = (xticks, xtickslabels), fmt=:png)
plot!(modeldata.timestampfloor, modeldata.ofi_5min_avg, label="OFI 5 minute Average")

The OFI values have been compressed from \((-50, 50)\) to \((-10,
10)\). From the average values we can see periods of positive and
negative regimes.

When building the model we split the data into a training and
testing sample, throwing away the early values where the was not
enough values for the rolling statistics to calculate.

We use a basic linear regression with just the normalised OFI value.

trainData = modeldata[(60*5):70000, :]
testData = modeldata[70001:(end-1), :]

ofiModel_predict = lm(@formula(CloseCloseReturn ~ ofi_norm), trainData)

StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}}}}, Matrix{Float64}}

CloseCloseReturn ~ 1 + ofi_norm

Coefficients:
────────────────────────────────────────────────────────────────────────────
                 Coef.  Std. Error      t  Pr(>|t|)    Lower 95%   Upper 95%
────────────────────────────────────────────────────────────────────────────
(Intercept)  0.0020086  0.00297527   0.68    0.4996  -0.00382293  0.00784014
ofi_norm     0.144358   0.00292666  49.33    <1e-99   0.138622    0.150094
────────────────────────────────────────────────────────────────────────────

A similar value in the coefficient compared to our previous model and
it remains statistically significant.

r2(ofiModel_predict)

0.033729601695801414

Unsurprisingly, a massive reduction on in-sample \(R^2\). A value of 3%
is not that bad, in the Deep Order Flow paper they achieve values of
around 1% but over a much larger dataset and across multiple
stocks. My 24 hours of Bitcoin data is much easier to predict.

returnPredictions = predict(ofiModel_predict, testData)

testData[!, :CloseClosePred] = returnPredictions

sst = sum((testData.CloseCloseReturn .- mean(testData.CloseCloseReturn)) .^2)
ssr = sum((testData.CloseClosePred .- mean(testData.CloseCloseReturn)) .^2)
ssr/sst

0.030495583445248473

The out-of-sample \(R^2\) is also around 3%, so not that bad really in
terms of overfitting. It looks like we’ve got a potential model on our
hands.

Does This Signal Make Money?

We can now go through a very basic backtest to see if this signal is
profitable to trade. This will all be done in pure Julia, without any
other packages.

Firstly, what happens if we go long every time the model predicts a
positive return and likewise go short if the model predicts a negative
return. This means simply taking the sign of the model prediction and
multiplying it by the observed returns will give us the returns of the
strategy.

In short, this means if our model were to predict a positive return
for the next second, we would immediately buy at the close and be filled
at the closing price. We would then close out our position after the
second elapsed, again, getting filled at the next close to produce a
return.

xticks = collect(minimum(testData.timestampfloor):Hour(4):maximum(testData.timestampfloor))
xtickslabels = Dates.format.(xticks, dateformat"HH:MM")

plot(testData.timestampfloor, cumsum(sign.(testData.CloseClosePred) .* testData.CloseCloseReturn), 
    label=:none, title = "Cummulative Return", fmt=:png, xticks = (xticks, xtickslabels))

Up and to the right as we would hope. So following this strategy would
make you money. Theoretically. But is it a good strategy? To measure
this we can calculate the Sharpe ratio, which is measuring the overall
profile of the returns compared to the volatility of the returns.

moneyReturns = sign.(testData.CloseClosePred) .* testData.CloseCloseReturn
mean(moneyReturns) ./ std(moneyReturns)

0.11599938576235787

A Sharpe ratio of 0.12 if we are generous and round up. Anyone with
some experience in these strategies is probably having a good chuckle
right now, this value is terrible. At the very minimum, you would
like a value of 1, i.e. that your average return is greater than the
variance in returns, otherwise you are just looking at noise.

How many times did we correctly guess the direction of the market
though? This is the hit ratio of the strategy.

mean(abs.((sign.(testData.CloseClosePred) .* sign.(testData.CloseCloseReturn))))

0.530163738236414

So 53% of the time I was correct. 3% better than a coin toss, which is
good and shows there is a little bit of information in the OFI values
when predicting.

Does a Threshold Help?

Should we be more selective when we trade? What if we set a threshold and
only trade when our prediction is greater than that value. Plus the
same in the other direction. We can iterate through lots of potential
thresholds and see where the Sharpe ratios end up.

p = plot(ylabel = "Cummulative Returns", legend=:topleft, fmt=:png)
sharpes = []
hitratio = []

for thresh in 0.01:0.01:0.99
  trades = sign.(testData.CloseClosePred) .* (abs.(testData.CloseClosePred) .> thresh)

  newMoneyReturns = trades .* testData.CloseCloseReturn

  sharpe = round(mean(newMoneyReturns) ./ std(newMoneyReturns), digits=2)
  hr = mean(abs.((sign.(trades) .* sign.(testData.CloseCloseReturn))))

  if mod(thresh, 0.2) == 0
    plot!(p, testData.timestampfloor, cumsum(newMoneyReturns), label="$(thresh)", xticks = (xticks, xtickslabels))
  end
  push!(sharpes, sharpe)
  push!(hitratio, hr)
end
p

The equity curves look worse with each higher threshold.

plot(0.01:0.01:0.99, sharpes, label=:none, title = "Sharpe vs Threshold", xlabel = "Threshold", ylabel = "Sharpe Ratio", fmt=:png)

A brief increase in Sharpe ratio if we set a small threshold, but
overall, steadily decreasing Sharpe ratios once we start trading
less. For such a simple and linear model this isn’t surprising, but
once you start chucking more variables and different modeling
approaches into the mix it can shed some light on what happens around
the different values.

Why you shouldn’t trade this model

So at the first glance, the OFI signal looks like a profitable
strategy. Now I will highlight why it isn’t in practice.

• Trading costs will eat you alive

I’ve not taken into account any slippage, probability of fill, or
anything that a real-world trading model would need to be
practical. As our analysis around the Sharpe ratio has shown, it wants
to trade as much as possible, which means transaction costs will just
destroy the return profile. With every trade, you will pay the full
bid-ask spread in a round trip to open and then close the trade.

• The Sharpe ratio is terrible

With a Sharpe ratio < 1 shows that there is not much actual
information in the trading pattern, it is getting lucky vs the actual
volatility in the market. Now, Sharpe ratios can get funky when we are
looking at such high-frequency data, hence why this bullet point is second to the trading costs.

• It has been trained on a tiny amount of data.

Needless to say, given that we are looking at seconds this dataset
could be much bigger and would give us greater confidence in the
actual results once expanded to a wider time frame of multiple days.

• I’ve probably missed something that blows this out of the water

Like everything I do, there is a strong possibility I’ve gone wrong
somewhere, forgotten a minus, ordered a time-series wrong, and various other errors.

How this model might be useful

• An overlay for a market-making algorithm

Making markets is about posting quotes where they will get filled and
collecting the bid-ask spread. Therefore, because our model appears to
be able to predict the direction fairly ok, you could use it to place
a quote where the market will be in one second, rather than where it
is now. This helps put your quote at the top of the queue if the
market does move in that direction. Secondly, if you are traded with
and need to hedge the position, you have an idea of how long to wait
to hedge. If the market is moving in your favour, then you can wait an
extra second to hedge and benefit from the move. Likewise, if this
model is predicting a move against your inventory position, then you
know to start aggressively quoting to minimise that move against.

• An execution algorithm

If you are potentially trading a large amount of bitcoin, then you
want to split your order up into lots of little orders. Using this
model you then know how aggressive or passive you should trade based on
where the market is predicted to move second by second. If the order
flow imbalance is trending positive, the market is going to go up, so
you want to increase your buying as not to miss out on the move and
again, if the market is predicted to move down, you’ll want to slow
down your buying so that you fully benefit from the lull.

Conclusion

Overall hopefully you now know more about order flow imbalance and how
it can somewhat explain returns. It also has some predictive power and
we use that to try and build a trading strategy around the signal.

We find that the Sharpe ratio of said strategy is poor and that
overall, using it as a trading signal on its own will not have you
retiring to the Bahamas.

This post has been another look at high-frequency finance and the
trials and tribulations around this type of data.