HFT: Basics of orderflow

If you don’t understand order flow, you’re trading blind. Here’s what happens when smart money enters the game

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Table of contents:

1. Introduction.

2. What is order flow?

3. Why flat lines are a myth?

4. The square-root law

5. The basics of reaction-diffusion—aka simulating the trading frenzy

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Introduction

Picture this: you’re selling ice cream on a sunny beach. Life’s good. But suddenly, a bus full of tourists arrives. Your cart is swarmed. Do you:
A) Keep prices the same and run out of ice cream in 5 minutes?
B) Raise prices because heck yeah, supply and demand?

If you picked B, congratulations! You’ve just mastered the first rule of trading: big demand → higher prices. But here’s the twist: in financial markets, it’s not quite that simple. Instead of tourists, imagine hedge funds with billion-dollar orders. Instead of ice cream, think Tesla shares. And instead of you, picture a grumpy market maker muttering, “Why can’t these traders just stick to small orders?”

Before we dig in, a friendly reminder: while many sophisticated models might mention topics like overfitting or the Kelly criterion, today we’ll leave those behind. Our focus is on clear, engaging insights and simple mathematics. Ready to trade some ice cream for knowledge? Let’s begin!

What is order flow?

Let’s start with a hypothesis: Big trades move prices more than small ones. Shocking, right? But how do we quantify this? Enter order flow—the heartbeat of markets that refers to the sequence of transactions. Every buy/sell order tells a story:

• Small orders are like casual conversations among friends.

• Big orders are the equivalent of shouting across the room, grabbing everyone’s attention.

In 1991, Joel Hasbrouck published a paper titled Measuring the Information Content of Stock Trades. His findings?

1. Price impact isn’t instant. Like bad news from your in-laws, it takes time to sink in.

2. Bigger trades = bigger impact. A 1M order hits harder than a 100 one.

3. Market makers hate surprises. They widen spreads after big trades.

4. Illiquid markets = drama queens. Thinly traded stocks overreact to everything.

Speaking of drama, let’s talk about order books — the market’s version of a Rorschach test.

Let’s put on our math professor hat for a moment—don’t worry I promise to keep it simple. Suppose we denote the order size by Q. In many trading scenarios, the impact of a trade isn’t directly proportional to Q. Instead, it’s more nuanced. Big orders tend to have a disproportionately large effect, but not as dramatically large as a one-for-one increase in size might suggest.

To illustrate, consider a simplistic relationship:

\(I = k \cdot Q^{\alpha}\)

where:

• I is the price impact.

• k is a constant that encapsulates market conditions.

• α is an exponent that is often less than 1, meaning the impact grows less than proportionally with order size.

For many markets, researchers have found that α is around 0.5. In plain English, if you quadruple your order size, the price impact roughly doubles rather than quadruples. It’s like our ice cream cart: if you suddenly serve four times as many cones, you might run out faster, but the extra excitement in the queue only increases so much.

Now that we’ve set the stage with order flow, let’s move on to something even more visual: the order book. Think of it as the market’s ledger—a detailed menu of who is buying and selling, at which prices, and in what quantities. Unlike the neatly drawn lines in textbooks, real order books are as quirky as they are informative. Ready to dive into this art form? Let’s go!

I must admit that this photo really looks like an avocado and strawberry ice cream 😋

Why flat lines are a myth?

Can a system truly remain in a state of perfect equilibrium, or is every flat line just an illusion hiding underlying fluctuations?

In some books, you might see an order book depicted as two perfectly horizontal lines—one for buy orders, one for sell orders. In the real world, however, order books are more like the zany scribbles of a mad artist: they form a V shape, reflecting the depth and liquidity of the market.

Let’s break down a real order book:

• Buy orders or Bids: These represent the quantities buyers are willing to purchase at different prices. The closer you are to the current market price, the fewer shares you might get, but at a more favorable price.

• Sell orders or Asks: These indicate the quantities sellers are ready to sell. As you move away from the current market price, you might find larger orders, but at a steeper cost.

Here’s a simplified visual. Fake order books look like this:

|   Buy Orders   |   Sell Orders  |  
|100 shares @ $10|100 shares @ $11|  
|100 shares @ $10|100 shares @ $11|
|100 shares @ $10|100 shares @ $11|  

But real order books? They look like this, just like in the picture above:

|   Buy Orders     |   Sell Orders     |
|50 shares  @ $9.90|50 shares  @ $10.10|
|200 shares @ $9.80|200 shares @ $10.20|
|500 shares @ $9.70|500 shares @ $10.30|

Notice the V shape? As you dig deeper into the order book—further from the market price. The available quantity increases, but the prices become less attractive. It’s like going to the back of an ice cream truck and finding a shelf full of flavors you wouldn’t normally choose—even though they’re plentiful.

If order books were actually flat, trading would be as easy as ordering pizza. But no, they’re more like a sushi conveyor belt—the good stuff gets snatched fast, leaving you with seaweed salad.

Okay! Having visualized the order book, let’s move on to one of the most fascinating discoveries in market impact research: the square-root law. If order flow is the market’s gossip and the order book its colorful ledger, then the square-root law is the neat mathematical formula that ties it all together. Let’s crack the code behind this intriguing relationship.

The square-root law

When we first looked at how orders affect prices, you might have expected a linear relationship: double the order, double the impact. But real markets are a bit more forgiving. Researchers discovered that the price impact I of an order of size Q scales with the square root of Q:

\(I = \sigma \sqrt{\frac{Q}{V}}\)

where:

• I is the price impact.

• σ represents the market’s volatility.

• Q is the trade size.

• V is the daily traded volume.

Let’s breaking down the formula:

• Volatility (σ) is like the market’s emotional barometer. A volatile market is like a roller coaster—fast and unpredictable.

• Trade size (Q): The bigger your trade, the more likely you are to nudge the market. But here’s the kicker—the impact grows with the square root of QQQ, not directly with QQQ itself.

• Daily volume (V) is the total activity in the market. If you’re trading in a market where millions of shares change hands, your 10,000-share trade might be a mere ripple.

A trade’s impact grows with the square root of its size. So, if you trade 4x as much, the impact is only 2x. Markets are kind of forgiving!

import math

def calculate_impact(Q, V, sigma):
    """
    Calculate the price impact using the square-root law.
    
    Parameters:
    Q (float): The size of the trade.
    V (float): The daily traded volume.
    sigma (float): The market volatility (as a decimal, e.g., 0.02 for 2%).
    
    Returns:
    float: The estimated price impact.
    """
    impact = sigma * math.sqrt(Q / V)
    return impact

# Example parameters
sigma = 0.02       # 2% daily volatility
V = 1_000_000      # 1,000,000 shares traded daily
Q = 10_000         # Trading 10,000 shares

price_impact = calculate_impact(Q, V, sigma)
print(f"Price Impact: {price_impact:.4f}")

# Output: 0.0020 → 0.2% price move  

Buying 10k shares with 1M daily volume moves the price by ~0.2%. Not bad! 😬

If you’re wondering why it’s a square root and not, say, a cube root, blame the V shape. Or thank it. Either way, send your complaints to the French mathematicians who discovered this.

Now that we’ve tamed the beast of market impact with our trusty square-root law, it’s time to introduce a more dynamic model: Bouchaud’s reaction-diffusion model. If you thought our previous topics were interesting, buckle up—this model takes us on a wild ride into the physics of trading. Let’s dive in!

The basics of reaction-diffusion—aka simulating the trading frenzy

Picture a bustling marketplace where buyers and sellers move around like particles in a fluid. Jean-Philippe Bouchaud, one of the rockstars of market impact research, likened traders to particles undergoing a reaction-diffusion process. In simple terms, imagine that every buy or sell order is a little particle moving around, occasionally colliding and reacting.

In physics, reaction-diffusion models describe how particles interact and spread out over time. When two different particles meet, they might react and change the state of the system. In our trading analogy:

• Buyers (particle A): Exclaim, I need this stock now!

• Sellers (particle B): Counter, sell, sell, sell!

When these particles collide, a trade happens. Over time, these interactions create ripples—much like throwing a stone into a pond.

The reaction-diffusion model is governed by partial differential equations (PDEs), which, don’t worry, we won’t fully derive here. The key takeaway is that the propagation of these “trading waves” can be modeled, and the impact of a big trade is analogous to throwing a rock into still water: the ripples spread out and gradually dissipate.

A simplified version of the model might look something like this—in continuous form:

\(\frac{\partial C(x,t)}{\partial t} = D \frac{\partial^2 C(x,t)}{\partial x^2} - R C(x,t) \)

where:

• C(x,t) represents the concentration of “trading pressure” at price level x and time t.

• D is the diffusion coefficient—how fast the impact spreads.

• R is a reaction term representing the decay of impact over time.

In case you want to play around here’s a very simplified Python snippet that simulates the diffusion of trading impact over time:

import numpy as np
import matplotlib.pyplot as plt

# Simulation parameters
L = 100           # Number of price levels
T = 50            # Number of time steps
D = 0.1           # Diffusion coefficient
R = 0.05          # Reaction (decay) rate

# Initial condition: a big trade at the center price level
C = np.zeros((T, L))
C[0, L//2] = 1.0  # Impact concentrated in the middle

# Simulate diffusion over time
for t in range(1, T):
    for x in range(1, L-1):
        diffusion = D * (C[t-1, x-1] - 2 * C[t-1, x] + C[t-1, x+1])
        reaction = -R * C[t-1, x]
        C[t, x] = C[t-1, x] + diffusion + reaction

# Plot the impact over time for select time steps
plt.figure(figsize=(30, 7))
for t in [0, 10, 20, 30, 40, 49]:
    plt.plot(C[t, :], label=f'Time {t}')
plt.xlabel('Price level')
plt.ylabel('Trading impact')
plt.title('Simplified reaction-diffusion model of trading impact')
plt.legend()
plt.grid(True)
plt.show()

We simulate a price space of 100 levels and let the impact evolve over 50 time steps. Then, the initial impact is simulated as an impulse in the center of our grid. For each time step, the impact diffuses spreads to neighboring price levels and decays, which is the reaction term.

Now that we’ve explored how big orders create ripples in the market, let’s shift gears to the realm of high-frequency trading. These are the cheetahs of the trading world—quick, agile, and always ready to pounce on any opportunity. How do they use our insights from order flow, order books, and diffusion models to make split-second decisions? Let’s find out.

They are like the espresso shots of the financial markets: fast, potent, and capable of waking up even the sleepiest market maker. Their algorithms constantly monitor order flow, track order book dynamics, and use mathematical models—like our trusty square-root law—to predict and capitalize on price movements.

HFT firms leverage several key strategies:

• Real-time order flow analysis: Every trade, no matter how small, provides data. HFTs analyze this data in real time to predict where the market is headed.

• Rapid quote adjustments: Based on the expected price impact—calculated via our square-root law. HFTs adjust their bid and ask prices faster than you can say latency arbitrage.

• Slicing big orders: Instead of executing a massive order all at once—and alerting the entire market—big orders are sliced into smaller chunks. This reduces immediate market impact and helps the order blend in with normal trading noise.

HFT algorithms are like that friend who always finishes your sentences—only instead of finishing your thoughts, they’re finishing off all the available liquidity. If HFTs were any faster, they’d be ordering pizza before the delivery guy even picked up the call!

Despite our neat formulas and elegant models, the reality of market impact is as messy as a dropped ice cream cone on a hot sidewalk. Here are some of the hurdles and nuances that even the most sophisticated models must face:

1. The metaorder puzzle.

A metaorder is a large order that is split into many smaller trades to minimize market impact. Imagine trying to eat an entire tub of ice cream in one go—you’d probably end up with brain freeze. Instead, you take it spoonful by spoonful. Similarly, big orders are sliced and diced to hide their true size.

• Challenge: Detecting whether a series of trades is part of a metaorder can be like trying to figure out if your neighbor’s multiple trips to the fridge are for a midnight snack or an all-out ice cream binge.

• Implication: If you know a big order is being executed in pieces, market makers might widen spreads, anticipating the continued pressure on price.

2. Informed vs. uninformed trading.

Not all trades are created equal. Some traders have insider insights, while others are simply following the herd.

• Informed trades: When traders have privileged information, their trades can have a larger impact on price.

• Uninformed trades: These are like a casual impulse buy at the ice cream truck—they might cause a minor stir but generally don’t move the market much.

• Challenge: Distinguishing between these types of trades is difficult, and models must account for the fact that not every trade is equally influential.

3. Time decay and impact fading.

Market impact is not static. Just as the excitement around a new ice cream flavor fades after the initial rush, the impact of a large order decays over time.

• Mathematical twist: Some models introduce a time decay factor to account for how the price impact dissipates. Though we won’t dive into the complex mathematics here, the idea is that the market forgets large orders over time.

• Practical implication: A trade that moved the market significantly a few minutes ago may have little impact now, allowing liquidity to recover.

4. Liquidity and market conditions.

Market liquidity can vary wildly. In highly liquid markets, even large orders might only cause a small ripple. In contrast, in a thinly traded market, the same order could create a tidal wave.

• Implication: Trading in a low-liquidity market is like trying to swim in a kiddie pool—every movement creates a big splash!

• Consideration: Traders must adjust their strategies based on current liquidity levels, and models must be flexible enough to handle these variations.

5. Noise and randomness.

Markets are influenced by countless factors—news, economic data, investor sentiment, and even the weather. This randomness—aka noise—makes it challenging to isolate the pure effect of a trade.

• Implication: Even the best models sometimes get it wrong because the market is a living, breathing entity full of surprises.

While these challenges might seem daunting, they also highlight the beauty of the markets: a place where mathematics meets human behavior, and every trade tells a story.

P.S. You don't mind helping me filter topics for the newsletter, do you? Let me know which ones you prefer!

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👍 Did you gain valuable insights and now want to empower others in the field?

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Appendix

Don't forget to check out this Rust example for inspiration on how to identify toxic flow by creating the VPIN metric [GitHub-VPIN].