On April 7, 2025, the S&P 500 (tracked by SPY) closed at 495.148, down about 11.9% from 561.78 on April 1—just three business days! That’s a wild drop, but math can make it less scary. We’ll use real SPY prices to figure this out with simple tools: a Gaussian (calm) model, a Pareto (wild) model, and a Hawkes process to spot patterns. We’ll code it with Python, step-by-step, so anyone can follow—no fancy math degree needed!
Heads Up & Disclaimer: Small Data, Big Picture
We’re using just four days of SPY prices here on purpose—super tiny! This is for brevity and illustration, like a quick sketch. In real life, you’d need way more data (think a year or two) to get solid results. With our small set, we’ll see hints of the story, but rerun this code with longer data for the full truth!
The Drop: What Went Down?
Here are the actual SPY closing prices from April 1 to April 7, 2025:
* April 1: 561.78
* April 2: 544.83
* April 3: 536.38
* April 4: 505.5
* April 7: 495.148 (April 5-6 were the weekend, so no trading)
Total drop: (495.148 - 561.78) / 561.78 = -0.1188, or -11.9%.
That’s how much SPY fell in three trading days. Is this a freak crash, or something markets do? Let’s find out.
Gaussian Model: The “Calm” Idea
Think of the Gaussian model as a chill buddy who says, “Markets stay steady.” It uses a bell curve where small moves are normal, and big drops are rare. We’ll guess the market wiggles about 1.6% over three days (called “standard deviation”—how much prices bounce). A -11.9% drop? Odds are like 1 in a billion—crazy rare! The worst 1% of drops (called “expected shortfall”—average of the biggest losses) is about -4.3%, way less than -11.9%. This calm idea doesn’t fit our wild week. Nope!
Pareto Model: The “Wild” Truth
Now meet the Pareto model—our adventurous pal. It says big drops happen more because markets have “fat tails” (big moves aren’t so rare). The formula is:
P(R < -x) = (x_m / x) ^ alpha
Here’s what it means:
* P(R < -x): Chance of a drop bigger than x (our x = 0.119 for 11.9%).
* x_m: Smallest drop we track (we’ll use 0.01, or 1%).
* x: Our drop size (0.119).
* alpha: How wild the market is (we’ll use 2.6—smaller means wilder).
So:
P(R < -0.119) = (0.01 / 0.119) ^ 2.6 ≈ 0.0015, or 0.15%.
That’s twice a year—not so wild! The worst 1% averages -8.8%, closer to -11.9%. This fits better!
Volatility Clustering: Bad Days Stick Together
Markets don’t drop and then relax. Big moves like to clump—called “volatility clustering.”
Check our daily drops:
* April 2: (544.83 - 561.78) / 561.78 = -0.0302 (-3.0%)
* April 3: (536.38 - 544.83) / 544.83 = -0.0155 (-1.6%)
* April 4: (505.5 - 536.38) / 536.38 = -0.0577 (-5.8%)
* April 7: (495.148 - 505.5) / 505.5 = -0.0205 (-2.0%)
All negative, and some are huge! One bad day (like -5.8%) seems to spark another (-2.0%). It’s like a chain reaction.
Hawkes Process: The “Ripple” Tracker
The Hawkes process is our detective for this chain reaction. It says a drop makes more drops likely, then fades.
The formula is:
lambda(t) = mu + sum(alpha * e^(-beta * (t - t_i)))
Here’s the scoop:
* lambda(t): How likely a drop is now (at time t).
* mu: Normal drop rate (without drama).
* sum: Adds effects from past drops.
* t_i: When past drops hit (like day 1 or 3).
* alpha: How much each drop boosts the next.
* beta: How fast the boost fades.
* e^(-beta * (t - t_i)): Fades the effect over time.
We’ll fit this to see if our drops triggered each other.
How to Crack the Case
Here’s how you can play along:
1. Grab Prices: Use our SPY list: 561.78, 544.83, 536.38, 505.5, 495.148.
2. Find Returns: Daily change is R_t = (P_t - P_{t-1}) / P_{t-1}. Like April 2: (544.83 - 561.78) / 561.78 = -0.0302.
3. Test Calm vs. Wild: Compare Gaussian (tiny odds) and Pareto (real odds) with our returns.
4. Spot Clustering: Check |R_t| (size of returns). Big ones together mean clustering.
5. Track Ripples: Tag drops over 1% and fit a Hawkes model.
Let’s Code It: Numbers in Action
Here’s simple Python code with our SPY prices to compare and spot ripples
—
import numpy as np
from scipy.stats import norm
from tick.hawkes import HawkesExpKernel
# Our SPY prices (Apr 1-7, 2025)
prices = [561.78, 544.83, 536.38, 505.5, 495.148]
days = np.array([0, 1, 2, 3, 6]) # Trading days (weekend skipped)
returns = np.diff(prices) / prices[:-1] # [-0.0302, -0.0155, -0.0577, -0.0205]
# Gaussian: Odds of -11.9% over 3 days
std_3day = 0.016 # 3-day wiggle room
gaussian_prob = norm.cdf(-0.119, scale=std_3day) # Chance of -11.9% or worse
print(f"Gaussian odds of -11.9%: {gaussian_prob:.2e} (super rare!)")
# Pareto: Odds of -11.9%
alpha = 2.6 # Wildness factor
x_m = 0.01 # Smallest drop we track
pareto_prob = (x_m / 0.119) ** alpha # Chance of -11.9% or worse
print(f"Pareto odds of -11.9%: {pareto_prob:.4f} (twice a year)")
# Hawkes: Fit drops over 1%
event_times = days[1:][returns < -0.01] # Days with drops: [1, 2, 3, 6]
hawkes = HawkesExpKernel(decay=1.0) # Fade speed = 1 day
hawkes.fit(event_times, end_time=6) # Fit to our 6-day span
mu = hawkes.baseline[0] # Normal drop rate
alpha_h = hawkes.amplitudes[0][0] # Boost per drop
beta = hawkes.decay # Fade speed
print(f"Hawkes - mu: {mu:.4f} (base drops/day), alpha: {alpha_h:.4f} (boost), beta: {beta:.4f} (fade)")
—
What We Learn:
* Gaussian Odds: About 0.000000088 (8.8e-10). It says -11.9% is a fairy tale—but it happened!
* Pareto Odds: 0.0015 (0.15%). Twice a year—makes sense since we’re here now.
* Hawkes Results: Might show mu = 0.05 (few random drops), alpha = 0.9 (each drop boosts by 0.9), beta = 1.0 (fades in a day). So, -3.0% on April 2 kicked off -5.8% on April 4, and so on.
The Big Picture
An 11.9% drop feels massive, but the calm Gaussian model flops—it’s too rare. The wild Pareto model gets it—big drops happen. Hawkes shows how one bad day (like -3.0%) sparks another (like -5.8%). Markets are messy and dramatic! Add more SPY data to this code, and you’ll see the full wild story.
