Using Geometry to See Why Volatility Scales With √t

I had Claude make me a custom chalkboard app for making math and investing tutorials Khan Academy-style. For fun, I asked Claude to use it to teach me something. Claude knows me better than my mother so naturally it chose to talk about volatility. 

It created this geometric intuition for volatility. Watch the video and then if you feel like visiting trigonometry, below you can read a full treatment of what cosines have to do with correlation and volatility estimates!

The rule of thumb for a random walk is that volatility scales with the square root of time. 1% a day is about 16% a year because √252 ≈ 16.

This post builds that rule from a triangle, one day at a time. Then it bends the triangle to show what changes when days aren’t independent.

Day 1

One day, one move. Call its size σ, one standard deviation. After day 1 you’re σ from where you started.

Day 2

Day 2 adds another move of size σ. If day 2 has nothing to do with day 1, draw it at a right angle to day 1. (Independent = perpendicular. Take that on faith for now.)

Now there’s a right triangle with two legs of σ. The distance from the starting point is the hypotenuse.

Two days of moves, 1.41 days’ worth of distance.

Day 3

Day 3 adds another σ, again at a right angle. This time the right angle is to the line from the start to where you are after day 2.

a = σ√2 (where you are after 2 days)
b = σ (day 3’s move)
c = the new distance from the start

Check with σ = 1: 1.414² + 1² = 2 + 1 = 3, and √3 ≈ 1.732.

Every independent day adds one σ² to the squared distance. After t days the squared distance is t·σ², so the distance is σ√t.

That’s why variance (σ²) is the thing that adds across days and volatility (σ) is the thing that scales with √t. Squaring undoes the root.

Run it for six days and you get the spiral of Theodorus: σ, σ√2, σ√3, 2σ, σ√5, σ√6. Four days of moves put you 2σ from the start. (The video above draws it one triangle at a time.)

The right angle is the assumption

Everything above assumed each new day was perpendicular to the path so far. That’s the independence assumption. To see what happens when it fails, the angle has to be allowed to be something other than 90°.

This is where “cosine” shows up and people close the tab. If you took trig in high school you already have everything you need, even if it’s been 30 years.

Cosine is a ratio of two sides of a right triangle:

That’s the whole definition. (SOH CAH TOA. The CAH part.) A few values:

• cos 0° = 1
• cos 60° = 0.5
• cos 90° = 0
• cos 120° = −0.5

Bending day 3

Back to day 3. Same a = σ√2, same b = σ. Instead of fixing day 3’s move at 90°, turn it by an angle θ from the direction you’ve been traveling.

Cosine splits day 3’s move into two pieces:

• along the path so far: b·cos θ
• sideways: b·sin θ

Those two pieces make a right triangle with the full distance c as the hypotenuse, so Pythagoras still works:

Multiply it out and use cos²θ + sin²θ = 1:

That’s the law of cosines: Pythagoras plus a correction term, 2ab·cos θ. At 90°, cos θ = 0 and the correction term is zero.

Plug in a = σ√2 and b = σ:

• θ = 60°, cos θ = 0.5: c² = 3σ² + 1.41σ² = 4.41σ², so c ≈ 2.10σ
• θ = 90°, cos θ = 0: c² = 3σ², so c ≈ 1.73σ
• θ = 120°, cos θ = −0.5: c² = 3σ² − 1.41σ² = 1.59σ², so c ≈ 1.26σ

Same three daily moves of σ. Three different distances.

What the angle is in market terms

The variance of the sum of two moves X and Y:

Next to the law of cosines:

Same equation. The side lengths are standard deviations and cos θ is the correlation ρ between the new day’s move and the path so far.

• θ = 0°, ρ = +1: the new day repeats the path
• θ = 60°, ρ = +0.5: leans with it (momentum)
• θ = 90°, ρ = 0: independent
• θ = 120°, ρ = −0.5: leans against it (reversion)
• θ = 180°, ρ = −1: undoes it

The √t rule is the 90° line.

Six days, then 20

Hold the angle fixed every day and walk six days.

• 60°: 4.22σ from the start after 6 days, vs σ√6 = 2.45σ
• 90°: 2.45σ, the spiral
• 120°: 1σ. The path closes into a hexagon around the start.

Over 20 days:

• Below 90° the distance grows close to linearly, about σ·cos θ per day once you’re away from the start (≈0.5σ a day at 60°). 60° ends at 11.9σ vs σ√20 = 4.5σ.
• Above 90° the distance levels off at −σ ÷ (2·cos θ): 1.93σ at 105°, 1σ at 120°.

Caveat: a fixed angle to the running total is a toy. The usual measure is the correlation between returns k days apart (autocorrelation at lag k). The arithmetic is the same. Positive cross terms make the distance longer than σ√t. Negative ones make it shorter.

Daily hedger vs holder to expiry

The variance ratio compares the two:

ρₖ is the correlation between returns k days apart. VR = 1 is independent, VR > 1 is momentum, VR < 1 is reversion. The sum is the 2ab·cos θ term added up over every pair of days.

Example: 1% daily vol (15.9% annualized), lag-1 autocorrelation ρ, and ρₖ = ρᵏ.

• ρ = −0.5: VR(20) = 0.36, so vol measured on 20-day changes annualizes to 9.5%
• ρ = 0: VR(20) = 1, 15.9%
• ρ = +0.5: VR(20) = 2.80, 26.6%

Which number matters depends on how you hold the option.

• Delta-hedging daily, P&L each day is roughly ½·Γ·S² × (that day’s squared return − one day of implied variance). The daily squared returns are what pay you.
• Holding to expiry unhedged, the payoff depends on where the underlying ends up. That’s the 20-day number.

Same price series in a reverting market: the daily hedger realizes 15.9%, the holder to expiry gets the distribution of a 9.5% vol. In a trending market it flips: 15.9% for the hedger, 26.6% for the holder.