The math trick quants use instead of prices

Why quants model log returns instead of prices, and how that leads to geometric Brownian motion.

Play with it

Log returns. See why +10% then −10% doesn't get you back, and why log returns simply add up.

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Up and back down is not a round trip

Move up, then down by10%

To get from $110.00 back to $100, the price only needs to fall 9.09%.

Start

$100.00

Up 10%

$110.00

Down 10%

$99.00

Step	Simple return	Log return
$100 → $110.00	+10.00%	+9.53%
$110.00 → $100	-9.09%	-9.53%
Round trip adds to	+0.91%	0.00%

A 10% gain followed by a 10% loss leaves you at $99.00, not $100. Simple returns are lopsided: up 10.00% but back down only 9.09%. Log returns are symmetric: +9.53% up and -9.53% back, adding to exactly zero.

Log returns add up, simple returns don't

Monthly volatility8%

$100 became$128.30

Real total return+28.30%

Simple return each month Log return each month

Sum of 12 simple returns

+28.58%

≠ real total return +28.30%

Sum of 12 log returns

+24.92%

= ln($128.30 ÷ $100), exactly

Adding up this year's monthly simple returns gives +28.58%, but $100 actually became $128.30, a +28.30% return. The gap of 0.28% comes from compounding. The log returns add to +24.92%, and e raised to that gives the real total back.

Model log returns and prices stay positive

Volatility50% a year

Dollar-step paths that went below $022%

Geometric Brownian motion paths below $00%

Random dollar steps

Geometric Brownian motion

Both panels use the same random shocks over three years. Added as fixed dollar amounts, 22% of prices fall below zero, which no real price can do. Treated as normally distributed log returns, the price is the exponential of the process, so it never reaches zero.

Try this

Set the move to 10%: $100 goes to $110, then back down to $99. Recovering only takes a 9.09% fall.
Try a 50% move. How far from $100 do you end up?
Draw a new year of returns a few times. The log sum always matches ln(end ÷ start); the simple sum never quite does.
Raise the volatility in the last part and count how many dollar-step prices go negative.

The math

r_t = \ln\frac{P_t}{P_{t-1}} \qquad \sum_{t=1}^{n} r_t = \ln\frac{P_n}{P_0}

$r_t$: log return in period t
$P_t$: price at the end of period t
$\ln$: natural logarithm
$P_n = P_0 e^{\sum r_t}$: prices are the exponential, so always positive

Quick check

Two questions to see if it clicked. Answers explain themselves.

A price goes from $100 to $110 and back to $100. What is the percentage loss on the way down?

What is the key advantage of log returns?