Euler And Milstein Schemes

Simulating a stock's path forward one time step at a time means picking a rule for turning "where I am now" plus "a random shock" into "where I am next." The simplest such rule, Euler-Maruyama, takes a straight step: it uses today's volatility for the whole step and never asks whether that volatility itself is changing within the step. When volatility doesn't depend on the price — plain Brownian motion — that's exact. The moment volatility depends on the price, as it does for a stock under geometric Brownian motion, the straight step introduces an error, and Milstein's scheme is the fix: one extra term that accounts for how much the path curves during the step it's taking.

Walking a hillside blindfolded

Walking blindfolded across a hillside in fixed-length strides, always stepping in the direction you were facing at the start of each stride, works fine on flat ground but drifts you off course on a slope, because the ground's steepness changes as you walk and your stride never adjusts mid-step. A hiker who adjusts each stride's direction slightly for how the slope curves underfoot as they walk stays truer to the real path. Euler-Maruyama is the blindfolded straight-stride walker; Milstein is the hiker who corrects for the curve within each step.

$$\text{Euler:} \quad S_{t+\Delta t} = S_t + \mu S_t \Delta t + \sigma S_t \Delta W$$ $$\text{Milstein:} \quad S_{t+\Delta t} = S_t + \mu S_t \Delta t + \sigma S_t \Delta W + \tfrac12 \sigma^2 S_t \left(\Delta W^2 - \Delta t\right)$$

In plain English: both start the same way — a drift term $\mu S_t \Delta t$ plus a random shock $\sigma S_t \Delta W$, where $\Delta W$ is a normal random variable with variance $\Delta t$. Milstein adds one more term, proportional to $\sigma^2$ and to how much $\Delta W^2$ overshoots or undershoots its own average value of $\Delta t$. That correction exists precisely because volatility here is $\sigma S_t$, which itself depends on $S_t$ — as the price moves during the step, the "correct" volatility to have used was subtly different from the volatility at the step's start, and this term captures the leading-order size of that discrepancy.

Worked example 1: one step, both schemes, same random draw. Take $S_0=100$ (a $100 stock), $\mu=0$, $\sigma=30\%$, $\Delta t=0.01$ years (about 2.6 trading days), and suppose the random draw happens to be $\Delta W = 0.05$ (a genuine, if slightly large, one-step normal draw with standard deviation $\sqrt{0.01}=0.10$). Euler: $S = 100 + 0 + 0.30\times100\times0.05 = 100 + 1.50 = 101.50$, i.e. $101.50. Milstein: the correction term is $0.5\times0.09\times100\times(0.05^2-0.01) = 4.5\times(0.0025-0.01)=4.5\times(-0.0075)=-0.03375$, so $S = 101.50 - 0.034 = 101.47$, i.e. $101.47. A small difference on this one step — about 3 cents — but it's systematic, not noise, and it doesn't average away across steps the way random error does.

Worked example 2: comparing to the exact answer. For geometric Brownian motion the exact update is known in closed form: $S_{t+\Delta t} = S_t \exp\left[(\mu-\tfrac12\sigma^2)\Delta t + \sigma \Delta W\right]$. With the same numbers: exponent $= (0-0.045)\times0.01 + 0.30\times0.05 = -0.00045+0.015=0.01455$, so $S=100\times e^{0.01455}=100\times1.01466=101.47$, i.e. $101.47. Milstein's $101.47 matches the exact answer to the cent; Euler's $101.50 is off by three cents on a single step — and that Euler bias compounds and grows (though it still shrinks toward zero as $\Delta t\to0$) across thousands of simulated paths and steps in a way Milstein's does not, at the same order.

Shrink the step size mentally as you look at a simulated path: Euler's error shrinks too, but more slowly (order $\sqrt{\Delta t}$ in the strong sense) than Milstein's (order $\Delta t$) — Milstein needs far fewer steps for the same accuracy whenever volatility depends on the price.

What this means in practice

For plain geometric Brownian motion, most desks just use the exact lognormal update above and skip discretisation error entirely. Milstein earns its keep on models where no exact update exists but volatility still depends on the state — local volatility models, CIR-type variance processes in Heston — where its extra term meaningfully reduces the number of steps needed for Monte Carlo Option Pricing to converge to a given accuracy.