The Milstein Scheme

The Euler-Maruyama scheme simulates a stochastic differential equation by taking small time steps and, at each step, adding a drift term and a noise term scaled by the current volatility — treating the volatility as if it stays constant across that one small step. But if volatility itself depends on the asset price (as it does in geometric Brownian motion and almost every realistic model), that approximation quietly loses accuracy: the noise term should really reflect how volatility is also moving during the step, not just its value at the start. The Milstein scheme adds exactly one extra term to correct for this, using a piece of Itô calculus that Euler-Maruyama leaves out.

An analogy: predicting a hiker's position on shifting terrain

Imagine forecasting a hiker's position after a step where each stride's length itself depends on the ground underfoot, and the ground's steepness is changing as the hiker moves. A naive forecast uses the steepness at the start of the stride to size the whole stride. A better forecast accounts for the fact that the steepness itself is drifting slightly as the stride happens, adding a small correction proportional to how fast the steepness changes with position. That correction is exactly what the Milstein scheme adds to the plain Euler forecast, using the derivative of volatility with respect to the asset price.

The scheme, one symbol at a time

For an SDE $dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t$, where $\mu$ is the drift, $\sigma$ is the volatility (which can depend on $X_t$), and $dW_t$ is a Brownian increment, the Euler-Maruyama update over a step of size $\Delta t$ is $X_{t+\Delta t} \approx X_t + \mu(X_t)\Delta t + \sigma(X_t)\Delta W_t$, where $\Delta W_t$ is a normal random draw with variance $\Delta t$. The Milstein scheme adds one term:

$$X_{t+\Delta t} \approx X_t + \mu(X_t)\Delta t + \sigma(X_t)\Delta W_t + \tfrac{1}{2}\sigma(X_t)\sigma'(X_t)\left[(\Delta W_t)^2 - \Delta t\right],$$

where $\sigma'(X_t)$ is the derivative of the volatility function with respect to the state. In plain English: the extra piece corrects for the fact that volatility isn't really constant across the step — it captures, to leading order, how the size of the random kick itself is affected by where the state ends up moving during the very step being simulated. This correction comes directly out of Itô's lemma: it's the same $\frac{1}{2}\sigma^2$ second-derivative adjustment that appears whenever Itô calculus is applied, here applied to $\sigma$ itself.

Worked example 1: one step of GBM by hand

Geometric Brownian motion has $\mu(X) = rX$, $\sigma(X) = \sigma_0 X$ (so $\sigma'(X) = \sigma_0$). Take $X_0 = 100$, $r = 0.05$, $\sigma_0 = 0.2$, $\Delta t = 0.01$, and a drawn Brownian increment $\Delta W = 0.05$. Euler-Maruyama: $X_{\Delta t} \approx 100 + (0.05)(100)(0.01) + (0.2)(100)(0.05) = 100 + 0.05 + 1.0 = 101.05$. Milstein adds $\tfrac{1}{2}(0.2 \times 100)(0.2)\left[(0.05)^2 - 0.01\right] = \tfrac{1}{2}(20)(0.2)(0.0025 - 0.01) = 2 \times (-0.0075) = -0.015$. So the Milstein estimate is $101.05 - 0.015 = 101.035$ — a small correction here because the step is small, but the correction term does not shrink as fast as the leading terms do, which is exactly why it improves long-run accuracy.

Worked example 2: why the correction matters over many steps

Simulating the same GBM path over 100 steps, Euler-Maruyama's local error at each step is of order $\Delta t$ but accumulates a global error of order $\sqrt{\Delta t}$ in the strong sense (path-by-path accuracy). Milstein's correction improves the strong global error to order $\Delta t$ — for $\Delta t = 0.01$, that's roughly a $\sqrt{0.01}/0.01 = 10\times$ improvement in path-tracking accuracy for the same number of steps. For pricing instruments that depend on the whole simulated path (barrier options, path-dependent payoffs) rather than just the terminal value, this directly translates into needing far fewer time steps to hit a given accuracy target.

Compare paths generated at coarse versus fine time steps above — Milstein's correction is precisely what keeps a coarse-step path close to what a very fine-step path would have produced, for models where volatility depends on the state.

What this means in practice

The Milstein scheme is used whenever path-level accuracy matters and volatility is state-dependent — barrier and lookback option pricing, local volatility model simulation, and any setting where Euler-Maruyama's cruder approximation would require impractically many time steps to reach acceptable accuracy. When $\sigma$ is constant (as in arithmetic Brownian motion with constant volatility), $\sigma'=0$ and the Milstein correction vanishes entirely — Euler-Maruyama is already exact in that special case, which is a useful sanity check that the extra term is implemented correctly.