Euler-Maruyama and Milstein schemes

Simulating a random equation

Most stochastic equations have no formula for their solution, so we simulate them. Cut time into steps of size $h$. The Euler-Maruyama scheme moves the state by the drift for one step plus the volatility times a random step $\Delta W$, a normal number with mean $0$ and variance $h$. For $dX=a(X)\,dt+b(X)\,dW$ it reads $X_{n+1}=X_n+a(X_n)\,h+b(X_n)\,\Delta W$. It is the simplest rule: the stochastic version of Euler's method.

The Milstein scheme adds one term, $\tfrac12\,b\,b'(\Delta W^{2}-h)$. It comes from the quadratic variation and fixes the part of the error that Euler ignores. For geometric Brownian motion $b=\sigma x$, so $b\,b'=\sigma^{2}x$. How good are the schemes? The strong error measures how close each simulated path is to the true path: Euler is of order $\tfrac12$ (the error shrinks like $\sqrt h$) and Milstein of order $1$ (like $h$). The weak error measures how close averages are, such as an option price: both are of order $1$. So to price a European option, Euler with many paths is usually enough. For hedging tests or path-dependent products, Milstein pays off.

• Euler-Maruyama: drift for one step plus volatility times a normal step. Strong order $\tfrac12$.
• Milstein: adds $\tfrac12\,b\,b'(\Delta W^{2}-h)$ and reaches strong order $1$.
• Strong vs weak error: paths versus averages. Pricing needs the weak error, which is order $1$ for both.

Lab · what's graded

• One Euler step15 pts
• One Milstein step15 pts
• Euler error when the step is cut15 pts
• Milstein error when the step is cut15 pts
• Which error matters for pricing15 pts
• Match the vocabulary10 pts
• Why Milstein beats Euler15 pts