Monte Carlo & the volatility smile.

A diffusion $d{S}_t=\mu (S_t)dt+\sigma (S_t)d{W}_t$ is simulated on a grid. The Euler scheme is the obvious one:

The Milstein scheme adds the Itô correction from expanding $\sigma$ along the path:

The payoff is theoretical orders of strong convergence - how fast the simulated path approaches the true one - of ½ for Euler and 1 for Milstein. The experiment below measures it: GBM paths simulated at six step sizes against the exact solution driven by the same Brownian increments, error on a log-log plot. The fitted slopes land on the theory within Monte Carlo noise.

Black-Scholes assumes constant volatility, so it prices every strike with the same implied vol - a flat line the market flatly refuses. The Heston model makes variance itself a mean-reverting diffusion, correlated with the spot:

with $d⟨W^S,W^v{⟩}_t=\rho dt$. The figure simulates Heston with a full-truncation Euler scheme (variance floored at zero inside the coefficients - the lab's fix for the scheme going negative), prices calls across strikes with antithetic paths, and then inverts Black-Scholes by bisection to express each price as an implied vol - the exact pipeline a vol desk runs in reverse.

Two parameters shape the smile. Vol-of-vol η controls its curvature: at $\eta \approx 0$ variance barely moves and the smile collapses onto the flat Black-Scholes line at $\sqrt{\theta }$. Correlation ρ controls the skew: equity markets live around $\rho \approx -0.7$ - spot down, vol up - which tilts the smile so low strikes (crash protection) carry the highest implied vol. Set $\rho =0$ and you get a symmetric smile; positive ρ tilts it the other way, the signature of some commodity markets.