Rough volatility · teaching module · Layer 1b — pricing (Q2)

Can you price it cheaply?

Pricing an option under rough volatility means heavy Monte Carlo. The celebrated speed-up is multilevel Monte Carlo (MLMC) — simulate on coarse-to-fine grids and telescope. But it only pays off if the correction between grids shrinks fast enough as you refine. Drag the roughness below and watch whether it does.

Start here — a two-minute guided tour

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01 · the premise

Start at H = 0.7

The teal line is how much the coarse-vs-fine correction shrinks at each grid level. Here it drops faster than the dashed cost line (β > γ) — so refining is cheap, and MLMC does exactly what the textbooks promise.

02 · go rough

Drag H down toward 0.1

Watch the teal line flatten. The rougher the vol, the slower the correction shrinks — coarse and fine rough paths barely line up. The decay rate is β ≈ 2H.

03 · the stall

Watch it cross the cost line

Once the correction shrinks slower than cost grows (β < γ), MLMC's whole advantage evaporates — for rough Asian options it even loses to plain Monte Carlo. What actually wins is in the Advanced panel.

Does the correction shrink fast enough? H = 0.70 · β = 1.40 · γ = 1.00
β · variance rate
1.00
how fast the correction shrinks (≈ 2H)
γ · cost rate
1.00
how fast work grows per level
β − γ
0.00
positive → MLMC pays · negative → it stalls
The correction shrinks faster than cost grows — MLMC's premise holds.
Set the roughness
0.70H
— smooth vol · MLMC thrives
0.05 · rough0.95 · smooth
0.1
0.15
0.5
0.7
0.9
Grid levels shown7
the coarse-to-fine hierarchy MLMC telescopes over

Advanced — what actually prices it cheaply

THE FINDING show ▼
MLMC complexity (Giles rates) with the project’s measured β ≈ 2H · cost ratios from the P2 matched-L runs · runs in your browser