Local volatility

Heston explains the smile with a random volatility. Dupire showed that a deterministic volatility that depends on the spot level and time can reproduce any arbitrage-free smile exactly. Pick a market shape, read off the local volatility his formula implies, and check by simulation that it really does give the smile back.

Implied surface

At-the-money volatility: 20%

Skew: -0.10

Curvature: 0.04

Term slope: +0.02 per year

Market and check

Spot (S)

Rate (r)

Yield (q)

Check strike (K)

Check maturity (T)

Paths

Time steps

Try this

0 of 3 done

Flat in, flat out

Choose the flat preset and compute.

What to look for: Both heatmaps show the same number everywhere and the round-trip check recovers it. With a flat smile, local volatility is just Black-Scholes.

The two-times rule

Choose the equity-skew preset and read the skew ratio under the slice chart at 0.1 and at 2 years.

What to look for: The local skew is roughly twice the implied skew at short maturities and closer to it at long ones. Implied volatility averages local volatility along the path.

Break the surface

Push the curvature slider to 0.5 and compute.

What to look for: Cells start failing in the warning: the butterfly spread has gone negative somewhere, meaning the implied surface would allow arbitrage. Real surfaces cannot look like that.

Implied vol at (K, T)

20.51%

From the shape

Local vol at (K, T)

22.57%

Dupire's formula

Simulated with local vol

20.30%

Implied vol of the simulated price, $5.69 ± 0.0704

Round trip

-0.21%

Recovered minus implied; Monte Carlo noise is a few tenths of a percent

Slice at maturity

The local skew is about twice the implied skew at short maturities: the implied volatility at a strike averages the local volatility over the paths that end there, which flattens it. Here the ratio at this maturity is 1.62.