Two ways to explain a smile

Heston fits the smile by making volatility random. Dupire (1994) asked a different question: is there a volatility that depends only on where the spot is and what time it is, \(\sigma(S, t)\), that reproduces the market's option prices exactly? The answer is yes, and it is unique. The model is

$$ dS_t = (r - q)\, S_t\, dt + \sigma(S_t, t)\, S_t\, dW_t $$

with a single Brownian motion, so the market is still complete and every option can be hedged with the stock alone.

Dupire's formula

Write call prices as a function of strike and maturity, \(C(K, T)\). The local variance at spot level \(K\) and time \(T\) is

$$ \sigma^2(K, T) = \frac{\dfrac{\partial C}{\partial T} + (r - q)\, K \dfrac{\partial C}{\partial K} + q\, C} {\tfrac{1}{2} K^2 \dfrac{\partial^2 C}{\partial K^2}} $$

The denominator is the risk-neutral density of the spot at \(T\), scaled: Breeden and Litzenberger showed that \(\partial^2 C / \partial K^2 = e^{-rT} \times\) density. So local variance is a ratio of calendar spread to butterfly spread. If either is negative the surface admits arbitrage and the formula fails, which is exactly what the calculator reports when a shape is too extreme.

Reading the surfaces

Implied volatility at a strike is a kind of average of the local volatility along the paths that finish there. Averaging flattens, so local volatility is more extreme than implied: for short maturities the local skew is about twice the implied skew, the "two times rule". Local volatility is also more curved and moves more with maturity. The calculator's two heatmaps make the comparison directly, and the simulation check confirms that pricing with the local surface gives the implied surface back.

Where it falls short

Local volatility fits today's smile perfectly but predicts its dynamics badly: when the spot moves, the model says the smile slides in the opposite direction to what is observed, and forward-starting smiles flatten out. Desks therefore use it for products that depend only on the terminal distribution, such as digitals and simple barriers, and reach for stochastic or stochastic-local volatility when the future smile matters. Bergomi's remark sums it up: local volatility is the right model for pricing, and the wrong model for hedging.