Local Volatility

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This page covers the Dupire local volatility model. For context on how it fits into the vol surface pipeline, see How Surfaces Are Built. For a comparison with other methods, see Interpolation Methods.

Local volatility is the instantaneous volatility of the underlying at a specific price level and point in time. It is the function $\sigma_{\text{loc}}(S, t)$ that, when plugged into a diffusion model, reproduces all observed European option prices exactly.

The idea was developed independently by Bruno Dupire (1994) and Derman & Kani (1994). It is the unique model that matches the full implied vol surface without any additional assumptions.

Implied Vol vs. Local Vol

The key insight: implied vol is a blended average of the local vols along the path to expiry. Local vol is the instantaneous, "pointwise" volatility.

Implied Vol vs. Local Vol

Typical equity/crypto smile. Local vol is higher and more peaked than implied.

Local vol (solid) is always more peaked than implied vol (dashed). Implied vol is a weighted average of the local vols along the path.

Toggle between smile shapes and notice:

• Local vol is always more peaked. Because implied vol averages over the path, it smooths out the extremes. Local vol shows the raw, un-averaged picture.
• Steeper implied skew = more dramatic local vol. In the steep skew case (crisis), local vol diverges in the left wing. This is the model saying: "If spot drops that far, instantaneous vol would need to be very high to match observed put prices."
• The relationship is like spot rates vs. forward rates. Implied vol is the spot rate (average from now to maturity). Local vol is the forward rate (the instantaneous rate at a future point).

What Local Vol Means

Think of the underlying price evolving through a landscape of volatilities. At each point in (price, time) space, there is a specific volatility. As the underlying wanders, it experiences different instantaneous vols.

A 30-day ATM option with 50% implied vol might traverse local vols ranging from 40% to 65% along its path. The 50% implied vol is the risk-neutral average over all those local vols, weighted by the time spent at each level.

This is why two options with different strikes can have different implied vols even though they depend on the same underlying process: they traverse different parts of the local vol landscape.

When to Use Local Vol

Exotic option pricing

Local vol's primary use case. The workflow:

1. Observe European option prices (or implied vols) in the market
2. Fit an arbitrage-free implied vol surface (using SVI, SSVI, or similar)
3. Derive the local vol surface via Dupire's formula
4. Build a numerical pricing engine (finite difference PDE or Monte Carlo) using the local vol surface
5. Price the exotic by evolving the underlying through the local vol landscape

The guarantee: any exotic priced under local vol is consistent with all observed European option prices. Your barrier option price does not contradict the vanillas, which matters for hedging.

Surface-consistent Greeks

Greeks computed under local vol account for the fact that vol changes as spot moves. Delta under local vol differs from Black-Scholes delta because the model "knows" that moving to a different spot level means experiencing a different local vol. This is conceptually similar to what Taleb calls "shadow gamma": the extra delta change that comes from vol changing because spot moved.

The Dynamics Problem

Local vol has one well-known weakness: it predicts wrong smile dynamics.

Under local vol, volatility is a deterministic function of spot. Once you know where spot is, you know exactly what vol is. There is no "surprise" in vol. This means:

• When spot drops, local vol says vol was always going to be this high at this price. The smile flattens.
• In practice, when spot drops, vol often increases more than what local vol predicts, and the smile steepens.

The result: local vol systematically underprices options that depend on future smile shape (barrier options, forward-starting options, cliquets).

Local Vol vs. Other Models

	Local Vol	SVI	SABR
What it is	Instantaneous vol at each (S, t)	Parametric smile shape	Stochastic vol model
Observable?	No (derived)	No (fitted)	No (fitted)
Exact calibration	Yes (by construction)	Approximate	Approximate
Smile dynamics	Wrong (deterministic)	Not specified	Better (stochastic)
Exotic pricing	Yes (primary use)	No	Limited
Speed	Slow (PDE/MC)	Fast	Fast (formula)
Best for	Barriers, Asians, exotics	Vanilla pricing, risk	Swaptions, FX vanillas

Connection to Other Models

Implied vol to local vol: Dupire's formula. Requires an arbitrage-free implied surface as input.

Local vol to implied vol: Run a forward PDE under the local vol surface, price Europeans, invert to get implied vols. This round-trips exactly by construction.

SABR and local vol: SABR's $\beta$ parameter controls the local vol backbone ($\sigma_{\text{loc}} \sim F^\beta$), while $\nu$ adds a stochastic layer on top. SABR can be viewed as a parametric approximation to local vol with extra dynamics.

SVI and local vol: SVI gives you the implied surface. Dupire then gives you local vol. The chain: market quotes -> SVI fit -> implied surface -> Dupire -> local vol -> exotic pricer.

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See also:

• SVI Parameterization - The model used to build the implied surface
• SABR Model - A stochastic vol alternative
• Interpolation Methods - All methods compared
• How Surfaces Are Built - The full pipeline