One Model That Fits Every Option: Dupire's Local Volatility

By the early 1990s, every equity options desk was living a contradiction. They used the Black-Scholes formula, which assumes one volatility, while feeding it a different volatility for every strike and every maturity. It worked as a quoting convention, but as a model it was incoherent: you were simultaneously asserting that the stock's volatility was 25 percent (for one option) and 32 percent (for another), about the same stock, over the same period.

For plain vanilla options this hypocrisy was survivable, because you were only using the formula to convert between price and a quoting unit. But the moment you wanted to price an exotic, a barrier option, a cliquet, anything path-dependent, you needed a real model of how the stock actually moves. And you had none that was consistent with the vanilla prices sitting right next to it on the screen. Sell an exotic priced off one model while the vanillas you hedge with are priced off another, and you have handed someone an arbitrage.

Bruno Dupire's short 1994 article in Risk solved this, and the solution is startling.

The problem: infinitely many models, and none that fit

Suppose you accept that volatility is not constant. What is it, then? There were candidate answers. Maybe volatility is random (Hull-White, Heston). Maybe there are jumps (Merton). Maybe volatility depends on the price level (Geske's leverage effect).

But here is the awkward question. Which of these is right? Each of them has parameters. Fit them to the market and you get several different models that all reproduce today's vanilla prices tolerably well but give wildly different prices for exotics, because they imply different dynamics. The choice of model determines the price of your exotic, and nothing in the vanilla market tells you which choice is correct.

Dupire asked a sharper question. Is there a model that fits the entire observed surface of vanilla option prices exactly, and if so, is it unique?

The key idea via analogy: the unique speed limit map

Imagine you know, for every destination and every arrival time, the probability that a car ends up there. From that alone, can you reconstruct the speed limit at every point on the map?

Dupire's answer is yes, if you insist that the car's motion is a diffusion whose speed depends only on where it is and what time it is.

Translate back to finance. Breeden and Litzenberger had already shown that the market's option prices contain the full implied probability distribution of the stock at each maturity. So the market is telling you not just one distribution but a whole family of them, one for every expiry date. Dupire's insight was that this family of distributions, taken together, pins down a unique volatility function.

Specifically, if you assume the stock follows a diffusion in which the volatility is not constant but is a deterministic function of the current stock price and the current time, call it local volatility, then there is exactly one such function consistent with all observed option prices. And Dupire gave the formula for it. You compute it directly from the market's call prices by taking derivatives with respect to strike and maturity. No fitting, no optimisation, no guessing at parameters. You read the model off the market.

That is the extraordinary claim. The volatility smile is not a puzzle to be explained away. It is data. It is the market telling you, precisely, how volatile it thinks the stock will be at every future price level and every future date, and Dupire's formula is the decoder.

What local volatility actually means

Local volatility is not implied volatility, and confusing the two is the classic beginner's error.

Implied volatility is an average: it is the single constant volatility that would reproduce the price of one particular option. It blends together everything that happens between now and that option's expiry, across all the price levels the stock might visit.

Local volatility is instantaneous and local: it is the volatility the stock experiences right now, given that it is at a particular price at a particular time. It is the raw ingredient. Implied volatility is the cooked dish, a kind of weighted average of local volatilities along the paths the stock might take.

This distinction has a practical consequence traders know well: because implied vol is an average of local vols, the local volatility surface is steeper than the implied volatility surface. A modest downward skew in implied vol corresponds to a much more dramatic tilt in local vol. Gatheral's "rule of two" captures this folk knowledge: near the money, the local volatility skew is roughly twice the implied volatility skew.

Why it mattered

• It made exotics pricing coherent. For the first time, a desk could price a barrier option with a model that exactly reproduced the price of every vanilla option it was going to hedge with. That closes the arbitrage door. This is the reason local volatility was adopted so fast: it is not that anyone believed it was true, it is that it was consistent, and consistency is what stops you being picked off.
• It became the standard exotics engine. For twenty years, local volatility (typically implemented as a finite-difference grid or a Monte Carlo on the Dupire diffusion) was the production model for equity exotics at essentially every investment bank.
• It gave the smile an interpretation. The smile is not noise or a market inefficiency. It is a statement about the stock's future volatility conditional on price level. Under Dupire's lens, the equity skew says: "if the market drops 20 percent, volatility will be much higher than it is now." Which, empirically, is exactly right.
• It is a beautiful piece of mathematics. The relationship Dupire found is a forward equation: rather than solving backwards from a single option's payoff, it evolves the whole distribution forwards through time, letting you price all strikes and all maturities at once. That change of direction is what makes calibration tractable, and it is a genuinely elegant move.

The honest limitations

• The dynamics are wrong, and traders know it. This is the central, well-understood, universally acknowledged flaw. Local volatility fits today's smile perfectly, by construction. But it makes an absurd prediction about how the smile will move when the stock moves. Because volatility is tied rigidly to the price level, the model says that if the stock falls 10 percent, the whole smile shifts sideways with it: the smile "sticks to the strike" in a way that produces exactly the opposite of what actually happens. Real smiles tend to move with the money, and real skews flatten as maturity lengthens in ways local vol cannot capture.
• That wrongness shows up in the hedge ratios. The deltas produced by a local volatility model are systematically off, because the model has a wrong view of how implied volatility responds to spot. You get today's price right and tomorrow's hedge wrong. For a desk that lives by its hedges, this is not a small complaint.
• It is a fitting device, not a description of reality. Volatility in the real world is genuinely random. It moves for its own reasons, driven by news, positioning and fear, not merely as a function of where the stock is. Local volatility deliberately assumes away that randomness in order to buy uniqueness and calibration. The trade is explicit and it is a real trade.
• It is numerically fragile. The formula requires differentiating market prices with respect to both strike and maturity. Real prices are quoted at a coarse grid of strikes and a handful of expiries, and they are noisy. Differentiating noisy, sparse data is dangerous, and a poorly interpolated surface can produce local volatilities that are wildly unstable or even imaginary. Building a properly smooth, arbitrage-free input surface is most of the work in a real implementation.
• It struggles with forward-starting products. Any exotic whose payoff depends on volatility starting at some future date, a cliquet, for instance, is priced badly, because the model's view of future volatility is rigidly determined rather than random.
• The industry's fix admits the problem. Modern desks mostly use stochastic local volatility: a stochastic volatility model (for realistic dynamics) with a local volatility correction layered on top (to force an exact fit to the vanillas). That hybrid exists precisely because neither pure approach is adequate on its own, and it is a candid admission that Dupire's model gets the price right for the wrong reason.

The one-line takeaway

Dupire showed that the volatility smile is not a defect to be patched but data to be inverted, and that there is exactly one volatility-as-a-function-of-price-and-time that reproduces every option price in the market, a discovery that gave the exotics industry its first internally consistent pricing engine, at the cost of a model whose dynamics almost nobody believes.