Local Volatility: Dupire

The implied volatility surface you have been reading for five chapters has a quiet contradiction buried inside it, and Bruno Dupire's insight in nineteen ninety-four was to turn that contradiction into a tool.

Here is the contradiction. Black-Scholes assumes the asset has one single volatility, constant for all time and all price levels. But when you inverted every strike and every expiry, you got a whole surface of different implied vols. Each of those numbers is an average — the implied vol of a given option is, loosely, the market's view of average volatility over the entire life of that contract and over all the paths the asset might take to reach that strike. So implied vol is a blunt, smeared-out quantity. It tells you the average, but it never tells you the instantaneous volatility the asset is actually experiencing at a particular price, on a particular day.

Dupire asked the natural inverse question. Suppose volatility is not constant, but is instead some deterministic function of where the asset is and what time it is — call it a function of spot and time. Is there a single such function that, when you run it through option pricing, reproduces the entire implied-vol surface you observe in the market, every strike and every expiry at once? The astonishing answer is yes, and it is unique. That function is the local volatility.

So hold the distinction firmly, because students confuse these two for years. Implied volatility is an average attached to a contract — one number per strike and expiry. Local volatility is an instantaneous spot-rate of variance attached to a point in price-time space — the volatility the asset experiences if it happens to be at this price at this moment. Implied vol lives on the option. Local vol lives on the asset. One is the smeared average; the other is the thing being averaged.

Now build the intuition for why this is recoverable before we write the formula. Think about a tiny calendar spread — two options on the same strike, one expiring an instant later than the other. The price difference between them is paid entirely for that extra sliver of time. So the rate at which the call price changes as you push the expiry out tells you how much variance the market is pricing into that next instant, right around that strike. Meanwhile, the curvature of the call price across strike — how sharply it bends — pins down the probability the asset actually finishes near that strike. Dupire's formula is, at heart, the ratio of those two things: the time-decay of the option divided by its strike-curvature. The first measures variance being added; the second measures the density of paths there to receive it. Divide one by the other and you isolate the local variance at that exact point.

Here is the precise statement, in the clean case where rates and dividends are zero. Let C be the call price written as a function of strike K and expiry T — that is the surface, expressed in prices. The local variance, the square of local volatility at strike K and time T, equals the partial derivative of C with respect to T, divided by one-half times K-squared times the second partial derivative of C with respect to K. In symbols, local variance equals the time-derivative of the call over one-half K-squared times the strike-gamma of the call. The numerator is that calendar-spread term — variance per unit time. The denominator is the strike convexity — the risk-neutral density at K (the strike-gamma of the call equals that density when rates are zero), scaled by one-half K-squared. Take the square root and you have the local volatility surface.

Two things about that formula decide whether it gives you anything usable. The denominator contains a second derivative across strike. Second derivatives are violently sensitive to noise — a little jitter in your quotes becomes a big wobble in curvature, and a big wobble in the denominator throws the whole quotient around. Worse, that strike-curvature is the probability density, so it must be positive; if your fitted surface ever bends the wrong way — if call prices are not convex in strike — the density goes negative, and local variance comes back negative, which is nonsense. A volatility cannot be the square root of a negative number. So the Dupire formula is only as trustworthy as the smoothness and the no-arbitrage cleanliness of the surface you feed it. Garbage convexity in, imaginary volatility out.

That is why every serious implementation does not differentiate raw market quotes. It first fits an arbitrage-free, smooth surface — calendar-monotone in time, butterfly-convex in strike — and only then takes the derivatives. The arbitrage conditions are not bureaucratic fussiness. They are exactly the conditions that keep the numerator and denominator of Dupire's formula on the right side of zero.