Local Volatility

Black Scholes assumes one constant volatility, but real option markets show different implied volatilities for different strikes and expiries: the volatility smile and surface. The local volatility model, developed independently by Bruno Dupire and by Emanuel Derman and Iraj Kani in 1994, fixes this by letting volatility depend on both the underlying price and time. With the right local volatility function, the model reproduces the price of every listed option exactly. It is widely used to price exotic options consistently with vanilla market prices.

The idea#

Instead of a single σ, the underlying follows:

dS / S = r dt + σ(S, t) dW

The volatility σ(S, t) is a deterministic function: at each price level and time, there is one specific volatility. If the stock falls to $80 in three months, the model knows exactly what volatility applies there.

Dupire's formula#

Dupire showed that if you know call prices C(K, T) for all strikes and expiries, the local volatility is:

σ²(K, T) = [∂C/∂T + r × K × ∂C/∂K] / [½ × K² × ∂²C/∂K²]

In words: local variance equals how fast option prices change with expiry, divided by how curved option prices are across strikes. Because market prices exist only at listed strikes and expiries, practitioners first build a smooth, arbitrage free implied volatility surface and then compute local volatility from it. See Volatility Surface and Volatility Interpolation and Extrapolation.

Implied vs local volatility#

Implied volatility is an average of local volatility over the paths that matter for an option. A useful rule of thumb for short expiries: implied volatility at a strike is roughly the average of local volatility between the current price and that strike. That means local volatility skew is about twice as steep as implied volatility skew near the money.

Why it is useful#

• Exact calibration: reproduces all vanilla option prices used to build it.
• Consistent exotic pricing: barrier, lookback and other path dependent options can be priced in a way that matches the vanilla market, avoiding arbitrage between them. See Barrier Options.
• Simple to simulate: volatility is a known function, so Monte Carlo and finite differences are straightforward. See Monte Carlo Option Pricing.

The weaknesses#

• Unrealistic dynamics: local volatility predicts that when the underlying moves, the smile shifts in a way that does not match how markets behave. In particular, it tends to imply that future smiles flatten, while real forward smiles stay steep.
• Wrong hedge ratios: because of these dynamics, local volatility deltas can be poor. The model implies the smile moves opposite to the underlying (sticky local volatility), which can misprice risk. Patrick Hagan and colleagues highlighted this problem in their 2002 paper introducing SABR. See SABR Model.
• Forward starting and cliquet options: products that depend on future smiles are mispriced.
• Noisy calibration: second derivatives of market prices amplify noise, so the input surface must be very smooth.

Local vs stochastic volatility#

Many banks use local stochastic volatility (LSV) models, which combine both: a stochastic volatility process scaled by a local volatility function so that the model fits today's surface and has more realistic dynamics. See Stochastic Volatility and the Heston Model.

Frequently asked questions#

What is local volatility?#

A model in which volatility is a deterministic function of the underlying price and time, calibrated so that it reproduces all observed option prices.

What is Dupire's formula?#

A formula that derives local volatility from how option prices change across strikes and expiries.

What is the main weakness of local volatility?#

It fits today's option prices exactly but predicts unrealistic future smile behaviour, which can lead to poor hedges and mispricing of some exotics.

Next, learn a model where volatility itself is random in Stochastic Volatility and the Heston Model.