What it means
The classic Black-Scholes model for pricing options assumes one constant volatility. Real markets disagree, because options with different strike prices (the price at which the holder may buy or sell) and expiry dates show different implied volatilities, creating a pattern called the volatility smile or skew.
A model with one number cannot match that pattern. Local volatility fixes this by treating volatility as a function of the asset price and time.
The idea was developed in the 1990s, notably by Bruno Dupire, and it gives a unique surface that matches every vanilla option price, meaning the standard calls and puts. Once the surface is built, it can be used to price more complicated options consistently.
In practice, a trader collects market prices for options across many strikes and maturities, smooths them into a continuous surface and then extracts the local volatility from the shape of that surface. The result tells the model how volatile the asset should be at each price level at each future date.
It is then used in simulations to value exotic options, such as barrier options that switch on or off at set prices. The approach has limits.
It assumes volatility is a function only of price and time, which means the smile tends to flatten in the model as time passes, unlike what is often seen in markets. Models that add random volatility, called stochastic volatility models, are often combined with it to address this.
For non-specialists the main message is that option prices carry information about the market's view of risk at different price levels. Local volatility is a tool for turning that information into a consistent pricing model that banks use for hedging and risk reporting.
In practice
Real-world examples.
Example
A bank trades a barrier option on a stock index that is cancelled if the index touches a set level. A flat volatility model prices it badly because it ignores the skew. The desk uses a local volatility surface, which prices the barrier in line with the quoted vanilla options.
Example
A risk team at an asset manager wants to estimate how a portfolio of options would behave if the market fell 10%. It uses a local volatility model to simulate price paths consistent with today's option prices. The results are used in a stress report.
Example
A corporate treasurer buys an option structure to hedge currency exposure and asks the bank how it was priced. The bank explains that it used local volatility calibrated to market option prices. The treasurer compares the price with a second quote.
Formula
Calculation
Dupire's formula, with interest rates and dividends set to zero for simplicity, is:
Local variance = (Change in call price over time) / (0.5 x Strike^2 x Second derivative of call price with respect to strike)
Assume values read from a smoothed option price surface for an asset, at a strike of $100: the call price rises by $0.80 per year of extra maturity, and the second derivative of the call price with respect to strike is 0.004. The denominator is 0.5 x $100 x $100 x 0.004 = 0.5 x 10,000 x 0.004 = 20. Local variance = 0.80 / 20 = 0.04. Local volatility is the square root of 0.04, which is 0.20, or 20%.Case study
Seen in the real world.
Kestrel Capital is an illustrative, fictional trading firm that priced knock-out options on a stock index using a single volatility of 18%. When it compared its prices with quotes from other banks, it found that its prices were consistently lower on contracts with barriers well below the current index level.
The quantitative team found that the market charged much higher volatility for low strikes, so the index was seen as more likely to fall sharply. The single-volatility model had ignored this skew and therefore underestimated the chance of the barrier being hit.
The team rebuilt the pricing with a local volatility surface calibrated to vanilla option prices across strikes and maturities. The new prices matched the market far better, and the firm stopped selling underpriced contracts. The team also reported that model risk remained, because local volatility does not capture every feature of real markets.
Watch out
Common mistakes.
- Assuming local volatility is a forecast of future volatility, when it is a consistent way of fitting today's option prices.
- Confusing local volatility with implied volatility, which is the single number that reproduces one option price in the Black-Scholes formula.
- Believing the model captures every feature of the market, when its smile dynamics are often unrealistic.
Questions
People also ask.
What is the difference between local and stochastic volatility?
Local volatility is a deterministic function of price and time, while stochastic volatility lets volatility itself move randomly.
Why do banks use it?
It matches all vanilla option prices, which makes pricing and hedging of more complex options consistent with the market.
Who developed the approach?
It is associated with Bruno Dupire, whose formula links the surface to option prices, and with related work by others in the 1990s.
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