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Key Takeaways
- Local volatility is a function of spot level and time used inside the underlying’s diffusion model.
- Implied volatility is the pricing input that reproduces one option’s market price.
- Dupire uses both maturity and strike derivatives of a smooth European call-price surface.
- In the time-only example, two models have the same 25% one-year IV but local volatility of 25% and 35.36% at year one.
- A calibrated surface supplies the model inputs for repricing and calculating its hedge ratios.
What is local volatility?
Local volatility is the instantaneous volatility in a diffusion model, expressed as a function of spot and time, so the model assigns one value to each spot-and-date pair. The Dupire formula backs that function out of European option prices across strikes and maturities, which lets a calibrated model reproduce today’s option prices while specifying how the underlying itself moves.
Local volatility vs implied volatility
Implied volatility is the constant pricing input that reproduces one option price. A volatility surface organizes those inputs across strikes and expirations. Local volatility converts the European price surface into a specification for the underlying process.
Formula
σ_LV(K,T)² = [ dC/dT + (r − q) K dC/dK + q C ] / [ 0.5 K² d2C/dK2 ], with underlying dynamics dS/S = (r − q) dt + σ_LV(S,t) dW- C(K,T) is today’s European call price as a smooth function of strike and maturity.
- Use deterministic rate r and continuous dividend yield q.
- The second strike derivative is the discounted terminal density.
- Use positive strike convexity and a nonnegative carry-adjusted maturity numerator for a valid local-variance value.
- For a time-only volatility model with total variance w(T), local variance at T equals w′(T).
Worked Example
Compare two time-only diffusion models with the same spot, deterministic rates and dividend yield. Model A has total variance w(T) = 0.0625T. Model B has w(T) = 0.0625T². Time T is measured in years.
- At T = 1, both models have total variance 0.0625 and implied volatility √(0.0625 / 1) = 25%
- Model A: local variance w′(T) = 0.0625, so local volatility at year one is 25%
- Model B: local variance w′(T) = 0.125T, so local volatility at year one is √0.125 = 35.36%
- At T = 0.5, model A implied volatility is 25% and model B implied volatility is √0.03125 = 17.68%
- The one-year option prices match across strikes; the six-month prices distinguish the models
The same one-year option slice supports different instantaneous volatilities. The maturity dimension identifies the difference: model A has constant local variance, while model B’s local variance increases with time. Use the full strike-and-maturity surface to calculate a Dupire surface.
How is local volatility calculated with the Dupire formula?
The Dupire formula uses the call-price maturity derivative, its strike slope and convexity, and the rate and dividend inputs. Strike convexity represents the discounted terminal density. The carry-adjusted maturity derivative and that density determine local variance.
Start with a smooth price surface satisfying the required no-arbitrage conditions. For numerical calibration, verify both the numerator and the strike-convexity denominator before taking their ratio and square root.
Local volatility example: two time-only models
Model A has total variance w(T) = 0.0625T. Model B has w(T) = 0.0625T². Use the same spot, deterministic rates and dividend yield for both.
At one year, each has total variance 0.0625 and implied volatility 25%. Their European option prices match at that maturity across strikes. At six months, A still has 25% implied volatility while B has 17.68%.
In a time-only model, local variance is w′(T). A therefore has 25% local volatility at year one. B has local variance 0.125 and local volatility 35.36%. The maturity derivative identifies the difference.
How traders use local volatility models
A fitted local-volatility model supplies a process for repricing contracts as spot and time change. Its delta follows from that repricing with the surface dynamics specified.
The short-maturity, near-at-the-money one-half skew relationship provides a focused analytical result for smooth local-volatility models. Calculate finite-maturity prices and hedge ratios from the calibrated model itself.
For a listed-option trade, start with the selected contract’s IV, delta and vega. Then reprice the position in the options profit calculator, which shows its value across price and date as you shift volatility up or down.
Limitations of local volatility
- Calibrate from a smooth, arbitrage-free price surface. Sparse or noisy quotes make the strike and maturity derivatives unstable, so check both the numerator and the convexity term.
- Calibrate across maturities. The worked models share the same one-year option prices and still differ in local volatility at year one, 25% against 35.36%.
- Test its future smiles and surface moves against market behavior before you hedge with it. Volatility depends only on spot and time, with no separate random factor, and the model fits today’s prices; stochastic volatility models cover that gap.
- Compare its hedge ratios with Black-Scholes delta and vega before you hedge. They follow from the surface dynamics, so the two sets diverge when the surface carries skew.
- Use the one-half skew rule only at short maturities near the money, and calibrate the full surface for the rest.
Bottom line
Local volatility turns the implied volatility surface into a volatility for every price level and date, giving one model that matches today’s European option prices. Calibrate it from a smooth, arbitrage-free surface across strikes and maturities, and check its hedge ratios and future smiles against market behavior.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Bruno Dupire, Pricing with a Smile, Risk 7(1), 1994
- Derman, Kani and Zou: The Local Volatility Surface, Unlocking the Information in Index Option Prices
- Gatheral and Jacquier: Arbitrage-Free SVI Volatility Surfaces
- Local Volatility Under Rough Volatility (asymptotics of the local volatility surface)
- Capturing Smile Dynamics with the Quintic Volatility Model: SPX, Skew-Stickiness Ratio and VIX
How entries are sourced, calculated and updated: editorial methodology.