Introductory Context
"Bruno Dupire derived the local volatility formula in 1994, showing that from any arbitrage-free implied volatility surface, a unique local volatility surface can be extracted. The Dupire formula expresses the local variance as a function of the market's implied volatility surface: σ²_local(K, T) = [∂C/∂T + rK∂C/∂K] / [½K²∂²C/∂K²]. This equation -- which uses the first and second partial derivatives of call option prices with respect to expiry and strike -- is one of the most important results in modern options theory. "
The Local Volatility Surface - What It Looks Like
The local volatility surface is a three-dimensional object: σ_local(S, t) specifies the instantaneous volatility of the underlying at every possible price level S and every future time t. The surface is typically visualised as a function of moneyness (S/K) and time to maturity -- showing the exact volatility the underlying would need at each point in its future price path to be consistent with today's options market prices. Key features of typical equity local volatility surfaces: (1) The local vol is highest for low price levels (consistent with the put skew -- the market expects high volatility when prices fall). (2) The local vol decreases as the underlying rises (consistent with the leverage effect). (3) Near-term local vol may be different from long-term local vol (reflecting the term structure). The local volatility surface's shape is entirely determined by the market's observable implied volatility surface.
Local vs Stochastic Volatility - The Critical Comparison
The Heston (stochastic) and Dupire (local) models both improve on Black-Scholes by allowing volatility to vary. Their fundamental difference: the Heston model makes volatility an independent random process that evolves over time, while the local volatility model makes volatility a deterministic function of the underlying's current level and time. This difference creates a critical practical divergence in how each model handles the volatility surface's forward dynamics -- specifically, how the model predicts the volatility surface will look in the future if the underlying moves.
The Dupire local volatility model has a known weakness: its predicted future volatility surfaces (how the vol surface will look after the underlying moves) do not match the empirical observation that vol surfaces tend to move together with the underlying rather than shifting dramatically with each underlying movement. The Heston model's stochastic vol process produces more realistic future vol dynamics than the local vol model. This is why institutional desks often use hybrid models (local-stochastic volatility models) that combine both approaches: the local vol component ensures calibration to the current market surface, while the stochastic vol component provides realistic forward dynamics.
Local vs Stochastic Volatility Comparison
Feature | Local Volatility | Stochastic Volatility (Heston). Vol process: deterministic σ(S,t) | random process for V. Calibration: exact fit to all market prices | approximate fit, calibrated to surface. Future dynamics: unrealistic (vol too 'sticky') | more realistic vol dynamics. Skew origin: from the shape of σ(S,t) surface | from ρ (price-vol correlation). Application: vanilla options hedging, exotic pricing (simple) | exotic options, vol trading, complex hedging. Computation: fast (analytical for vanilla) | slower (characteristic function methods). Industry use: universal for vanilla market making | standard for exotics and vol trading.
Applications in Indian Markets
For the Indian options market context: the local volatility model is the backbone of how NSE options market makers price and hedge their vanilla Nifty options books. When a market maker quotes a 22,500 PE or a 24,000 CE, their pricing system uses the current Nifty implied volatility surface to extract the local volatility function and then uses this function to compute precise deltas, gammas, and vegas for their position hedging. The result: a delta hedge that correctly accounts for the non-constant volatility across strikes -- unlike the simple Black-Scholes delta which ignores the skew's contribution to the option's directional sensitivity.
For retail options traders, the practical relevance of local volatility is indirect: (1) The delta you see on Sensibull or your broker's platform for options that are significantly OTM may incorporate some skew-adjusted delta correction that approximates local volatility effects. (2) Understanding that OTM put deltas are 'sticky' (they don't change as the market moves toward them in the way Black-Scholes predicts) reflects the local volatility surface's prediction of higher vol at lower price levels -- making the OTM put's delta larger than the naïve Black-Scholes calculation would suggest.
The local volatility model is the market's self-portrait: a mathematical representation of exactly what volatility must be at each point in the underlying's future price space for the current collection of market prices to be internally consistent. It is the tautological solution -- it doesn't predict anything about the future, it simply says: 'given the current options market, this is the volatility model that is perfectly consistent with every price you see.' Its power is this consistency; its limitation is this same consistency's static nature. The market's implied vol surface today was priced under different assumptions than tomorrow's surface will be -- and the local vol model, perfectly calibrated today, has no mechanism to predict tomorrow's surface.