Implied Volatility: What the Market Believes

Implied volatility (IV) is inferred from market prices using models like Black-Scholes. It’s a forward-looking measure of expected volatility and depends on the specific strike and time to maturity of the option. Plotting this across various strikes and maturities gives us the volatility surface—a key input for traders and risk managers.

This surface often exhibits patterns like skew and smile, where implied volatilities are not flat but vary significantly based on moneyness. These variations encode important information about investor sentiment and risk aversion, especially in equity markets.

Local Volatility: What the Model Assumes

Local volatility, by contrast, is a deterministic function of both the underlying asset price and time. Introduced via the Dupire framework, local volatility is the unique volatility function that, when plugged into a pricing model, perfectly reproduces the observed implied volatility surface—assuming no stochastic volatility.

It represents the instantaneous volatility conditional on being at a specific price and time. That means local volatility captures the marginal contribution to volatility if the underlying were to move to a given level right now.

Why Are They Different?

Even if local volatility can be engineered to match current market prices, the model it underpins assumes deterministic evolution. In contrast, implied volatility reflects stochastic expectations from a broad range of market participants.

  • Implied vol is useful for quoting, valuation, and trading.
  • Local vol is better suited for simulation, path-dependent pricing, and exotic options.

This matters because hedging behavior differs: local volatility may suggest different sensitivities to spot moves compared to what the market prices into implied vol.

Visual Comparison

local vs implied
Volatility: Local vs Implied 5

A Useful Analogy: Forward Rates

One of the most insightful comparisons made in the Goldman Sachs research is the analogy between local volatility and forward interest rates. Just as forward rates represent expectations of future short-term rates, local volatility represents expected spot volatility conditional on reaching specific price levels.

This forward-looking nature gives local volatility conceptual weight—it tells us how the market’s risk outlook changes with the price path.

Applications in Practice

Understanding the local vs implied vol distinction is crucial for:

  • Pricing barrier and path-dependent options.
  • Constructing hedges that are sensitive to dynamic vol risk.
  • Designing strategies that exploit vol surface mispricings.
  • Calibrating models like local-stochastic volatility (LSV) hybrids.

Conclusion: Two Sides of the Same Coin

Local and implied volatilities may stem from the same surface, but they play different roles. Implied vol is what you see in the market—it’s tradable and reflective of sentiment. Local vol is what a model needs to replicate that reality—and it’s essential for simulation and exotic pricing.

Mastery of both concepts is critical for any serious derivatives trader or quant. As volatility products grow more complex, understanding how these measures interact becomes not just useful—but necessary.