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Advanced Derivatives and Exotic Options / Local Volatility

The Dupire PDE and the Local Vol Formula

The Black-Scholes PDE propagates a price backward from terminal payoff to today. Dupire (1994) discovered a twin PDE that propagates market prices forward from today's option prices to all future strikes and maturities. From this forward PDE, one can read off — for any given implied vol surface — the unique deterministic function σL(K,T)\sigma_L(K, T) that makes the model price every European option correctly. This is the local vol formula, and it is the operational form of the conditional-expectation identity derived in the derivation sub-note.

The result matters not just for fitting the surface but for understanding what any given calibration implies: the local vol surface is a rich object — it encodes the market's view of vol at each time-space location — and reading it carefully reveals term-structure and spot-vol dynamics that differ substantially from what stochastic vol models produce.

Learning objectives. By the end of this note you should be able to:
  • state the Dupire forward PDE and explain how it differs from the backward Black-Scholes PDE;
  • write the local vol formula in terms of market call prices C(K,T)C(K,T) and in terms of implied vol Σ(K,T)\Sigma(K,T);
  • verify the formula in the Black-Scholes special case;
  • identify which numerical pitfalls arise when applying the formula to real market data.

In this lesson

  1. 1.1 The backward and forward PDEs side by side
  2. 1.2 The local vol formula
  3. 1.3 Local vol in terms of implied vol
  4. 1.4 Verification: flat surface → constant vol
  5. 1.5 Numerical pitfalls

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