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Advanced Derivatives and Exotic Options / The Volatility Smile and the Implied Vol Surface

Derivation — Local Variance as a Conditional Expectation of Instantaneous Variance

The implied volatility surface Σ(K,T)\Sigma(K,T) encodes market prices for all European options on a given underlying. Dupire (1994) and Derman-Kani (1994) independently showed that any admissible surface uniquely determines a local volatility function σL(S,t)\sigma_L(S, t) — a deterministic, state-dependent vol — under which all those option prices are consistent with no-arbitrage.

The central identity this derivation establishes is:

σL2(K,T)=EQ ⁣[vT | ST=K]\boxed{\sigma_L^2(K, T) = \mathbb{E}^{\mathbb{Q}}\!\left[v_T \,\middle|\, S_T = K\right]}
Local variance at strike KK and maturity TT is the risk-neutral expectation of instantaneous variance vTv_T conditional on the stock ending at KK. This identity is the mathematical root of local vol: it tells us that local vol is not arbitrary but is exactly the average instantaneous vol that the market assigns to paths ending at each strike.

This is a derivation sub-note. The result is used in the parent topic; the derivation here is for readers who want to see why it holds.

In this lesson

  1. Setup
  2. The Dupire PDE: differentiate the call price
  3. The conditional expectation identity
  4. Expressing local variance in implied vol coordinates

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