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Advanced Derivatives and Exotic Options / Local-Stochastic Volatility and Rough Volatility

Gyöngy's Lemma and the Effective Local Volatility

The Dupire derivation established that for any risk-neutral stochastic volatility model, the local variance satisfies σL2(K,T)=EQ[vT ∣ ST=K]\sigma_L^2(K,T) = \mathbb{E}^{\mathbb{Q}}[v_T \,|\, S_T = K]. This identity is the financial instance of a deeper probabilistic result: Gyöngy's theorem (1986) says that any Itô process — regardless of how complicated its non-Markovian dynamics are — has the same one-dimensional marginal distributions as a Markov diffusion whose squared diffusion coefficient equals the conditional expectation of instantaneous variance given the current state. The theorem is why every stochastic vol model that correctly calibrates to today's surface will agree on every vanilla option price, even though the models disagree on path-dependent prices such as barriers and cliquets.
This note's deliverable is the SLV leverage constraint. In the local-stochastic volatility (SLV) model, the spot process carries both a stochastic variance vtv_t and a deterministic leverage function L(St,t)L(S_t, t). Gyöngy's theorem constrains LL exactly:
L2(S,t)=σL2(S,t)EQ[vt ∣ St=S]L^2(S, t) = \frac{\sigma_L^2(S, t)}{\mathbb{E}^{\mathbb{Q}}[v_t \,|\, S_t = S]}

This formula — the ratio of the Dupire local variance to the model's conditional variance — is what SLV calibration algorithms must solve numerically. Everything else in this note is the argument that the formula is correct.

Learning objectives. By the end of this note you should be able to:
  • state Gyöngy's theorem precisely and identify its two components (the abstract result and the financial application);
  • explain why any stochastic vol model and the corresponding local vol model price all vanilla options identically;
  • derive the SLV leverage constraint from Gyöngy's theorem;
  • identify what the leverage function measures and what makes its computation hard.

In this lesson

    1. Gyöngy's theorem
    1. The financial application: from Gyöngy to the DKD identity
    1. The SLV leverage constraint
    1. The projection principle

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