Intuition for Gyöngy’s Theorem from European Option Prices

Summary

The response points to an intuitive route to understanding Gyöngy’s theorem through European call prices and generalized derivatives. It starts from an asset whose drift follows a risk-neutral specification and whose volatility may depend on the adapted path, then applies a generalized Itô expansion to the call payoff. Taking expectations connects the time evolution of call prices to the conditional expected variance of the asset given its current level.

This relationship motivates a local-volatility interpretation: the conditional variance at a given asset level can be inferred from the way call prices vary with strike and time, linking the result to Dupire’s formula. The excerpt is only a brief derivation sketch, not a full proof of the theorem. Its notation and displayed expression are compressed and may contain transcription issues, so readers should consult a careful mathematical source for assumptions, regularity conditions, and the exact formula.

Key ideas

• A generalized Itô expansion of a call payoff links option-price dynamics to asset variance.
• The derivation conditions volatility on the asset level at which the option payoff changes slope.
• Strike derivatives of call prices connect the argument to local-volatility recovery.
• The response gives intuition rather than a complete proof of Gyöngy’s theorem.
• The compact displayed formula should be checked against a rigorous source before use.

Tags

# Gyöngy Theorem Proof


# Gyöngy Theorem Proof












Can you please point me to a publicly available text that discusses the proof for the Gyöngy Theorem?

Gyöngy, I. (1986), “Mimicking the One-Dimensional Marginal Distributions of Processes Having an Ito Differential,” Probability Theory and Related Fields, 71, 501-516.

Any text that gives just the intuition of why this theorem is true would also be great.

## Answer by Sebapi (score 2)

https://quant.stackexchange.com/a/80971

Maybe the first part of Antoine Savine's 1998 "Theory of Volatility" paper for an intuitive derivation from european option prices:

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3173772

Useful generalized derivatives:

- $d(S-k)^+ = 1_{\{S>K\}}$

- $d1_{\{S>K\}} = \delta_{\{S=K\}}=d_K(S)$

- $d_K(S)$ can be derived $n$ times

Using $dS_t = (r-q) S_t dt + \sigma_t S_t dW_t$ with any adapted $\sigma_t$: $$d(S_t-K)^+ = 1_{\{S_t>K\}}dS_t + \frac{1}{2} d_K(S_t) \sigma_t^2 S_t^2 dt$$ Introducing $D_t=e^{-rt}$ and taking expectation, with $E(dS_t)=(r-q)S_t dt$: \begin{eqnarray*} dC(t,K) &=& - C rdt + D_t E( 1_{\{S_t>K\}}(r-q)S_tdt) + \frac{1}{2} D_t d_K(S_t) K^2 E(\sigma_t^2|S_t=K) dt \\ (r-q)S_t &=& (r-q)(S_t-K)+(r-q)K \\ dC(t,K) &=& -q C(t,K)dt - (r-q)K\frac{\partial C}{\partial K} dt + \frac{1}{2}\frac{\partial^2 C}{\partial K^2} K^2 E(\sigma_t^2|S_t=K) dt \end{eqnarray*}

Theorem: Expected Volatility and Call Prices - Dupire $$ E(\sigma_t^2|S_t=K) = 2 \frac{\frac{\partial^2 C}{\partial K^2}(t,K)+q C(t,K)dt + (r-q)K\frac{\partial C}{\partial K}(t,K) }{K_t^2 \frac{\partial C}{\partial t}(t,K)} $$

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