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Conditions for a Local Volatility Model to Fit an Option Surface

Article Quant Q&A · Author: Lefair
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Summary

The document asks whether every twice-differentiable call-price surface across maturities and strikes can be generated by a local volatility function of time and the underlying asset price. It points to the assumption used in Dupire’s approach and seeks a formal result establishing when that assumption holds. The central topic is the relationship between an observed option surface and the existence of a local volatility model that reproduces it.

No derivation, cited proof, or answer is provided, so the document does not establish that smoothness alone is sufficient. In practice, existence depends on whether the prices satisfy appropriate financial and mathematical constraints, and regularity conditions may also matter. The value of the question is to distinguish a formal smoothness assumption from the additional conditions needed for a valid, reproducible option surface.

Key ideas

  • The document asks whether a smooth call-price surface can always be represented by a local volatility function.
  • Dupire’s framework motivates the question of existence for a model that reproduces option prices across strikes and maturities.
  • The document offers no proof or resolution and does not specify sufficient conditions.
  • Smoothness alone should not be treated as evidence that an option surface is arbitrage-consistent or realizable by the model.

Tags

OptionsDerivatives pricingVolatility
Full text
# For any twice differential continuous function C(T, K), does there exist a sigma(t, S) that can reproduce C(T, K)?


# For any twice differential continuous function C(T, K), does there exist a sigma(t, S) that can reproduce C(T, K)?












In the Dupire's paper, he assumes that there exits a function $\sigma(t,S)$ that can reproduce $C(T, K)$. My question is that: is the assumption true for any twice differential continuous function $C(T, K)$? Since Dupire's paper came out in 1994, is there any literature formally prove the assumption? I did some research and did not find any correlated materials.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.