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Deriving Local Volatility from Implied Volatility and Diagnosing Negative Denominators

Article Quant Q&A · Author: Randor
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Summary

The document asks how to convert an implied volatility surface into local volatility and why some versions of the formula produce a negative denominator. It compares alternative denominator terms and questions a derivation after applying it to a two-strike example. The author notes that the example is believed to be arbitrage-free and evaluates local volatility at one of the quoted strikes, where the second-derivative contribution is taken to vanish.

The discussion highlights that the result may depend on the precise formula and derivation used, but it does not supply a resolution or establish which expression is correct. It is therefore best read as a problem statement about implementing and checking the implied-to-local volatility transformation, not as a complete derivation. The example alone cannot settle whether the inputs satisfy all relevant surface conditions or identify the source of the sign issue.

Key ideas

  • Local volatility can be derived from an implied volatility surface, but formula conventions need careful checking.
  • Different expressions for the denominator may reflect different definitions or derivation assumptions.
  • A negative denominator in an apparently arbitrage-free example raises a question about the formula or its inputs.
  • The document poses the issue without resolving the derivation or validating the example.

Tags

OptionsVolatilityDerivatives pricingStatistics
Full text
# what is the formula for local vol , as a function of implied vol, AND why am i getting a negative denominator?


# what is the formula for local vol , as a function of implied vol, AND why am i getting a negative denominator?












i am seeing different formulas at different places on stack exchange, so not sure! eg if denominator has a term y^2/w^2, or y/w^2 or 1/w or -1/w

and also, whatever formula i use, i get a negative denominator for the following (which has no arbitrage), so i wonder why it doesnt work? T=1,F=100,K1=95,K2=100,VOL1=27%,VOL2=22% and i am evaluating local vol at K1,T (ie so the 2nd derivative term is zero)

i guess AFK's answer at Local volatility surface corresponding to the implied volatility surface may be correct, as for his, it does not fail in my above example!

i wonder why this derivation would be wrong! https://www.frouah.com/finance%20notes/Dupire%20Local%20Volatility.pdf

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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