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How Strike and Maturity Interpolation Can Create Option Arbitrage

Article Quant Q&A · Author: user6703592
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Summary

The post asks whether linear interpolation of implied volatility can create arbitrage, distinguishing interpolation across maturities from interpolation across strikes. It notes the concern that interpolating volatility over time may violate calendar no-arbitrage conditions, even when the input points themselves satisfy those conditions. The author then asks whether the same concern applies when maturity is fixed and the interpolation is across strike.

The document provides no answer, derivation, or example demonstrating an arbitrage. Its value is in identifying an important distinction for constructing an option volatility surface: interpolation choices along maturity and strike need not have the same implications. It frames the issue as a question rather than a resolved result, so readers should not treat its stated concern or assumptions as a general conclusion. A complete treatment would need to relate the interpolated prices or volatility surface to the relevant no-arbitrage conditions.

Key ideas

  • The post distinguishes interpolation across option maturities from interpolation across strikes.
  • It raises calendar arbitrage as a concern when implied volatility is interpolated over time.
  • It asks whether strike interpolation at fixed maturity can also produce arbitrage.
  • No answer or proof is included, so the question remains unresolved in this document.

Tags

OptionsDerivatives pricingVolatilityArbitrage
Full text
# Wrong understanding of implied vol cannot be linear interpolated


# Wrong understanding of implied vol cannot be linear interpolated












There is a well known conclusion that `implied vol cannot be linear interpolated.` But I am not sure this `interpolation` is for strike $K$ or time $T$?

That's definitely true for time. When strike $K$ is fixed, interpolation of implied vols respect to time $T$ can make the `Calendar arbitrage`. Surely we assume the interpolation points satisfy the no arbitrage condition.

But is it true when time $T$ is fixed, interpolation of implied vols respect to strike $K$ can make a arbitrage.

I really struggled for related problems, can anyone give me a precise answer?

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