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No-Arbitrage Bounds for Implied Volatility Across Maturities

Article Quant Q&A · Author: quantoad
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Summary

The document explains a constraint on implied volatility across option maturities under a Black–Scholes term-structure assumption. It considers overnight and two-week at-the-money implied volatilities that are equal, and asks how to bound the one-week implied volatility. The response gives the key condition: cumulative variance must not decrease as maturity increases. In practice, compare implied volatility squared times maturity at adjacent points; the difference must be nonnegative. This supplies a no-arbitrage consistency check for a volatility term structure.

The note states the condition but does not work through the specific numerical bounds or discuss assumptions behind applying it to market quotes. It also does not address market frictions, interpolation choices, or other option-price constraints. Readers can use the variance condition to derive bounds, but must apply it consistently with the maturity units and the model assumptions.

Key ideas

  • Cumulative variance, calculated as implied volatility squared times maturity, must be nondecreasing across maturities.
  • The condition can be applied between each pair of adjacent maturities in a term structure.
  • Equal implied volatilities at two maturities do not by themselves specify the intermediate volatility; the variance constraint supplies bounds.

Tags

OptionsVolatilityDerivatives pricingArbitrage
Full text
# No-arbitrage bounds on Implied Volatility under Black-Scholes


# No-arbitrage bounds on Implied Volatility under Black-Scholes












Suppose the overnight (1-day) at-the-money implied volatility is X% and the two week (14-day) at-the-money implied volatility is also X%.

How would I go about finding the upper and lower no-arbitrage bounds for the 1-week (7-day) implied volatility assuming a term-structure following the Black-Scholes model?

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/63685

Simply make sure the forward variances remain non negative: $\Sigma(T_{i+1})^2 T_{i+1} - \Sigma(T_{i})^2 T_{i} \geq 0$ for all $i$.

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.