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Why Iso-Delta Volatility Interpolation May Allow Arbitrage

Article Quant Q&A · Author: Special Sauce
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Summary

The document asks whether interpolating implied variance through time at fixed option delta points preserves the no-arbitrage property associated with interpolation along iso-moneyness lines. The motivation is a volatility surface whose market observations are available at fixed deltas for several maturities. The questioner reasons that Black–Scholes call delta is monotonic in its input, but that property alone does not establish that the interpolated surface remains arbitrage free.

The answer distinguishes calendar-spread arbitrage from butterfly arbitrage. Linear interpolation of implied variance along iso-moneyness lines can preserve calendar-spread consistency under stated assumptions about arbitrage-free input data, but it does not establish that every interpolated slice avoids butterfly arbitrage. Iso-delta interpolation may be financially useful, yet it does not guarantee freedom from calendar-spread arbitrage because implied volatility itself need not vary monotonically. The answer notes that pathological counterexamples may be possible while suggesting the issue is often not material in realistic cases. It points readers toward further analysis rather than offering a proof or a general interpolation recipe.

Key ideas

  • Fixed-delta interpolation is not guaranteed to preserve calendar-spread arbitrage constraints.
  • Monotonicity of the normal cumulative distribution does not prove arbitrage freedom when volatility varies.
  • Iso-moneyness variance interpolation and iso-delta interpolation have different guarantees.
  • An interpolation method can avoid calendar-spread issues yet still create butterfly arbitrage.
  • The answer describes potential concerns but does not provide a proof or a concrete counterexample.

Tags

OptionsVolatilityArbitrageDerivatives pricing
Full text
# Interpolating implied volatility term structure when IV is sampled at fixed delta points


# Interpolating implied volatility term structure when IV is sampled at fixed delta points












According to the accepted answer to a question in this site on the interpolation in the term structure of volatility surface:

> A simple linear interpolation on implied variance along iso-moneyness lines is enough to guarantee that there is no arbitrage between maturities as long as the inputted market data is arbitrage free.

In my case, however, I have implied volatilities sampled at fixed delta points for a set of maturities $\{ T_i \}$. If I linearly interpolate the implied variance at a time $T$, where $T_i \leq T \leq T_{i+1}$, along iso-delta lines, will the time-interpolated results at $T$ for all delta points be arbitrage free?

My guess is yes, but I'm hoping someone can confirm. In the Black-Scholes model, the delta of a call option is $\Delta = N(d_1)$ where $N()$ represents the cumulative normal probability density function with $$d_1 = \frac{\log(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}.$$

Because $N()$ is monotonic and non-decreasing, I'm hoping the arbitrage-free result would still be maintained. I would much appreciate if anyone can confirm or refute this.

## Answer by jherek (score 2, accepted)

https://quant.stackexchange.com/a/73332

The iso-moneyness approach guarantees no arbitrage in terms of calendar spreads, but it is not proven it does not introduce some butterfly spreads at some interpolated time. See Arbitrages in the Volatility Surface Interpolation and Extrapolation.

Using iso-delta is sometimes done, not for arbitrage concerns, but because it may make more sense from a financial perspective. It does not guarantee the absence of calendar spread arbitrage. It would be interesting to find a counter example. $N()$ is monotonic but $\sigma$ is not. The no-arbitrage condition is derived in Arbitrage-free conditions for implied volatility surface by Delta does not look nice

Arbitrages with iso-delta interpolation is usually not a concern even if it may happen in not-so-realistic/manufactured examples.

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