# 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.

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