Numerically efficient conditions for calendar-arbitrage-free strike-specific variances

Determine numerically efficient conditions on the strike-specific log-normal variances V_{j,i} used in the generalized strike-wise interpolation model Ĉ_j(K) = Σ_i q_{j,i} Call(K_i, K, V_{j,i}) that ensure absence of calendar arbitrage (monotonicity in expiry) for all strikes, including extrapolated strikes beyond those quoted in the market.

Background

The paper proposes a smooth, strictly arbitrage-free option surface constructed from convex combinations of Black–Scholes call payoffs anchored at quoted strikes. In Remark 3.4, the authors discuss a generalization that replaces a common background martingale with strike-dependent martingales, which in the log-normal case corresponds to using different variances per strike.

While they provide an implementation that sets V_{j,i} to market bid variances at each strike and expiry, they note a key unresolved issue: identifying numerically efficient conditions on these per-strike variances that guarantee no calendar arbitrage across all strikes, not just at the observed market strikes. They further illustrate that simply ordering expiries at market strikes is insufficient via a synthetic counterexample (Figure 1).