Exact reference construction for overlapping VIX windows

Construct an exact martingale reference measure for overlapping VIX windows that consistently satisfies the dispersion identities associated with multiple VIX contracts sharing subintervals, including a model for the variance allocated to the overlap and martingale constraints at every adjacent event time.

Background

For non-overlapping VIX spans, the paper constructs a chronological product reference using conditionally independent lognormal SPX increments driven by the relevant VIX level. This construction works because no merged-timeline subinterval is assigned two different VIX states.

When VIX spans overlap, assigning a single volatility state to their shared subinterval generally prevents the product reference from satisfying both dispersion identities automatically. The authors identify the need to model the variance allocated to the overlap and to impose martingale rows at every adjacent event time, but leave the exact construction unresolved.

References

If two VIX spans overlap, assigning one volatility state to their shared subinterval generally prevents the product reference from satisfying both dispersion identities automatically. A consistent extension must model the variance allocated to the overlap and impose martingale rows at every adjacent event time. We leave the corresponding exact reference construction for future work.

— Global Multi-Maturity SPX-VIX Calibration Beyond Markovian Stitching  (2609.04087 - Acharya et al., 3 Sep 2026) in Section 5, paragraph 'Overlapping windows'