Global Multi-Maturity SPX-VIX Calibration Beyond Markovian Stitching

RESUMO

We develop a global framework for joint S&P 500 (SPX)-VIX smile calibration across multiple maturities without the conditional-independence restriction induced by Markovian stitching. Exact local and global feasibility are equivalent: every globally feasible law has a block-preserving SPX-Markovization that leaves each monthly (S_i,V_i,S_{i+1}) law unchanged. Nevertheless, stitched laws can form a strict subset of globally feasible path laws because Markovization discards dependence on earlier history beyond the current SPX level. Adjacent smiles therefore cannot identify this dependence, and with identical monthly calibrations can price multi-period claims differently. Under the standard Markov reference, relative entropy selects minimum-information completion; non-Markov dependence requires cross-period information, an appropriate objective, or a history-dependent prior. For finite discretizations, we introduce an augmented-Bregman mirror-descent scheme. It preserves the fit to observable quote moments while controlling martingale and dispersion residuals. In a controlled infeasible affine system, this split keeps prescribed marginals about 25 times tighter than cyclic row projection exposing the discrepancy in the conditional rows. An exact finite-state example verifies block preservation and exhibits material changes after Markovization. On smoothed SPX VIX surfaces, numerical calculations illustrate a finite-budget penalty path: the worst fitted-smile error remains below 0.70 volatility points reported across the sweep while bulk conditional diagnostics improve substantially.