Paper: arXiv 2609.04087

Authors: Atithi Acharya, Yue Sun, Brandon Augustino, Shouvanik Chakrabarti, Shree Hari Sureshbabu, Charlie Che

Abstract

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 laws with identical monthly calibrations can price multi-period claims differently. Under the standard Markov reference, relative entropy selects the stitched 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 by exposing the discrepancy in the conditional rows. An exact finite-state example verifies block preservation and exhibits material cross-period price changes after Markovization. On smoothed SPX and VIX surfaces, numerical calculations illustrate a finite-budget penalty path: the worst fitted-smile error remains below $0.70$ volatility points across the reported sweep while the bulk conditional diagnostics improve substantially.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical and novel approach to a complex calibration problem in quantitative finance. While it includes numerical calculations and addresses practical issues, the core contribution is theoretical and methodological, demonstrating strong empirical rigor in its validation and comparison to existing methods.

Research Flowchart

  flowchart TD
    A[Research Goal: Global Multi-Maturity SPX-VIX Smile Calibration] --> B{Methodology: Global Framework & Augmented-Bregman Mirror Descent};
    B --> C{Inputs: SPX & VIX Smile Data (Multi-Maturity)};
    C --> D[Computational Process: Exact Feasibility, Block Preservation, Optimization];
    D --> E{Finding 1: Stitched Laws are a Strict Subset of Globally Feasible Laws};
    D --> F{Finding 2: Non-Markov Dependence Affects Multi-Period Claim Pricing};
    D --> G{Finding 3: Augmented-Bregman Achieves Tighter Marginal Fits & Improves Diagnostics};