Paper: arXiv 2404.16295

Authors: Liexin Cheng, Xue Cheng, Xianhua Peng

Abstract

The Chicago Board Options Exchange Volatility Index (VIX) is calculated from SPX options and derivatives of VIX are also traded in market, which leads to the so-called ``consistent modeling" problem. This paper proposes a time-changed Lévy model for log price with a composite change of time structure to capture both features of the implied SPX volatility and the implied volatility of volatility. Consistent modeling is achieved naturally via flexible choices of jumps and leverage effects, as well as the composition of time changes. Many celebrated models are covered as special cases. From this model, we derive an explicit form of the characteristic function for the asset price (SPX) and the pricing formula for European options as well as VIX options. The empirical results indicate great competence of the proposed model in the problem of joint calibration of the SPX/VIX Markets.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a mathematically sophisticated framework using time-changed Lévy processes with composite time changes to solve the joint SPX/VIX pricing problem. It derives explicit characteristic functions for pricing options and demonstrates solid empirical utility via joint calibration, though code/data availability is limited.

Research Flowchart

  flowchart TD
    A[Goal: Joint Pricing & Consistent Modeling of SPX and VIX Options] --> B[Proposed Model: Time-Changed Lévy Model with Composite Change of Time]
    B --> C[Derive Explicit Characteristic Functions for SPX & VIX Derivatives]
    D[Input: SPX Options & VIX Options Market Data] --> E[Computational Process: Joint Calibration of SPX and VIX Implied Volatilities]
    C --> E
    E --> F[Outcomes: Captures Volatility of Volatility & Outperforms Benchmark Models]

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1 of 96 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.