Is (independent) subordination relevant in option pricing?

Paper: arXiv 2307.08628

Authors: Michele Azzone, Roberto Baviera

Abstract

Monroe (1978) demonstrates that any local semimartingale can be represented as a time-changed Brownian Motion (BM). A natural question arises: does this representation theorem hold when the BM and the time-change are independent? We prove that a local semimartingale is not equivalent to a BM with a time-change that is independent from the BM. Our result is obtained utilizing a class of additive processes: the additive normal tempered stable (ATS). This class of processes exhibits an exceptional ability to accurately calibrate the equity volatility surface. We notice that the sub-class of additive processes that can be obtained with an independent additive subordination is incompatible with market data and shows significantly worse calibration performances than the ATS, especially on short time maturities. These results have been observed every business day in a semester on a dataset of S&P 500 and EURO STOXX 50 options.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a strong theoretical contribution by proving a limitation of Monroe’s theorem under independence assumptions, utilizing advanced stochastic calculus and additive processes. It complements this rigorous mathematical proof with daily empirical calibration across major equity indices to demonstrate the real-world failure of independent subordination for short maturities.

Research Flowchart

  flowchart TD
    A[Research Question: Is independent subordination relevant in option pricing?] --> B[Theoretical Analysis: Semimartingales & Independent Time-Changed BM]
    B --> C[Proof: Local semimartingales != BM with independent time-change]
    
    D[Data Inputs: S&P 500 & EURO STOXX 50 Daily Option Surfaces] --> E[Model Calibration & Empirical Comparison]
    
    C --> E
    F[Additive Normal Tempered Stable ATS Process] --> E
    
    E --> G[Findings: Independent subordination fails on short maturities & misses market dynamics]