Paper: arXiv 2302.11423

Authors: Li Lin, Didier Sornette

Abstract

We propose a formulation to construct new classes of financial price processes based on the insight that the key variable driving prices $P$ is the earning-over-price ratio $γ\simeq 1/P$, which we refer to as the earning yield and is analogous to the yield-to-maturity of an equivalent perpetual bond. This modeling strategy is illustrated with the choice for real-time $γ$ in the form of the Cox-Ingersoll-Ross (CIR) process, which allows us to derive analytically many stylised facts of financial prices and returns, such as the power law distribution of returns, transient super-exponential bubble behavior, and the fat-tailed distribution of prices before bubbles burst. Our model sheds new light on rationalizing the excess volatility and the equity premium puzzles. The model is calibrated to five well-known historical bubbles in the US and China stock markets via a quasi-maximum likelihood method with the L-BFGS-B optimization algorithm. Using $φ$-divergence statistics adapted to models prescribed in terms of stochastic differential equations, we show the superiority of the CIR process for $γ_t$ against three alternative models.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces an innovative theoretical framework by modeling price processes via an inverse CIR earnings yield, successfully deriving key analytical properties like power-law returns and bubble dynamics. It complements this analytical depth with robust empirical calibration across multiple historical market bubbles using quasi-maximum likelihood estimation and model selection metrics.

Research Flowchart

  flowchart TD
    A[Goal: Model financial prices & bubbles using inverse CIR process] --> B[Core Concept: Earning Yield γ_t ≈ 1/P_t modeled via CIR SDE]
    B --> C[Data: 5 Historical Bubbles in US & China Stock Markets]
    C --> D[Calibration: Quasi-Maximum Likelihood with L-BFGS-B Algorithm]
    D --> E[Model Validation: φ-divergence statistics vs 3 alternative models]
    E --> F[Key Outcomes: Analytically derives power-law returns, super-exponential bubbles & resolves equity premium/volatility puzzles]