The inverse Cox-Ingersoll-Ross process for parsimonious financial price modeling

TL;DR

This paper introduces a novel financial price modeling approach using the inverse Cox-Ingersoll-Ross process for the earning yield, capturing stylized facts and explaining market puzzles.

Contribution

It develops a new class of price processes based on the inverse CIR process for the earning yield, providing analytical insights and empirical validation.

Findings

Successfully models stylized facts like fat tails and bubbles

Calibrated to historical bubbles in US and China markets

Demonstrates the superiority of the inverse CIR model over alternatives

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 $\gamma \simeq 1/P$, 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 $\gamma$ 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 $\phi$-divergence statistics adapted to models prescribed in terms of stochastic differential equations, we show the superiority of the CIR process for $\gamma_t$ against three alternative models.