A Parsimonious Inverse Cox-Ingersoll-Ross Process for Financial Price Modeling

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 ≃ 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 γt against three alternative models.