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

We 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, interpreted as the earning yield and analogous to the yield-to-maturity of a perpetual bond. Modelling γ as a Cox-Ingersoll-Ross (CIR) process, we derive analytically several stylised facts of financial markets, including power law distributed returns, transient super-exponential bubble dynamics, and fat-tailed price distributions prior to crashes. The framework explains excess volatility and the equity premium as emergent consequences of stochastic fluctuations in the discount rate, whose nonlinear inversion into prices generates an amplification factor and a positive risk premium absent in constant-rate models. 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 optimisation 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.