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From debt crises to financial crashes (and back): a stock-flow consistent model for stock price bubbles

Matheus R. Grasselli, Adrien Nguyen-Huu

Abstract

We develop a stochastic macro-financial model in continuous time by integrating two specifications of the Keen economic framework with a financial market driven by a jump-diffusion process. The economic block of the model combines monetary debt-deflation mechanisms with Ponzi-type financial destabilization and is influenced by the financial market through a stochastic interest rate that depends on asset price returns. The financial market block of the model consists of an asset with jump--diffusion price process with endogenous, state-dependent jump intensities driven by speculative credit flows. The model formalizes a feedback loop linking credit expansion, crash risk, perceived return dynamics, and bank lending spreads. Under suitable parameter restrictions, we establish global existence and non-explosion of the coupled system. Numerical experiments illustrate how variations in credit sensitivity and jump parameters generate regimes ranging from stable growth to recurrent boom--bust cycles. The framework provides a tractable setting for analyzing endogenous financial fragility within a mathematically well-posed macro--financial system.

From debt crises to financial crashes (and back): a stock-flow consistent model for stock price bubbles

Abstract

We develop a stochastic macro-financial model in continuous time by integrating two specifications of the Keen economic framework with a financial market driven by a jump-diffusion process. The economic block of the model combines monetary debt-deflation mechanisms with Ponzi-type financial destabilization and is influenced by the financial market through a stochastic interest rate that depends on asset price returns. The financial market block of the model consists of an asset with jump--diffusion price process with endogenous, state-dependent jump intensities driven by speculative credit flows. The model formalizes a feedback loop linking credit expansion, crash risk, perceived return dynamics, and bank lending spreads. Under suitable parameter restrictions, we establish global existence and non-explosion of the coupled system. Numerical experiments illustrate how variations in credit sensitivity and jump parameters generate regimes ranging from stable growth to recurrent boom--bust cycles. The framework provides a tractable setting for analyzing endogenous financial fragility within a mathematically well-posed macro--financial system.
Paper Structure (21 sections, 4 theorems, 101 equations, 8 figures, 2 tables)

This paper contains 21 sections, 4 theorems, 101 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The real economy
  3. Accounting Framework
  4. Sector behaviour
  5. Production and Financing for Firms
  6. The Labour Market and Inflation
  7. Banks and Households
  8. Dynamical system
  9. The financial market
  10. Special cases and fundamental properties
  11. The deterministic case
  12. The case $\bar{\lambda}^\pm =0$
  13. The case $f^+ = \bar{f} >0$
  14. The case $f^- = \bar{f} >0$
  15. Numerical Simulations
  16. ...and 6 more sections

Key Result

Proposition A.1

Let $r=(r_t)_{t\ge 0}$ be a bounded function of time such that Let $(\omega_0,e_0,m_{f,0},\ell_0)\in (0,+\infty)\times(0,+\infty)\times\mathbb{R}^2$. Let $\pi$ be given by eq:profit share, namely Let $g(\pi)$ be defined by eq:growth, and let $f_t=\Psi(g(\pi_t)+i(\omega_t))$ with $\Psi$ defined in eq:Psi. Assume that $\Phi$ is affine as in eq:philips, and that $i(\omega)$ is given by inflation. T

Figures (8)

  • Figure 1: Wage share $\omega$ and employment rate $e$ (top row), net financial charges $\bar{r}_L \ell -\bar{r}_M m$ and speculative flow $f$ (middle row), and discounted stock price $\tilde{S}$, trend indicator $\mu$ and effective rate $r$ (bottom row) in the deterministic case with $\bar{\rho}_1 = \bar{\lambda}^{\pm}=0$. A low interest rate $\bar{r}_L=0.02$ (left) leads to the interior equilibrium for economic variables and $\bar{\mu}_0 = 0.025$ for the long-term mean of the trend indicator. A higher interest rate $\bar{r}_L=0.15$ (right) leads to a collapse of the real economy and a higher value $\bar{\mu}_0 = 0.145$ for the long-term mean of the trend indicator. In both cases the stock price follows a geometric Browning motion.
  • Figure 2: Wage share $\omega$ and employment rate $e$ (top row), net financial charges $\bar{r}_L \ell -\bar{r}_M m$ and speculative flow $f$ (middle row), and discounted stock price $\tilde{S}$, trend indicator $\mu$ and effective rate $r$ (bottom row) in the case with nonzero jump intensities $\bar{\lambda}^\pm = 1$ but $\rho_1 = 0$. A low interest rate $r_L=0.02$ (left) still leads to the interior equilibrium for economics variables, but stock price now has downward jumps of relative size $\bar{J}^+ = 0.1$ occurring on average once every $(\bar{\lambda}^+\bar{f})^{-1}\approx 17.67$ years, and $\bar{\mu}^+ \approx 0.02469 < \bar{\mu}_0$. A high interest rate $r_L=0.15$ still leads to a collapse of the economy, with the stock price exhibiting upward jumps of relative size $\bar{J}^- = 0.1$ occurring on average once every $(\bar{\lambda}^-\bar{f})^{-1}\approx 6.67$ years, and $\bar{\mu}^- \approx 0.1435 < \bar{\mu}_0$. (right)
  • Figure 3: Wage share $\omega$ and employment rate $e$ (top row), net financial charges $\bar{r}_L \ell -\bar{r}_M m$ and speculative flow $f$ (middle row), and discounted stock price $\tilde{S}$, trend indicator $\mu$ and effective rate $r$ (bottom row) in the case with $\bar{\lambda}^\pm = 0$ and $\rho_1 = 0.01$. A low baseline interest rate $r_L=0.02$ (left) can now be associate with a collapse in the economy because of the influence of the stock price on the effective rate $r$, even in the absence of jumps. The higher baseline interest rate $r_L=0.15$ (right) simply makes this collapse occur sooner.
  • Figure 4: Wage share $\omega$ and employment rate $e$ (top row), net financial charges $\bar{r}_L \ell -\bar{r}_M m$ and speculative flow $f$ (middle row), and discounted stock price $\tilde{S}$, trend indicator $\mu$ and effective rate $r$ (bottom row) for baseline case with all parameters as in Table \ref{['table']} (left) and case with higher stock price volatility $\bar{\sigma} = 0.25$ (right).
  • Figure 5: Wage share $\omega$ and employment rate $e$ (top row), net financial charges $\bar{r}_L \ell -\bar{r}_M m$ and speculative flow $f$ (middle row), and discounted stock price $\tilde{S}$, trend indicator $\mu$ and effective rate $r$ (bottom row) for the cases with $\bar{\eta}_\mu = 5$ (left) and $\bar{\eta}_\mu = 0.2$.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Proposition A.1
  • proof
  • Proposition A.2
  • proof
  • Proposition A.3
  • proof
  • Corollary A.1
  • proof