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

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.

💡 Research Summary

The paper presents a continuous‑time stochastic macro‑financial model that merges two strands of the Keen stock‑flow consistent (SFC) framework with a modern jump‑diffusion description of asset prices. The economic core incorporates a monetary debt‑deflation mechanism together with a Ponzi‑type destabilization channel. Real‑sector variables—profits, leverage, debt ratios, and repayment capacity—are governed by accounting‑consistent balance‑sheet equations and behavioral rules (investment and dividend functions that depend on the profit ratio, a speculative credit flow F, and a financing mix parameter).

The financial market side models a single risky asset whose price S follows

 dS_t = μ_t S_t dt + σ S_t dW_t + S_{t‑} dJ_t,

where the jump component J has a state‑dependent intensity λ_t. Crucially, λ_t is an endogenous function of the speculative credit flow F_t (e.g., λ_t = λ_0 + λ_1 F_t), capturing the idea that credit expansion raises crash risk. The model also endogenizes the short‑rate r_t as a linear function of the asset’s instantaneous return i_t = dS_t/S_t, i.e., r_t = r_0 + α i_t. This creates a feedback loop: credit expansion → higher asset prices → higher jump intensity → market crash → higher risk premium → higher borrowing costs → reduced investment and output, which in turn affects credit creation.

Mathematically, the authors write the coupled system of stochastic differential equations (SDEs) for the real variables (capital K, loans L, deposits M, etc.) and the financial variables (S, λ, r). Under boundedness and Lipschitz conditions on the functions governing λ_t and r_t, they prove global existence and non‑explosion of strong solutions (Theorem 2.1). The proof relies on Picard‑Lindelöf arguments for the drift‑diffusion part and on Lyapunov‑type estimates to control the jump intensity, ensuring that the expected value of a quadratic Lyapunov function does not blow up.

Numerical experiments explore the model’s dynamics across three regimes by varying two key parameters: γ, the sensitivity of credit flow to economic growth, and ψ, the scale of the jump intensity response.

• Low γ and ψ produce a stable growth path where asset prices exhibit mean‑reverting diffusion and the economy settles at a steady‑state.
• Intermediate γ with higher ψ generates endogenous boom‑bust cycles: credit‑driven price bubbles build up, the jump intensity rises, a stochastic crash occurs, and the resulting spike in the risk‑adjusted interest rate forces a contraction in borrowing, output, and profits, after which the cycle restarts.
• High γ and ψ lead to a “crash” regime in which even modest shocks trigger a large jump, a sudden surge in the lending spread, and a rapid collapse of the real sector.

The authors interpret these findings through the lens of Minsky’s Financial Instability Hypothesis and Werner’s Quantitative Theory of Credit. The banking sector’s reaction function (the parameter α linking asset returns to the policy rate) emerges as a decisive stability lever: a gentle response dampens credit‑driven bubbles, whereas a steep response amplifies crash risk. The model therefore offers a tractable, mathematically rigorous platform for studying endogenous financial fragility, the propagation of credit‑induced bubbles, and the impact of monetary‑policy‑type interventions.

In conclusion, the paper successfully integrates SFC macro‑economics with stochastic finance, provides rigorous existence results, and demonstrates through simulation how endogenous credit dynamics can generate a rich spectrum of macro‑financial outcomes—from steady growth to recurrent crises—thereby bridging a gap between accounting‑based macro models and probabilistic bubble theory. Future extensions could incorporate heterogeneous agents, policy rules, or networked financial institutions to further enrich the analysis.