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

Imagine the economy as a giant, complex garden. For a long time, economists have tried to understand why this garden sometimes flourishes and other times suddenly withers.

This paper, written by Matheus Grasselli and Adrien Nguyen Huu, builds a new kind of weather forecast model for this garden. Instead of just looking at the soil (the real economy) or just the sky (the stock market), they built a model that shows how the two talk to each other, often in a dangerous loop that leads to storms.

Here is the story of their model, broken down into simple parts:

1. The Two Gardens: The "Real" and the "Speculative"

The authors combine two different ways of looking at the economy:

The Real Garden (Keen Model): This is the part where people work, factories make things, and banks lend money to build factories. It's grounded in reality. If you borrow too much money to build a factory you can't afford, you get into trouble.
The Speculative Garden (The Stock Market): This is the part where people trade shares of companies. Usually, models treat this like a smooth, predictable river. But the authors argue it's more like a stormy ocean with sudden, massive waves (crashes) and tsunamis.

2. The Secret Connection: The "Credit Leash"

The big innovation in this paper is how they connect these two gardens. They imagine a leash made of credit (loans).

When the Stock Market is Happy: Imagine the stock market is like a party. Prices are going up, and everyone feels rich. Because prices are rising, banks feel confident. They loosen the leash and say, "Sure, borrow more money! The future looks great!"
The Feedback Loop: This extra borrowing money flows back into the stock market, pushing prices even higher. It's a self-fulfilling party.
The Danger: But this party is built on debt. The more people borrow, the more fragile the system becomes.

3. The "Jump" in the Ocean

Most models assume stock prices change slowly, like a gentle tide. This paper says: No, they jump.

Imagine a stock price is a boat. Usually, it bobs up and down gently. But when too many people are borrowing money to buy the boat (speculation), the boat becomes top-heavy.

The Crash: Suddenly, a small wave hits, and the boat doesn't just rock; it flips over.
The Model's Magic: The authors use math to show that the more people borrow to buy stocks, the higher the chance of the boat flipping over. They call this "endogenous jump intensity." It means the crash isn't caused by an outside enemy (like a war or a virus); the crash is caused by the system's own greed.

4. The Banker's Panic Button

Here is the most critical part of the story: How the banks react.

The Good Times: When the stock market is rising, banks are relaxed. They charge low interest rates.
The Panic: When the stock market starts to wobble or crash, the banks get scared. They think, "Oh no, these borrowers might not pay us back!"
The Squeeze: To protect themselves, banks suddenly tighten the leash. They raise interest rates sharply.
The Death Spiral: This is the killer. When banks raise rates, the companies in the "Real Garden" suddenly have huge bills to pay. They can't afford it. They stop building factories, fire workers, and stop borrowing. The economy shrinks. Because the economy is shrinking, the stock market crashes even harder. The banks get even more scared and raise rates even more.

It's a vicious circle:

Stock market goes up $\rightarrow$ Banks lend more.
Banks lend more $\rightarrow$ Stock market goes up more (bubble).
Bubble bursts $\rightarrow$ Banks panic and raise rates.
High rates kill the real economy $\rightarrow$ Stock market crashes harder.

5. What the Computer Simulations Show

The authors ran thousands of computer simulations (like running a video game 5,000 times with slightly different rules) to see what happens.

The "Slow Reversion" Trap: If the banks are slow to calm down after a scare (they keep rates high for too long), the economy almost always collapses.
The "Fast Reversion" Hope: If the banks can calm down quickly and lower rates when the panic is over, the economy can survive the shock.
The Volatility Factor: The more "jumpy" the stock market is (more wild swings), the more likely the whole system is to blow up.

The Big Takeaway

This paper tells us that financial crises aren't just "accidents" that happen from the outside. They are built into the system.

When we let banks lend too freely during a boom, we are essentially building a house of cards. When the wind blows (a small drop in stock prices), the house doesn't just shake; the banks' fear causes them to pull the rug out from under the whole economy.

In simple terms: The paper warns us that if we don't manage how much credit flows into the stock market, we are setting up a trap where a small stumble turns into a massive fall, and the banks' attempt to save themselves is what actually destroys the economy.

Here is a detailed technical summary of the paper "From debt crises to financial crashes (and back): a stock–flow consistent model for stock price bubbles" by Grasselli and Nguyen Huu.

1. Problem Statement

The paper addresses a gap in macro-financial modeling regarding the bidirectional causality between real economic debt dynamics and financial market instability.

The Gap: Existing literature often treats these domains separately. "Rational bubble" models (strict local martingales) typically rely on exogenous parameters for volatility and jumps, failing to link them to credit creation. Conversely, Stock-Flow Consistent (SFC) macroeconomic models (like the Keen framework) often treat financial markets as reduced-form or exogenous, lacking a rigorous stochastic representation of crashes and bubbles.
The Objective: To develop a unified, tractable, continuous-time stochastic model that integrates a Keen-style SFC macroeconomic block (featuring endogenous credit creation and debt-deflation) with a stochastic financial market block (featuring jump-diffusion asset prices). The goal is to formalize how speculative credit fuels asset bubbles, how crash risk feeds back into the real economy via bank lending spreads, and how this creates endogenous boom-bust cycles without external shocks.

2. Methodology

The authors construct a coupled system of stochastic differential equations (SDEs) with state-dependent jump intensities.

A. The Real Economy Block (SFC Framework)

Based on the Keen model (extended by Nguyen-Huu et al.), this block ensures accounting consistency across three sectors: Households, Firms, and Banks.

Variables: Wage share ( $\omega$ ), employment rate ( $e$ ), firm deposit ratio ( $m_f$ ), and loan-to-output ratio ( $\ell$ ).
Mechanisms:
- Production: Linear production function with constant capital-output ratio.
- Finance: Firms finance investment via retained profits and bank loans. A key innovation is the inclusion of a speculative flow ( $F$ ), representing debt-financed purchases of existing assets (not new production), which drives the link to the financial market.
- Dynamics: The system is driven by profit ratios, wage bargaining (Phillips curve), and inflation dynamics. The speculative flow $F$ is a function of the nominal output growth rate.

B. The Financial Market Block (Jump-Diffusion)

The asset price $S_t$ follows a compensated jump-diffusion process:
$dS_t = \bar{r}_L S_t dt + \bar{\sigma} S_t dW_t - \bar{J}_+ S_t (dN^+_t - \lambda^+ dt) + \bar{J}_- S_t (dN^-_t - \lambda^- dt)$

Endogenous Intensities: The jump intensities $\lambda^+$ $λ^{+}$ (downward jumps/crashes) and $\lambda^-$ $λ^{-}$ (upward jumps) are state-dependent, driven by the normalized speculative flow $f = F/(pY)$ $f = F / (p Y)$ .
- High positive speculative flow ( $f > 0$ ) increases the intensity of downward jumps ( $\lambda^+$ ), modeling the buildup of systemic fragility during credit booms.
- Negative flow ( $f < 0$ ) increases upward jump intensity ( $\lambda^-$ ), modeling short-covering or policy rebounds during deleveraging.
Trend Indicator: A mean-reverting trend indicator $\mu_t$ tracks the predictable component of log-returns, inheriting the same Brownian and jump shocks as the asset price.

C. The Coupling Mechanism

The two blocks are linked via the bank lending rate ( $r_t$ ):

Banks adjust the effective lending rate based on the trend indicator $\mu_t$ : $r_t = \bar{r}_L + \bar{\rho}_1 e^{-\bar{\rho}_2(\mu_t - \bar{r}_L)}$ .
Feedback Loop:
1. Economy $\to$ Market: High growth/profits $\to$ increased speculative credit ( $F$ ) $\to$ higher crash intensity ( $\lambda^+$ ).
2. Market $\to$ Economy: Market turbulence (low $\mu_t$ ) $\to$ banks increase lending spreads ( $r_t$ ) $\to$ higher debt servicing costs for firms $\to$ reduced investment and potential economic collapse.

3. Key Contributions

Mathematical Unification: The paper successfully merges the accounting discipline of SFC models with the stochastic calculus of modern bubble theory (strict local martingales and jump processes).
Endogenous Fragility: It demonstrates that financial fragility and crash risk are not exogenous shocks but emerge endogenously from the credit expansion process. The model formalizes Minsky's hypothesis that stability breeds instability.
Global Existence and Non-Explosion: The authors provide rigorous proofs (Appendix A) that the coupled system admits a unique global strong solution. They establish that the system does not explode in finite time and that asset prices remain strictly positive, ensuring the model is mathematically well-posed for simulation.
Mechanism of Transmission: The paper isolates the specific transmission channel: the bank lending spread. It shows that the reaction function of banks to market trends is the critical determinant of whether the economy absorbs market volatility or enters a self-reinforcing debt-deflation spiral.

4. Results (Numerical Simulations)

Using Monte Carlo simulations in Julia, the authors explore various regimes:

Decoupled Cases:
- Without feedback ( $\rho_1=0$ ), the economy behaves like a standard Keen model (converging or exploding based on interest rates), while the stock price follows a geometric Brownian motion with state-dependent jumps.
- Without jump intensities ( $\lambda=0$ ), the stock market still influences the economy via the interest rate, potentially causing collapse even with low baseline rates.
Fully Coupled Regimes:
- Boom-Bust Cycles: The system naturally generates cycles of credit expansion, asset price appreciation, and subsequent crashes.
- Crash Risk: Higher speculative flows increase the frequency of downward jumps.
- Bank Reaction Sensitivity:
  - Fast Mean Reversion ( $\bar{\eta}_\mu$ ): If banks quickly normalize rates after a shock, the economy is more resilient.
  - Slow Mean Reversion: If banks keep rates high for long periods after a market dip, the economy is prone to collapse.
  - Rate Sensitivity ( $\bar{\rho}_2$ ): A highly sensitive reaction (banks drastically raising rates when $\mu_t$ drops) significantly increases the probability of economic collapse.
Parameter Sensitivity:
- Higher baseline interest rates ( $\bar{r}_L$ ) and higher asset volatility ( $\bar{\sigma}$ ) increase the probability of crisis.
- The combination of slow trend reversion and high rate sensitivity creates the most unstable environment.

5. Significance

Theoretical: The model provides a rigorous mathematical foundation for the "Quantitative Theory of Credit," showing how money creation for financial transactions directly impacts asset prices and real economic stability. It bridges the gap between Post-Keynesian macroeconomics and quantitative finance.
Policy Implications: The framework serves as a testbed for macroprudential regulation. It suggests that:
- Stabilizing the lending spread reaction function is crucial.
- Policies that dampen speculative credit flows during booms can reduce crash intensity.
- The speed at which banks adjust rates to market conditions is a critical stability parameter.
Methodological: It offers a tractable, non-explosive stochastic framework for analyzing systemic risk, moving beyond linearized models or purely exogenous shock scenarios.

In summary, the paper demonstrates that financial crises are structural features of a credit-based economy where asset bubbles and debt dynamics are inextricably linked. The model successfully captures the transition from debt-driven growth to financial crash and the subsequent feedback loop that deepens the real economic crisis.