Capillarity Model of Financial Stability (CMFS): Evidence of Non-Linear Systemic Tension in Global Financial Markets

© Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce Capillarity Model of Financial Stability (CMFS): Evidence of NonLinear Systemic Tension in Global Financial Markets 1. Introduction Existing volatility and stress models treat liquidity as a continuous flow, failing to account for possible structural ruptures in the skin surrounding the Market (or the firm). Models such as VaR, GARCH, CCAR (Comprehensive Capital Analysis and Review), or DFAST (Dodd-Frank Act Stress Test) cannot fully explain sudden nonlinear structural collapses (i.e., Flash Crashes, Bank Runs). These standard linear models fail to explain how a stable state in financial markets can become a crisis in an instant. VaR and GARCH-like models are based primarily on the idea that events follow continuous probability distributions (Bell-Shaped Curves). Thus, a crash is merely a "statistical outlier" or a "3-sigma event." Hence, during a collision or bank failure, the same market behavior dynamics exist, but at different stages of intensity. Pioneering academic work has long recognized that linear models are unable to identify sudden shifts in financial markets and in the business cycle in general. Hamilton (1989) argued that standard autoregressive (ARMA) and basic GARCH models cannot account for abrupt (unobserved) shifts between regimes (e.g., boom/recession, quiet/volatile). Tong (1990) states that non-linear threshold effects govern economic phenomena; the behavior of the observed variable changes after crossing a critical point. He further noted that linear models cannot explain such a sudden structural change. In a comprehensive review, Cont (2001) demonstrates that volatility clustering, heavy tails, and long-term dependences are inherently nonlinear. He further argues that simple ARMA or GARCH models are not compatible with such phenomena; they fail to capture sudden, explosive increases in volatility during crises. Bekaert and Harvey (2002) examined weather models of global market integration that capture sudden and significant increases in return correlations observed during flight-to-equity or contagion episodes. They state that the dynamics of international financial linkages are not constant but follow a regime-switching model in which stable periods follow one linear process. In contrast, during unstable periods, they follow a different, unpredictable, and highly correlated process. © Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce 2 To address this gap, we employ the Threshold Autoregressive Framework (Tong, 1990) model and the Amihud Illiquidity Ratio to illustrate the practical relevance of the Capillarity Model of Financial Stability (CMFS) and to provide a more comprehensive understanding of global financial Stability. The layout of the paper is as follows: Section 2 presents the data and methodology. Section 3 presents empirical findings and suggestions for future research. Section 4 Concludes. 2. Data, Theoretical Framework, and Methodology 2.1 Data We collect data for three major global indices representing G3 countries: the S&P 500, the DAX, and the Nikkei 225. For market depth, we collect the volume of the S&P 500 ETF (SPY) and the highly liquid US-listed ETFs EWG and EWJ, respectively. Data starts on the 4th of January, 1993 (the launch of the S&P 500 index), and ends on the 30th of September 2025—a total of 11957 days and 17080 weeks. 2.2 Theoretical framework The debate boils down to a comparison between Exogenous Shock vs. Endogenous Fragility: that is, what happens to the financial markets (FM) if the economy crashes, versus what internal pressures cause the FM's structure to spontaneously rupture. Table 1 summarizes the main features of the comparison between Standard Finance's volatility modeling and the proposed Paradigm of CMFS: Table 1: Comparison between the current Standard Finance's volatility modeling and the proposed theory of CMFS: Feature Standard Finance CMFS Physics Metaphor Fluid Dynamics: Flow, pipes, turbulence Surface Science: Sessile drop, wetting, tension. Crash Definition Statistical Tail Event: Extreme value on a continuous curve. Structural Rupture: caused by pressure differences. Key Variable Volatility: the vibration of prices. Surface Tension: The cohesion of market depth. Liquidity View Continuous: Assumed to exist, costs more. Discrete/Finite: Can vanish instantly (Liquidity Vacuum). Institutional View Volume-based: Capital Adequacy (Quality and Quantity) Structure-based: Cohesion Ratios (Quality and Integration). Panic Role Noise: Temporary irrational deviation Nucleation: The trigger for a phase transition. Note: The Young-Laplace Equation serves as the basis for the proposed CMFS theory. By applying this fundamental physical law of water dynamics to FM, we can estimate the pressure difference between the inward forces exerted by the inner liquidity pressure © Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce 3 (Trade Volume) and the outward forces exerted by the water's outer surface, such as short-term/long-term fund movements. To strengthen the model's credibility, we identify and discuss the underlying assumptions, limitations, and contexts in which CMFS provides the most accurate insights into financial stability, leading to a flash crash or rupture rather than a correction. Looking at market depth data confirms that high price concentration with few market participants (i.e., beading) is a leading indicator of financial instability and possible crash. In contrast, low prices with a large number of market participants (i.e., wetting). The Contact Angle as a stability indicator in financial markets (FM) and financial institutions (FI). Fig. 1. The "Wetting principle" "Hydrophobic state vs. the "Beading" Hydrophilic state. In Fig. 1, the Wetting principle is analogous to the case of FM, in which the Hydrophobic state corresponds to a shorter contact area, indicating a less stable sessile drop; meanwhile, in the Hydrophilic state, the contact area is wider, suggesting a more stable sessile drop. The angle of θ > 90° in the case of the Hydrophobic state, which indicates a high probability of the water drop rolling over due to the tall volume of water in the center and shorter contact area. In such a case, the analogy is that an asset's price is high because of too much Capital in particular sectors, assets, markets, or geographic locations. Thus, it has a narrower market contact area, less economic integration, and fewer asset owners. Any tilt caused by an event or unexpected economic or financial numbers would cause significant capital to move away from this highly priced market (i.e., never catch a falling knife). A practical application of the CMFS proposed theory concerns the mismatch between the investment horizons of short-term (speculators) and long-term regulators. In such a case, FM faces a high probability of financial failure due to a sudden, unexpected cash outflow when © Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce 4 cash/investment owners believe the regulators may take a decision that does not meet their expectations. In the Hydrophilic states, the angle θ < 90°, suggesting a broader area of contact with the surface (i.e., a higher degree of integration with the economy and a broader ownership base). In this case, capital spread across different sectors and investment classes, such as infrastructure and production facilities, as well as long-term fund commitments. In such a state, an asset can withstand external pressure while maintaining its cohesion. Cash/investment owners feel relatively safe due to investment diversification and the avoidance of "putting all eggs in one basket." By linking the concept of physics variables to standard financial variables, we can highlight the underlying principles behind CMFS. Table 2 provides a matrix consisting of physical variables and their standard finance counterparts at the Macro (FM) and Micro (FI) 1 levels. Table 2: Physical Variables and their Standard Finance counterparts at Macro and Micro levels: Physics Variable Standard Finance CMFS Macro (FM) Micro (FI) Sessile Drop Market/Firm Stocks, Bonds, Foreign Exchange, and Commodities. Banks, insurance, and investment firms/ Balance Sheet/ Capital Base Surface Tension (Gamma) Global capital mobility / Loans-toDeposit Ratio Market Depth: Inverse Amihud Ratio/ Order Book Depth Equity Buffer: Tier 1 Capital / High-Quality Liquid Assets (HQLA) Internal Pressure (Delta) Investment cost (bid/ask spread Aggregate Volatility (VIX) Credit Risk / NonPerforming loans (NPLs) Substrate (Mu) Long-Term stability Economic Fundamentals Core Business /Franchise Value Contact Angle (Theta) Funding Structure Number of buyers/sellers Ownership size/type Rupture Event Flash Crash/Regime Switch Failure/Distress Note: these variables are used to map out the theory and how it connects physics to finance. In the empirical analysis (in the data and methodology sections), we use the following variables: Gamma, Delta, Mu, and Theta. 2.3 Methodology We specified the physics concepts and their respective financial roles as follows: 1) Surface Tension / Cohesion (Gammad): The force of holding the structure together (Liquidity and Equity Buffer). 2) Internal Pressure (Delta P): The outward pushing forces (Volatility/Operational Risk). Finally, 3) Contact Angle (Theta): The degree of integration of the FM into its fundamentals. Based on these foundations, we constructed input variables to develop the Macro STI (Systemic Tenacity Index) for the FM. 1 The case of FS would be the subject for future research. © Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce 5 In the Macro STI, we construct daily time series to predict flash crashes (or corrections). 𝑆𝑇𝐼𝑚𝑎𝑐𝑟𝑜,𝑡 = 𝐿𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝐶𝑜ℎ𝑒𝑠𝑖𝑜𝑛 𝑉𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 𝑆𝑡𝑟𝑒𝑠𝑠 = 1 𝐴𝑚𝑖ℎ𝑢𝑑𝑡 × 𝜎 (1) Where Amihud is the daily illiquidity measured by |𝑅𝑒𝑡𝑢𝑟𝑛 𝑡| 𝑉𝑜𝑙𝑢𝑚𝑒 𝑡 ×𝑃𝑟𝑖𝑐𝑒 𝑡 , and σ is estimated by GARCH (1, 1) for day t. Once the STI drops, we can infer that the surface tension "the skin" of the market is too weak to hold on and is facing volatile surroundings "Volatility." We apply the Self-Exciting Threshold Autoregressive Model (SETAR) in Equation (2 to test the relationship between past returns and future return changes and to determine whether it is structurally different when the STI crosses a specific point. 𝑅𝑡= {∝1+ 𝛽1 𝑅𝑡−1 + ∈𝑡 𝑖𝑓 𝑆𝑇𝐼𝑡−1 ≥ 𝛾 (𝑆𝑡𝑎𝑏𝑙𝑒 𝐿𝑖𝑞𝑢𝑖𝑑 𝑆𝑡𝑎𝑡𝑒) ∝2+ 𝛽2 𝑅𝑡−2 + ∈𝑡 𝑖𝑓 𝑆𝑇𝐼𝑡−1 < 𝛾 ( 𝑈𝑛𝑠𝑡𝑎𝑏𝑙𝑒 𝑅𝑢𝑝𝑡𝑢𝑟𝑒𝑑 𝑆𝑡𝑎𝑡𝑒) (2) In the Stable Liquid State, β1 should be close to zero; no structural difference between the current and previous returns. However, in a Ruptured State, β2 would be more volatile, indicating negative momentum and leading to panic 2 . 3. Results and discussion By applying the SETAR model and Amihud Illiquidity Ratio to G3 markets, we identify four primary areas for improvement offered by CMFS: First, the dynamics of market crash (Outlier vs. Rupture), which CMFS explains that the initial dynamics behind a FM crash are a transitional phase (i.e., rupture), in which the market's fundamental dynamics change and cause different market dynamics. Hence, surface tension arises from internal forces that are no longer capable of resisting external tension, occurring when pressure exceeds tension. That would explain why a "black swan" event happens more often than classical statistical models predict. Second, the Definition of Liquidity (Volume vs. Tension): In the banking sector, Basel III and the CCAR view liquidity in terms of volume and the measurement of High-Quality Liquid Assets (HQLA). However, CMFS argues that capital (or asset) assessment alone is not enough; you may have very high-quality assets, but if not held together through strategic, enforced links, they could, under pressure, rupture and be broken into pieces. In other words, a bank could hold a large volume of capital; 2 MCFS suggests that panic is a transitional phase, not a temporary deviation from long-term equilibrium, see 3rd in the introduction of this paper. © Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce 6 however, if that capital has high duration mismatches or "flighty deposits", it has no cohesion and cannot withstand increasing external pressure for long. A full bucket of water is likely to spill. In contrast, a drop of water (High Tension) holds its shape because surface tension resists external pressure. Third, the Role of Psychology (Noise vs. Nucleation), Efficient Market Hypothesis (EMH), and Standard Finance view panic as noise; a byproduct of market participants attempting to analyze and observe new information, which leads to wider spreads. In comparison, CMFS suggests that panic acts as a catalyst or a trigger for a crash. Therefore, the market is superheated and reaches a metastable state under pressure, remaining efficient until a nucleation event triggers phase changes and moves into crash mode. The final area of improvement is in the Timing of Signals and Indicators. Volatility indices (VIX) and GARCH models provide concurrent or lagging indicators of market behavior. As in car accidents, we typically cannot see the crash until it happens in front of us, even though we may still hear or see signs of a possible accident. Likewise, we cannot see a market crash occurring unless prices are collapsing. However, CMFS provides a leading indicator of crashes by estimating the Surface Tension Index (STI), which measures the thinning of the order book or the beading up of Capital (known as Hydrophobic wetting) before prices actually move down once a water drop starts wobbling and distorting due to low surface tension, which causes an expected burst to fallow, even if volatility has not yet begun. 4. Conclusion For future research and applications, we see a wide range of opportunities in banking and corporate finance. At the corporate level, CMFS has a wide range of applications in decisions such as M&A and portfolio organization. In terms of market depth, once a Letter of Intent (LOI) or Memorandum of Understanding (MOU), a nonbinding agreement announcing the intent of a Merger and Acquisition (M&A), is announced, market depth (measured by order book) would narrow as investors wait for the next step, and for additional information that would either strengthen the strategic bonds resulting in stronger than pre-M&A linkages between the assets of the two entities. In terms of banking, equity buffer: tier 1 capital, High-Quality Liquid Assets (HQLA), and Credit Risk/Non-Performing Loans (NPLs) are tension creators within the banking system. Once these indicators behave unexpectedly, a rupture is likely, as © Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce 7 external forces, mainly bank runs and regulators' interventions, would cause a rupture in the organization's skin or surface. In this case, we could address the issue of external and internal forces from both market-based and accounting-based perspectives. Both examples form the bases for future work fit with CMFS core's analogy of financial entities as sessile drops, in which stability depends on surface tension (γ, cohesion among assets), internal pressure (ΔP, risks or uncertainties), curvature (R, concentration of value), and wetting (W, integration with the economic substrate). References: © Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce 8 Appendix 1: Comparison of the three global indices (S&P 500 – DAX – Nikkei 225) © Rayan S. Hammad, ORCID: 0009-0007-9384-4905Confidential Draft: MCFS Theory - Created December 15, 2025 - Do Not Distribute or Reproduce 9 Appendix 2: Illiquidity ratio during a shorter time span for a closer picture.