The Three-Body Dynamics Hypothesis of Capital Markets

2026-02-07

Why does market behavior exhibit both patterns and profound uncertainty?

• Why are markets always full of opportunity yet uncertain?
• Why do markets always repeat but never exactly replicate?
• Why are markets sometimes predictable and sometimes not?
• Why do capital markets exhibit phenomena like volatility clustering, market crashes, and recoveries?
• Why do markets experience sudden regime shifts?
• Why do markets exhibit patterns of sharp rallies followed by steep declines?
• Why is there a risk of assets going to zero?
• Why are the statistical properties of markets (e.g., fat tails in return distributions, volatility clustering) robust?

Core Idea

Assume the capital market is a three-body system, composed of three fundamentally different types of capital. Their interactions generate complex dynamic behaviors, including emergent phenomena like volatility clustering and market crash/recovery cycles.

Similar to the three-body problem in celestial mechanics, this system may have no steady-state solution, instead exhibiting limit cycles, quasi-periodic, or chaotic behavior.

Definition of the Three Bodies

The essential difference among market participants lies not in their identity (retail, institutional, market maker) but in the nature of their feedback to price changes.

Momentum Capital M (Momentum Capital)

Definition: Trading capital that provides positive feedback to price changes.

$$\frac{d(\text{Position})}{dS} > 0$$

Behavioral Characteristics:

• Buy on strength, sell on weakness (chase rallies, cut losses)
• Use leverage to amplify gains/losses
• Tendency for short-term holding
• Momentum trading, trend following

Impact on the System:

• Amplifies price movements
• Destabilizing force
• Creates trends and crashes

Typical Representatives: Speculators, trend traders, high-leverage traders, holders with passive stop-losses

Value Capital V (Value Capital)

Definition: Trading capital that provides negative feedback to price changes.

$$\frac{d(\text{Position})}{dS} < 0$$

That is: Reduce position when price rises, increase position when price falls.

Typically anchored by an intrinsic value $S^*$:

• When $S < S^*$, tendency to buy
• When $S > S^*$, tendency to sell

Behavioral Characteristics:

• Buy low, sell high (contrarian action)
• Trade based on value judgment
• Tendency for long-term holding
• Mean reversion, value investing

Impact on the System:

• Dampens price movements
• Stabilizing force (active)
• Provides market support and resistance

Typical Representatives: Value investors, contrarian investors, arbitrageurs

Liquidity Capital L (Liquidity Capital)

Definition: Liquidity-providing capital with no directional response to price changes.

$$\frac{d(\text{Position})}{dS} \approx 0$$

Behavioral Characteristics:

• Quote both sides, earn the bid-ask spread
• Hold no directional exposure (or hedge quickly)
• Continuously provide buy/sell liquidity
• Limited risk tolerance

Impact on the System:

• Reduces transaction costs
• Stabilizing force (passive)
• Buffers price shocks

Typical Representatives: Market makers, liquidity providers, high-frequency traders (market-making type)

Interactions Among the Three Bodies

Interaction Matrix

Detailed Mechanisms

M → L: Consumes Liquidity

The buy-on-strength/sell-on-weakness behavior of Momentum Capital generates large one-sided order flow, depleting market makers' inventory and forcing them to take on larger risk exposure. When volatility becomes too high, market makers may withdraw.

L → M: Constrains Shock Impact

Ample liquidity cushions the price impact of Momentum Capital. In deep markets, even high-leverage trades struggle to move prices significantly. Liquidity acts as a "shock absorber."

M → V: Creates Trading Opportunities

Momentum Capital's chasing of trends pushes prices away from intrinsic value, creating opportunities for Value Capital:

• Panic selling → Price below intrinsic value → Buying opportunity for V
• Frenzied buying → Price above intrinsic value → Selling opportunity for V

This represents an opportunity transfer from Momentum Capital to Value Capital.

V → M: Suppresses Extreme Behavior

Value Capital's contrarian actions provide price support, reducing the probability of one-sided price collapses, thereby lowering the expected return of trend-chasing strategies. Rational Momentum Capital may thus reduce its scale.

V → L: Restores Confidence

Value Capital's intervention signals that "the market has a floor," reducing market makers' fear of extreme losses and attracting liquidity back. Value Capital acts as "insurance" for market makers.

L → V: Facilitates Trading

Ample liquidity allows Value Capital to establish positions at low cost; large orders do not incur excessive slippage, improving the capital efficiency of Value Capital.

Feedback Loops

Positive Feedback Loop (Destabilizing)

$$M \uparrow \to \sigma \uparrow \to L \downarrow \to \text{Price Impact} \uparrow \to \sigma \uparrow \to M \uparrow \text{ (or forced liquidation)}$$

Increase in Momentum Capital → Rise in volatility → Liquidity withdrawal → Intensified price impact → Further rise in volatility → Further increase in Momentum Capital or forced liquidation

This is the mechanism of a crash spiral.

Negative Feedback Loop (Stabilizing)

$$|S - S^*| \uparrow \to \sigma \uparrow \to V \uparrow \to |S - S^*| \downarrow \to \sigma \downarrow \to L \uparrow$$

Price deviation from intrinsic value → Rise in volatility → Value Capital intervenes → Price regression → Decline in volatility → Liquidity recovery

Specific paths:

• After a sharp decline: $S < S^*$ → V buys → Price recovers
• After a sharp rally: $S > S^*$ → V sells → Price retraces

This is the recovery mechanism.

Phase Transitions of the System

The system's state depends on which loop dominates:

• Positive feedback > Negative feedback: System tends towards a crash
• Negative feedback > Positive feedback: System tends towards stability
• Critical point: System is at the boundary of a phase transition

Phase transitions are emergent outcomes of the system, not preset thresholds. The system spontaneously generates phase transitions through the competition between positive and negative feedback.

Describing the market state requires three core variables:

Return-Risk-Cost Matrix

There exists an axisymmetric relationship between the three types of capital (M, V, L) and the three market variables (δ, μ, σ):

Where:

• Return: When this variable increases, this capital directly profits.
• Risk: When this variable increases, this capital may incur losses.
• Cost: When this variable increases, the operational efficiency of this capital declines.

Symmetry

This matrix exhibits perfect axisymmetric structure:

• Each row: One return, one risk, one cost.
• Each column: One return, one risk, one cost.
• Diagonal: M-μ, V-δ, L-σ each correspond to the core source of return.

Three-Body Balance

From a column perspective, an increase in each market variable creates winners, losers, and those whose resources are consumed:

No single variable is beneficial or harmful to all capital types, which embodies the three-body balance.

Detailed Justification

Relationship between M and the three variables:

• $\mu$ (Return): Trend continuation = Profit for M. This is M's core return source.
• $\delta$ (Risk): Excessive premium signals reversal, posing a loss risk for M.
• $\sigma$ (Cost): High volatility makes stop-losses more likely to be triggered, increasing trading costs.

Relationship between V and the three variables:

• $\delta$ (Return): Large premium = Opportunity for V. This is V's core return source.
• $\sigma$ (Risk): High volatility means V may face larger unrealized losses after establishing a position, enduring pain even if the price eventually reverts. High volatility also implies the intrinsic value $S^*$ itself may be changing, destabilizing V's anchor.
• $\mu$ (Cost): Trend continuation forces V to wait longer, reducing capital efficiency.

Relationship between L and the three variables:

• $\sigma$ (Return): High volatility = More trading opportunities, higher market-making profits. This is L's core return source.
• $\mu$ (Risk): Strong trends cause L's inventory to accumulate persistently in one direction, exposing it to directional losses.
• $\delta$ (Cost): Large premium forces L to widen spreads for self-protection, reducing market-making efficiency.

Market Phases

Based on the high/low states of the three variables δ, μ, σ, the market exhibits $2^3 = 8$ typical phases.

Naming convention:

• Only one variable high: Name of the return party for that variable + "Beneficial Period"
• Only one variable low: Name of the return party for that variable + "Detrimental Period"
• All low / All high: Cold / Hot

Four dual pairs, codes are bitwise complements, names perfectly symmetric:

• Cold (000) ↔ Hot (111)
• Value Beneficial Period (100) ↔ Value Detrimental Period (011)
• Momentum Beneficial Period (010) ↔ Momentum Detrimental Period (101)
• Market-Maker Beneficial Period (001) ↔ Market-Maker Detrimental Period (110)

Detailed Phase Analysis

Cold (000): δ Low, μ Low, σ Low

Characteristics: All three parties unprofitable, market shrinks, low trading volume.

Typical Scenarios: Unpopular stocks, near-delisting, despair phase at the end of a bear market.

Market-Maker Beneficial Period (001): δ Low, μ Low, σ High

Characteristics: Golden period for L, price oscillates within a reasonable range with high frequency.

Typical Scenarios: Consolidation periods in mature markets, markets dominated by high-frequency trading.

Momentum Beneficial Period (010): δ Low, μ High, σ Low

Characteristics: M profits, L harmed, V watches; δ will gradually increase.

Typical Scenarios: Early stages of a trend, start of a slow bull/bear market.

Value Detrimental Period (011): δ Low, μ High, σ High

Characteristics: M dominates the market, high volatility and high trend, δ will increase rapidly.

Typical Scenarios: Early stages of MEME coins, initial stages of thematic speculation, breakout moves.

Value Beneficial Period (100): δ High, μ Low, σ Low

Characteristics: V sees opportunity but market is stagnant, awaiting a catalyst.

Typical Scenarios: Undervalued but neglected stocks, deep value investment targets.

Momentum Detrimental Period (101): δ High, μ Low, σ High

Characteristics: Battleground for V and L, high volatility but no clear direction.

Typical Scenarios: Around earnings reports, periods of major event uncertainty, standoff between bulls and bears.

Market-Maker Detrimental Period (110): δ High, μ High, σ Low

Characteristics: Clear trend but low volatility, M profits steadily, V waits painfully.

Typical Scenarios: Mid-phase of a one-sided bull/bear market, main rising/falling wave after trend establishment.

Hot (111): δ High, μ High, σ High

Characteristics: All three parties face extreme conditions, high return/high risk, system at a critical point.

Typical Scenarios: Peak of a bubble, moment of a crash, black swan events.

Phase Transitions

Each phase can transition to 3 adjacent phases by changing one dimension (δ, μ, or σ). All transitions are bidirectional.

graph TD
    S0["Cold<br/>(δ low, μ low, σ low)"]
    S1["Market-Maker Beneficial Period<br/>(δ low, μ low, σ high)"]
    S2["Momentum Beneficial Period<br/>(δ low, μ high, σ low)"]
    S3["Value Detrimental Period<br/>(δ low, μ high, σ high)"]
    S4["Value Beneficial Period<br/>(δ high, μ low, σ low)"]
    S5["Momentum Detrimental Period<br/>(δ high, μ low, σ high)"]
    S6["Market-Maker Detrimental Period<br/>(δ high, μ high, σ low)"]
    S7["Hot<br/>(δ high, μ high, σ high)"]

    S0 <--> |"σ"| S1
    S2 <--> |"σ"| S3
    S4 <--> |"σ"| S5
    S6 <--> |"σ"| S7

    S0 <--> |"μ"| S2
    S1 <--> |"μ"| S3
    S4 <--> |"μ"| S6
    S5 <--> |"μ"| S7

    S0 <--> |"δ"| S4
    S1 <--> |"δ"| S5
    S2 <--> |"δ"| S6
    S3 <--> |"δ"| S7

This is a three-dimensional hypercube (3-cube) structure: 8 vertices correspond to the 8 phases, 12 edges correspond to the 12 single-dimension transitions.

Typical Evolution Paths

Bubble Formation and Crash:

Cold → Momentum Beneficial Period → Market-Maker Detrimental Period → Hot → Momentum Detrimental Period → Value Beneficial Period → Cold
(000) → (010) → (110) → (111) → (101) → (100) → (000)

Healthy Market Oscillation:

Market-Maker Beneficial Period ↔ Value Detrimental Period ↔ Market-Maker Beneficial Period
(001) ↔ (011) ↔ (001)

Value Discovery:

Value Beneficial Period → Momentum Detrimental Period → Market-Maker Beneficial Period
(100) → (101) → (001)

Symmetry Between Cold and Hot

An important strategic insight from the phase transition diagram:

All three exits from Hot (111) lead to Detrimental Periods:

• δ↓ → Value Detrimental Period (011)
• μ↓ → Momentum Detrimental Period (101)
• σ↓ → Market-Maker Detrimental Period (110)

No matter which variable declines first, one type of capital is harmed, and it's unpredictable which variable will change first. Therefore, in an overheated state, any directional bet is gambling. The optimal strategy is non-participation or reduced leverage.

All three exits from Cold (000) lead to Beneficial Periods:

• δ↑ → Value Beneficial Period (100)
• μ↑ → Momentum Beneficial Period (010)
• σ↑ → Market-Maker Beneficial Period (001)

No matter which variable rises first, one type of capital benefits. Therefore, in a cold state, any participation is potentially profitable. The key is to stay in the market.

Ecological Niches of the Three Bodies

Ecological Balance: A healthy market requires the coexistence of all three.

• Lack of Momentum Capital: Market stagnant, no volatility, no trading opportunities.
• Lack of Liquidity Capital: High transaction costs, low market efficiency.
• Lack of Value Capital: Market fragile, prone to crashes, even risk of going to zero.

Relation to Traditional Classifications

Note: The same participant may play different roles at different times. The essence of classification is behavioral pattern, not identity label.

Three-Body Analogy

The market's three bodies share a profound similarity with the three-body problem in celestial mechanics.

Key to the Analogy: The essence of the three-body problem is the interaction of three celestial bodies with comparable mass. Precisely because they are evenly matched, no single body can dominate the system, leading to chaotic behavior.

In markets, M, V, L are similarly evenly matched:

• If M >> V and L: Market experiences one-sided rallies/crashes then goes to zero (bubble burst).
• If V >> M and L: Market barely fluctuates (stagnant).
• If L >> M and V: Price determined entirely by external information (perfectly efficient market).

Only when the three are evenly matched does the market exhibit genuine complex dynamics.

Lessons from Celestial Three-Body Problem:

• Two-body problem has analytical solutions (elliptical orbits).
• Three-body problem generally has no analytical solution, sensitive to initial conditions.
• Orbits can be periodic, quasi-periodic, or chaotic.

Inferences for Market Three-Body Problem:

• Long-term prediction impossible: System sensitive to initial conditions, random disturbances amplify, leading to long-term unpredictability.
• Short-term characteristics predictable: Trends and volatility clustering are short-term phenomena.
• Statistical laws robust: Macro statistical properties like fat tails in return distributions, volatility clustering are stable.

Supplement: Long-Term Evolution of Capital Scale

The core of the three-body model is the interaction among M, V, L. But a prerequisite question exists: Why can the three types of capital coexist and remain evenly matched in the long run?

This relies on a subordinate mechanism: Return-Driven Natural Selection.

Mechanism Description

Within the same capital type, individual returns follow a distribution. Individual behavior (expansion, contraction, exit) is highly correlated with their return:

• High-return individuals tend to remain or expand.
• Low-return individuals tend to contract or exit.

In the statistical effect of a large number of individuals, this tendency manifests as a change in the total scale of that capital type.

Self-Regulation

When a certain capital type is in excess:

1. Internal competition intensifies.
2. Average return declines.
3. Marginal exits increase.
4. Total scale of that capital type contracts.

When a certain capital type is insufficient:

1. Internal competition weakens.
2. Average return rises.
3. Attracts new capital inflow.
4. Total scale of that capital type expands.

Return Sources and Competition for Each Capital Type

Theoretical Status

Return-Driven Natural Selection is a slow variable mechanism (weekly to yearly timescale), while the three-body interaction is a fast variable mechanism (second to daily timescale).

This subordinate mechanism explains the existence and persistence of the three-body system—why the market does not evolve into a state dominated by a single capital type—but does not alter the core dynamics of the three-body interaction.

Research Directions

1. Phase Space Structure: Attractors, repellors, separatrices.
2. Timescale Separation: Fast variables (price), slow variables (capital structure).
3. Statistical Properties: Ergodicity, invariant measure, dwell time distribution.
4. Dynamic Equations: SDE system based on this framework (detailed in a separate article).

References

• Volatility and Leverage
• Volatility Effect Superior to Leverage Effect
• Capital War of Attrition