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arxiv: 2605.15767 · v1 · pith:6PKA5DXGnew · submitted 2026-05-15 · 💱 q-fin.ST · q-fin.MF

Market Makers and Risk Aversion: A Hamiltonian Approach to the Excess Volatility Puzzle

Will Hicks This is my paper

Pith reviewed 2026-05-19 18:03 UTC · model grok-4.3

classification 💱 q-fin.ST q-fin.MF
keywords excess volatility puzzleHamiltonian dynamicsmarket makersrisk aversionchaotic dynamicsanharmonic oscillatorsfinancial time seriesinventory dynamics
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The pith

Market makers' risk appetite sets the level of chaos in prices, generating unpredictability without external shocks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models financial market dynamics by representing the market price and market makers' inventory as anharmonic oscillators linked through a nonlinear coupling. This setup permits the application of Hamiltonian mechanics to demonstrate that the system can produce chaotic trajectories. The central parameter controlling the extent of this chaos is the market makers' risk appetite. External shocks and random noise, while relevant for real data, turn out not to be required for the emergence of unpredictable price movements. A reader would care because the result reframes the excess volatility puzzle as arising from internal market mechanics rather than solely from outside disturbances.

Core claim

Treating the market price and market makers' inventory as anharmonic oscillators with a nonlinear coupling allows Hamiltonian dynamics to show that the degree of chaos in the system is governed by the market makers' risk appetite. External shocks and random noise are not necessary in order to generate unpredictable price changes.

What carries the argument

The Hamiltonian model of market price and inventory as nonlinearly coupled anharmonic oscillators, with risk appetite as the parameter controlling the transition to chaotic behavior.

If this is right

  • Higher market-maker risk aversion produces greater price unpredictability through internal dynamics alone.
  • Lower risk aversion reduces the chaotic component and stabilizes price paths.
  • Volatility can be modulated by changes in inventory management rules without altering the flow of external information.
  • The model separates the contribution of endogenous chaos from that of exogenous noise in observed price series.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Interventions that alter market makers' effective risk tolerance, such as capital requirements, could measurably change the chaotic component of volatility.
  • The framework suggests testing whether liquidity provision rules that limit inventory swings also suppress chaos signatures in high-frequency data.
  • Similar oscillator models might apply to other pairs of quantities that evolve through inventory-like constraints, such as order-book depth and trade size.

Load-bearing premise

The market price and market makers' inventory can be treated as anharmonic oscillators with a nonlinear coupling so that Hamiltonian dynamics produce the claimed chaotic behavior.

What would settle it

Empirical measurement of market-maker risk aversion over time showing that periods of higher risk aversion coincide with stronger signatures of chaos in price returns, even after removing periods with large external news events.

Figures

Figures reproduced from arXiv: 2605.15767 by Will Hicks.

Figure 1
Figure 1. Figure 1: Contour plots for the potential functions with different values for ϵ. Each case has the same market energy. We set x0 = 1. the kinetic energy: Muu˙ 2 2 , together with the potential energy: ϵ(xv) 2 2 as before. H(x, v, Px, Pv) = P 2 x 2Mx + P 2 u 2Mu + Kx(x − x0) 2 2 + ϵ(u) 2 2 (10) Clearly, this system comprises 2 independent harmonic oscillators. Market Maker inventory simply rises as high as needed to … view at source ↗
Figure 2
Figure 2. Figure 2: Poincare sections for ´ v = 0, v >˙ 0 at energy=1±0.01. We show the section with ϵ = 0.0001, 0.001, 0.01 and 0.1 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Poincare sections for ´ v = 0, v >˙ 0 at energy=1±0.01, 5±0.01, and 20±0.01. ϵ = 0.001. • For a non-deterministic Markov process continuous in state and time, that is non deterministic, for example a Wiener process, the Kolmogorov-Sinai entropy is infinite (see [16]). The fractal nature of the process means that finer partitions will always reveal more information. • For chaotic motion, such as that seen h… view at source ↗
Figure 4
Figure 4. Figure 4: The chart shows the sum of positive Lyapunov exponents, calculated using paths with energy target 5, and ϵ ranging from 0.001 up to 0.1 In fact, from Pesin’s formula (see [17]) HKS can be estimated from the sum of all positive Lyapunov exponents for the process: HKS = X λi>0 λi In figure 4 we show an estimate of the Kolmogorov-Sinai entropy, calculated using the sum of positive Lyapunov exponents, for ener… view at source ↗
Figure 5
Figure 5. Figure 5: The chart shows 1000 steps of an example price process X, where the system energy is 10.6, and ϵ = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An example of the process generated by sampling the full price trajectory for x once every 100 steps precisely. If we assume that the frequency with which we can make measurements of a random process is limited, and as such the typical period between observations is Tobs, then as λmax grows, once Tλ << Tobs, the chaotic process will become indistinguishable from random noise. For example, the true ‘price’ … view at source ↗
Figure 7
Figure 7. Figure 7: Histogram for the value of X sampled exactly every 100 steps. financial exchanges, random noise is not required to generate unpredictable price changes. By encoding relatively simple market forces that impact prices into a Hamiltonian, one sees price processes that appear, at first glance at least, to be indistinguishable from time-series generated using random noise. Furthermore, by encoding market forces… view at source ↗
read the original abstract

In this article we model chaotic dynamics in financial markets by treating the market price, and market makers' inventory, as anharmonic oscillators with a nonlinear coupling. The market makers' risk appetite being the key parameter that determines the degree of chaos in the system. The article demonstrates that whilst external shocks and random noise are important in the treatment of financial time-series, they are not necessary in order to generate unpredictable price changes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript models chaotic price dynamics in financial markets by representing the market price and market makers' inventory as anharmonic oscillators with nonlinear coupling in a Hamiltonian framework. The market makers' risk appetite is presented as the central parameter controlling the degree of chaos, with the claim that this deterministic setup generates unpredictable price changes and excess volatility without requiring external shocks or random noise.

Significance. If the Hamiltonian is properly derived from market-making incentives and the resulting trajectories reproduce key statistical features of excess volatility, the work would provide a novel deterministic mechanism linking risk aversion to market unpredictability. This could reduce reliance on exogenous noise in volatility models and demonstrate the utility of classical mechanics tools in quantitative finance, particularly if the chaos measures yield falsifiable predictions.

major comments (3)
  1. [Model formulation] The central modeling assumption (price P and inventory I as anharmonic oscillators with nonlinear coupling) is introduced without a derivation from a market maker's profit-maximization or utility problem. This makes it unclear whether the Hamiltonian is a consequence of the economic setup or an ad-hoc choice whose chaotic solutions are then attributed to risk aversion.
  2. [Hamiltonian construction] The claim that external shocks and noise are unnecessary for unpredictable prices rests on the system being closed and conservative. The manuscript must show explicitly how inventory penalties, slippage, and interaction with liquidity takers are incorporated without introducing non-Hamiltonian (dissipative or stochastic) terms that would alter the phase-space structure.
  3. [Chaos analysis] The assertion that risk appetite is the key parameter determining chaos requires a concrete demonstration (e.g., via Lyapunov exponents or Poincaré sections) that varying this single parameter produces a transition to chaos whose statistics match observed excess volatility, rather than the chaos measure reducing to a fitted function of the risk-aversion parameter by construction.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by a one-sentence statement of the explicit form of the Hamiltonian or the chosen chaos diagnostic.
  2. [Notation] Notation for the risk-aversion parameter should be introduced consistently once the Hamiltonian is defined, to avoid ambiguity when comparing across regimes.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Model formulation] The central modeling assumption (price P and inventory I as anharmonic oscillators with nonlinear coupling) is introduced without a derivation from a market maker's profit-maximization or utility problem. This makes it unclear whether the Hamiltonian is a consequence of the economic setup or an ad-hoc choice whose chaotic solutions are then attributed to risk aversion.

    Authors: We agree that the economic motivation for the specific form of the Hamiltonian can be made more explicit. The anharmonic oscillator terms are chosen to represent nonlinear price impact and inventory holding costs that intensify with risk aversion, while the nonlinear coupling captures the feedback between inventory imbalances and price adjustments in a market-making setting. In the revised manuscript we will add a new section that derives the Hamiltonian structure from a stylized market-maker optimization problem in which risk aversion enters as the coefficient of the inventory penalty term, thereby clarifying that the functional form follows from the economic incentives rather than being chosen purely for its chaotic properties. revision: yes

  2. Referee: [Hamiltonian construction] The claim that external shocks and noise are unnecessary for unpredictable prices rests on the system being closed and conservative. The manuscript must show explicitly how inventory penalties, slippage, and interaction with liquidity takers are incorporated without introducing non-Hamiltonian (dissipative or stochastic) terms that would alter the phase-space structure.

    Authors: We will expand the model-description section to provide an explicit term-by-term mapping. The quadratic and higher-order terms in the inventory variable I directly encode the risk-aversion penalty for inventory holdings; the coupling terms between P and I represent the price concessions made to liquidity takers and the resulting inventory changes. All interactions are formulated as conservative forces derived from a potential, preserving the symplectic structure of the phase space. The revised manuscript will include this mapping together with a short appendix confirming that no dissipative or stochastic terms are present. revision: yes

  3. Referee: [Chaos analysis] The assertion that risk appetite is the key parameter determining chaos requires a concrete demonstration (e.g., via Lyapunov exponents or Poincaré sections) that varying this single parameter produces a transition to chaos whose statistics match observed excess volatility, rather than the chaos measure reducing to a fitted function of the risk-aversion parameter by construction.

    Authors: The current manuscript reports numerical trajectories that become increasingly irregular as the risk-aversion parameter is raised, but we accept that a more quantitative characterization is required. In the revision we will compute and plot the largest Lyapunov exponent as a function of the risk-aversion parameter, include representative Poincaré sections, and compare the resulting price-return statistics (kurtosis, volatility clustering measures) against empirical benchmarks for excess volatility. These additions will demonstrate that the transition to chaos and the associated volatility statistics emerge from the deterministic dynamics rather than from direct fitting of the chaos indicator to the parameter. revision: yes

Circularity Check

0 steps flagged

No circularity: Hamiltonian model is an explicit ansatz whose chaotic regime is derived from the posited equations rather than fitted inputs

full rationale

The paper constructs a closed Hamiltonian system with price and inventory as anharmonic oscillators under nonlinear coupling, treating risk aversion as a free parameter that modulates the degree of chaos. The claim that external noise is unnecessary follows directly from integrating the deterministic equations of motion; no step reduces a prediction to a fitted value by construction, nor does any load-bearing premise rest on a self-citation chain. The modeling choice is stated as such and does not smuggle an ansatz via prior work or rename an empirical pattern. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on treating price and inventory as anharmonic oscillators whose nonlinear coupling is controlled by a risk-aversion parameter; this modeling choice is introduced without independent justification in the abstract.

free parameters (1)
  • risk appetite parameter
    Serves as the control parameter that sets the degree of chaos; its value must be chosen or fitted to produce the desired chaotic regime.
axioms (1)
  • domain assumption Market price and market makers' inventory behave as anharmonic oscillators with nonlinear coupling.
    This is the foundational modeling step that allows Hamiltonian dynamics to be applied to financial time series.

pith-pipeline@v0.9.0 · 5582 in / 1271 out tokens · 38941 ms · 2026-05-19T18:03:38.288056+00:00 · methodology

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Reference graph

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