Sequence-Dependent Arbitrage (SDA) Dynamics

• Sequence-Dependent Arbitrage (SDA) is a phenomenon where the order of trades and information revelations creates persistent cycles and non-ergodic market behaviors.
• The concept is rigorously modeled using matrix products, geometric invariant subspaces, and stochastic processes that quantify how sequential arbitrage operations impact economic states.
• Applications of SDA span currency markets, functionally generated portfolios, and high-frequency trading, offering actionable insights for risk management and market strategy.

Sequence-Dependent Arbitrage (SDA) refers to the phenomenon in economics and finance where the exploitation and elimination of arbitrage opportunities depend intrinsically on the order or sequence in which trades, updates, or information revelations occur. Rather than arbitrage operations instantaneously balancing prices across markets, the specific path of actions can generate persistent cycles, non-trivial equilibria, or targeted re-steering of market outcomes. This topic synthesizes advanced linear algebraic and geometric perspectives, as well as stochastic process and optimal transport theories, to characterize the impact and structure of SDA across different economic systems.

1. Mathematical Framework for SDA: Matrix Products and Economic States

SDA is rigorously formalized within finite-dimensional models by representing the state of an economy as a vector of principal exchange rates. For a four-good "mini-economy" (e.g., Food-Arms-Relics-Medicine), the ensemble of exchange rates is denoted by $$R = (r_{FA}, r_{FR}, r_{FM}, r_{AR}, r_{AM}, r_{RM})$$, where each $r_{XY}$ is a strictly positive rate. A logarithmic transformation is applied, $$v(R) = (\log r_{FA}, \log r_{AR}, \log r_{RM}, \log r_{FR}, \log r_{AM}, \log r_{FM})$$, converting the multiplicative economic balance conditions into a linear additive structure.

Each arbitrage transformation—such as a producer correcting a price discrepancy—is modeled as a linear operator: for an arbitrage operation $\hat{A}$, there is a $6 \times 6$ matrix $B$ such that $v(R\hat{A}) = v(R) B$. For a chain of arbitrages, this generalizes to

$$v(R \hat{A}_1 \hat{A}_2 \cdots \hat{A}_n) = v(R) \prod_{i=1}^n B_{k(i)} .$$

This representation enables precise analysis of the cumulative impact of sequences of arbitrages through matrix product theory, including spectral radius, invariant subspaces, and joint behavior of arbitrage matrices (Kozyakin et al., 2010).

2. Geometric Interpretation and Invariance Properties

The algebraic analysis reveals a geometric substrate: the transformation matrices associated with arbitrage operations share a common invariant subspace corresponding to balanced economic states. The balance conditions (e.g., $r_{FA} \cdot r_{AR} = r_{FR}$, $r_{AR} \cdot r_{RM} = r_{AM}$) are linearized in log coordinates, manifesting as invariance equations such as $v^{(1)} + v^{(2)} = v^{(4)}$, etc. Changing to suitable coordinates via an invertible matrix $Q$, the matrices become block triangular:

$$D_n = Q^{-1} B_n Q = \begin{bmatrix} I & 0 \ F_n & G_n \end{bmatrix} ,$$

where $I$ acts on balanced components and $G_n$ governs deviations. The action of $G_n$—all with spectral radius one—on the deviation space describes the evolution and control of imbalances.

The geometric action preserves a polyhedral unit ball $\mathbb{P}$ (the convex hull of signed "corner vectors"), whose structure is isomorphic to an octahedron graph. Arbitrage transformations accordingly move economic states within this bounded region, characterizing the persistent or recurrent nature of discrepancies under sequences of arbitrages.

3. SDA in Currency Markets: Periodicity and Non-Ergodicity

In foreign exchange (FX) markets with three currencies, arbitrage operations immediately adjust all rates to satisfy the law of one price—the system is uniquely balanced after one correction. The introduction of a fourth currency leads to six principal rates but only three independent balancing conditions, greatly increasing the degrees of freedom.

Periodic chains of arbitrage, especially when orchestrated by an external "Arbiter" controlling the order of information revelation, induce cycle phenomena. Rather than smooth convergence, the system can evolve periodically, with non-ergodic behavior and persistent mispricings. For example, a repeated 24-step chain of arbitrage yields periodic sequences of FX ensembles, and the set of attainable balanced states is described by parameterizations of the form $(\alpha^{n_1} \bar{r}_{\$€}, \alpha^{n_2} \bar{r}_{\$£}, ...)$, with$\alpha$a mark-up factor and$n_i \in \mathbb{Z}$ dictating the sequence (<a href="/papers/1112.5850" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Cross et al., 2011</a>).</p> <p>Moreover, the outstanding discrepancies evolve as orbits among vertices of distorted polyhedrons (truncated octahedra or tetrahedra) under matrix products, with the actual trajectories strongly dependent on the sequence of arbitrage events.</p> <h2 class='paper-heading' id='stochastic-and-binary-market-models-pathwise-sda-and-asymptotics'>4. Stochastic and Binary Market Models: Pathwise SDA and Asymptotics</h2> <p>Fractional binary markets model asset paths as traversals on a binary tree; &quot;arbitrage points&quot; are <a href="https://www.emergentmind.com/topics/neural-ordinary-differential-equations-nodes" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">nodes</a> where the local non-arbitrage condition fails, defined via inequalities on history-dependent increments:</p> <p>$d_n^{(N)}(x) < -a_n^{(N)} < u_n^{(N)}(x) ,$</p> <p>where$a_n^{(N)}$is drift,$Y_n^{(N,H)}$a sum over past increments, and$g_n^{(N,H)}$$quantifying current moves (<a href="/papers/1401.7850" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Cordero et al., 2014</a>). Arbitrage paths, sequences through at least one arbitrage point, directly operationalize SDA.</p> <p>Asymptotic analysis leveraging rescaled disturbed random walks shows that the proportion of arbitrage points converges to the probability that a limiting process$$|\mathcal{Y}_H| > g_H$:</p> <p>$\lim_{n\to\infty} \frac{|\mathcal{A}_n^{(N_n,H)}|}{2^{n-1}} = P(|\mathcal{Y}_H| > g_H) > 0 .$</p> <p>For large Hurst parameter$H$$(long-range dependence), almost every path is an arbitrage path ($$P=1$). Thus, SDA is deeply entwined with the underlying memory structure of the market process.</p> <p>Critical transaction cost analysis reveals that eliminating arbitrage in discrete models requires costs approaching unity as discretization increases, even as limiting continuous-time models lack arbitrage under arbitrarily small friction (<a href="/papers/1407.8068" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Cordero et al., 2014</a>). Only optimally scaled 1-step trading strategies, exploiting sequence-dependent price histories, yield non-vanishing &quot;asymptotic arbitrage&quot; in the limit.</p> <h2 class='paper-heading' id='sda-via-functionally-generated-portfolios-and-optimal-transport'>5. SDA via Functionally Generated Portfolios and Optimal Transport</h2> <p>In equity markets, the geometric theory of relative arbitrage identifies <a href="https://www.emergentmind.com/topics/functionally-generated-portfolios" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">functionally generated portfolios</a>—rules that assign portfolio weights deterministically as functions of current market weights via a concave generating function $\Phi$:</p> <p>$\pi_i(p) = p_i [1 + D_{(e(i) - p)} \log \Phi(p)] ,$$</p> <p>with value process evolution decomposed as</p> <p>$$\log V(t) = \log \frac{\Phi(\mu(t))}{\Phi(\mu(0))} + A(t) ,$</p> <p>where$A(t)$ is the cumulative nonnegative &quot;L-divergence.&quot; The property of multiplicative cyclical monotonicity (MCM) is necessary and sufficient for arbitrage: for closed market weight cycles,</p> <p>$\prod_{t=0}^{m-1} [1 + \langle \pi(\mu(t))/\mu(t), \mu(t+1) - \mu(t)\rangle ] \geq 1 .$$</p> <p>Optimal transport characterizations further identify these portfolios as optimal maps minimizing a logarithmic transport cost, with c-cyclically monotone structure (<a href="/papers/1402.3720" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Pal et al., 2014</a>). Thus, the market path (sequence) determines portfolio outperformance, tightly binding SDA to concave function generation and optimal transport theory.</p> <p>Empirical implementations leverage these results by constructing portfolios via historical data and matching distributions (e.g., diversity-weighted portfolios with explicit formulas), robustly capturing sequence-dependent gains from market volatility and diversity.</p> <h2 class='paper-heading' id='geometric-and-quantum-mechanical-formalisms-for-sda'>6. Geometric and Quantum-Mechanical Formalisms for SDA</h2> <p>The geometric arbitrage framework recasts market dynamics as a stochastic principal fibre bundle, with connection curvature measuring instantaneous arbitrage capability. The parallel transport along time and portfolio directions encodes the effect of sequence ordering:</p> <p>$$R(t, x, g) = g dt \wedge A_x \left[ D \log(D_E) + r \right] ,$$</p> <p>where ordering of operations (parallel transports in bundle directions) generates distinct accumulations of curvature, and thus arbitrage (<a href="/papers/1906.07164" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Farinelli et al., 2019</a>).</p> <p>Quantization of the stochastic Lagrangian system, under self-financing constraints, yields a quantum-mechanical description via the Schrödinger equation:</p> <p>$$i \partial_t \psi(q, t) = H \psi(q, t) ,$

ensuring that both stochastic variational and quantum mechanical perspectives reproduce the same SDA-sensitive dynamics. The framework precisely links the sequence (or path) in market operation to the measurable arbitrage potential via curvature and quantum expectation values. This analysis demonstrates the equivalence and consistency of geometric and quantum treatments of sequence-dependent arbitrage.

7. Implications, Applications, and Broader Connections

SDA is directly implicated in practical market microstructure phenomena—limits to arbitrage, persistent mispricings, and periodic inefficiencies. The law of one price may fail not due to external frictions but via sequence-structured information flow and trade execution. Models with partial information, asynchronous operation, or fragmentation naturally manifest SDA.

Applications include portfolio algorithms that exploit or suppress sequence-dependent volatility, high-frequency trading adaptations sensitive to discretization and transaction cost scaling, and risk management strategies distinguishing pointwise arbitrage from asymptotic or pathwise opportunities. The results bridge abstract matrix analysis, geometric invariant structure, stochastic process asymptotics, and optimal transport theory, providing a unified lens through which the complex effects of arbitrage sequencing in financial systems can be understood and strategically engaged.