Statistical Arbitrage for Multiple Co-Integrated Stocks

A. Papanicolaou; T. N. Li

REVIEW 3 major objections 3 minor 34 references

Statistical Arbitrage for Multiple Co-Integrated Stocks

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For risk-averse investors, optimal multi-stock arbitrage portfolios are bounded and stable, and backtests show they profit most in volatile markets.

desk verdict The unconstrained HJB solution is a solid, useful extension of the Avellaneda–Lee model, but the market-neutral stability proof rests on a rank condition that is algebraically impossible for any nontrivial factor count, so the constrained claims currently lack proof. read the letter →

arxiv 1908.02164 v5 pith:QAZE5RQP submitted 2019-08-06 q-fin.PM math.OCmath.PR

T. N. Li , A. Papanicolaou This is my paper · ORCID

classification q-fin.PMmath.OCmath.PR MSC 62P0591B2893E20

keywords statisticalarbitrageco-integratedstockseigenportfoliosHamilton-Jacobi-BellmanequationmatrixRiccatimarket-neutralportfolioOrnstein-Uhlenbeckprocessstochasticoptimalcontrol

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that optimal statistical arbitrage among multiple co-integrated stocks, driven by eigenportfolios as untradeable factors, is mathematically well-behaved for risk-averse investors: with risk-aversion parameter $\gamma<0$, the value functions of the stochastic control problems stay finite and optimal portfolios converge to steady states as the horizon grows. The authors solve the resulting Hamilton-Jacobi-Bellman equations through an exponential ansatz, reducing each problem to a matrix Riccati equation and two linear ordinary differential equations, and prove boundedness and stability for both an unconstrained portfolio and one forced to be market-neutral. In sliding-window backtests on S&P 500 constituents from 2000 through 2021, the strategies are reported to be sensitive to parameter estimation and to perform best when overall market volatility is high. If the stability result is right, it gives a theoretical reason why multi-asset mean-reversion strategies can grow steadily without admitting an arbitrage that would make the value function blow up.

What carries the argument

The load-bearing object is the exponential ansatz $g(t,z)=\exp(a(t)+b(t)^\top z+z^\top C(t)z)$, which converts the nonlinear HJB PDE into a system of ODEs: a matrix Riccati equation for $C(t)$, a linear ODE for $b(t)$, and a scalar ODE for $a(t)$. Stability of the whole system is governed by the Riccati equation; the proof applies Wonham's existence-and-boundedness theorem for matrix Riccati equations, whose hypotheses for $\gamma<0$ reduce to the coefficient matrix $Q$ being symmetric positive definite and $-P$ being symmetric positive (semi)definite. For the constrained portfolio the same structure appears with $\Sigma_1^{-1}$ replaced by $\Sigma_1^{-1}-\Sigma_c$, and the observability half of Wonham's conditions is what forces the rank condition on $\beta$ and $\delta$. The optimal portfolio is then the myopic term plus a hedging term proportional to $\Sigma_1^{-1}\Sigma_2(2C(t)z+b(t))$.

What would settle it

Estimate $\beta$ and $\delta$ on any sliding-window training set of S&P 500 returns and check whether $\beta^\top\beta$ has full rank and whether $\delta-\beta(\beta^\top\beta)^{-1}\beta^\top\delta$ has rank $d$; if either fails, the constrained portfolio's boundedness theorem does not apply, and the Riccati ODE can be solved numerically to see whether $C(t)$ remains bounded or blows up.

Watch

Extended reading notes

Core claim

The paper's central claim is that for $\gamma<0$, the HJB equations for both the unconstrained and the market-neutral constrained portfolio have global, bounded solutions, which the authors interpret as absence of arbitrage and stable long-term growth. Proposition 2.3 uses Wonham's theorem on matrix Riccati equations to establish that $C(t)$ is unique, negative semi-definite, and bounded on $(-\infty,T]$, with a unique limit $\bar C$; Remark 2.3 states that this stability rules out finite-time singularities that would correspond to a 'Nirvana' arbitrage. For the constrained case, Proposition 2.8 proves the same boundedness provided the rank conditions in (2.42) hold. The paper further shows that the long-term certainty-equivalent growth rate is a constant, and that the market-neutral constraint $\pi^\top\beta=0$ makes the portfolio adapted to the spread processes and therefore immunised against factor moves. In backtests, optimal portfolios are reported to be more profitable and more volatile than myopic ones, while constrained portfolios reduce risk relative to unconstrained ones.

Load-bearing premise

The stability claim for the market-neutral portfolio rests on the rank condition in (2.42), which the paper assumes but never verifies on data; if real S&P 500 estimates violate it, the proof of boundedness and stable growth for the constrained strategy no longer holds.

Editorial extensions

If this is right

If the stability theorem is correct, neither the unconstrained nor the market-neutral model can produce a finite-time blow-up in the value function for $\gamma<0$, so the 'Nirvana' arbitrage is excluded within the model.
The steady-state limits $\bar C$ and $\bar b$ give explicit long-run optimal portfolios that can be computed by solving an algebraic Riccati equation, which is what the backtests use.
The market-neutral constraint lowers volatility, maximum drawdown, and profit relative to the unconstrained strategy, but still outperforms the S&P 500 ETF in high-volatility subperiods.
Backtest results vary substantially with training and testing window lengths, which the authors read as sensitivity to parameter estimation.
High-volatility periods, including 2000--2003, 2008, 2010--2012, and 2020, are where the strategies show their best relative performance against the ETF.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step, not taken in the paper, is to test the constrained theorem's rank condition on each sliding-window estimate; if it fails often, the boundedness guarantee for market-neutral portfolios would not apply to real data.
Because the stability proof relies mainly on $\gamma<0$ and on sign properties of diffusion matrices, the same Riccati-based argument could be carried over to other state dynamics, such as Ornstein-Uhlenbeck processes with stochastic volatility, provided the analogous coefficient matrices keep their signs.
The backtests deliberately exclude transaction costs and liquidity constraints; adding a 5--10 basis point penalty, which the paper notes is feasible, would show whether the high-volatility outperformance survives realistic frictions.
The paper's sensitivity finding implies that choosing window lengths and factor counts is itself a risk factor; an adaptive hyperparameter rule, rather than a fixed grid, is the implicit practical recommendation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the unconstrained half is a competent extension of Avellaneda–Lee to multi-stock stochastic control, but the constrained half has a load-bearing gap. Proposition 2.8's rank condition can never hold when there is at least one factor, so the paper's proof of boundedness and stable growth for the market-neutral portfolio is unsupported. This isn't a minor technicality; it's the theorem that the constrained stability claims rest on.

What the paper does well: it takes the eigenportfolio factor model seriously, treats factors as non-tradeable, derives the HJB equations for power utility, reduces them to a matrix Riccati system, and gives a clean Wonham-based stability argument for the unconstrained case. That derivation is standard but carefully done, and the unconstrained stability result appears correct. The empirical section is also honest: it flags survivorship bias, shows sensitivity to estimation windows, and explicitly says transaction costs are not included. The backtest setup is reproducible in spirit and the high-volatility profitability claim is at least plausible.

Where it falls down: the constrained stability analysis. In Proposition 2.8 the paper requires rank(δ − β(βᵀβ)⁻¹βᵀδ) = d. Since δ is diagonal and invertible and β is d×m full rank, that matrix equals (I − P)δ where P is the orthogonal projection onto the column space of β. Its rank is d−m, not d. So for m ≥ 1 the hypothesis is empty. The paper's Remark 2.7 says a necessary condition is δ not proportional to identity, which is true but far from sufficient. As a result the observability condition needed for Wonham's theorem is never satisfied, and the claimed boundedness for the constrained Riccati equation has no proof. Proposition 2.9 has a separate gap: it asserts that sums and products of positive semi-definite matrices have positive eigenvalues, which is not generally true. Minor issues include a sign typo in equation (2.37), where the constrained myopic control should read µ − δz, and backtests that lack transaction costs and statistical significance tests—though the authors acknowledge the first omission.

Who should read this: anyone working on stochastic control for statistical arbitrage, especially with multiple co-integrated assets and non-tradeable factors. The unconstrained part and the empirical framework are worth engaging with. But the market-neutral part needs real rework, likely a different argument than the rank condition or a reformulated constraint.

Recommendation: send to peer review, but with the expectation of major revision. The unconstrained contribution is real; the constrained theorem must be fixed or substantially revised before the paper's central claims can be accepted.

Referee Report

3 major / 3 minor

Summary. The paper develops optimal statistical arbitrage strategies for multiple co-integrated stocks whose spreads are driven by untradeable eigenportfolio factors. The authors derive HJB equations for both an unconstrained portfolio and a market-neutral constrained portfolio, reduce them via an exponential ansatz to a matrix Riccati ODE plus two linear ODEs, and claim long-term stability and absence of arbitrage for risk aversion gamma<0. They then present sliding-window backtests on S&P 500 constituents from 2000 to 2021, reporting that the strategies are sensitive to parameter estimation and more profitable in high-volatility periods.

Significance. If the theoretical claims were valid, the paper would provide a multivariate extension of Avellaneda-Lee co-integration trading with explicit optimal controls and sufficient conditions for long-term stable growth in both unconstrained and market-neutral settings. The unconstrained HJB reduction and the use of Wonham's theorem for the unconstrained Riccati equation are standard and mostly correct, and the empirical study is transparent about data cleaning, eigenportfolio construction, survivorship adjustment, and hyperparameter sensitivity. The paper also honestly acknowledges transaction costs, liquidity, and practical implementation issues. However, the constrained market-neutral stability theorem rests on an impossible rank condition, so the paper's central theoretical contribution for the constrained portfolio is not established as written.

major comments (3)

[§2.4, Proposition 2.8 and Eq. (2.42)] The first full-rank condition in (2.42) is impossible for any m >= 1. Since beta(beta^T beta)^{-1} beta^T is the orthogonal projection onto the column space of beta, it has rank m, so I - beta(beta^T beta)^{-1} beta^T has rank d - m. Because delta is diagonal and invertible (delta_i > 0 by the OU stationarity assumption), rank((I - beta(beta^T beta)^{-1} beta^T) delta) = rank(I - beta(beta^T beta)^{-1} beta^T) = d - m. Hence rank(delta - beta(beta^T beta)^{-1} beta^T delta) = d requires m = 0, contradicting the factor model with m >= 1. Consequently Proposition 2.8 is vacuous, and the boundedness, observability, and convergence conclusions for the constrained Riccati equation are unproved for every nontrivial factor model. The necessary condition stated in Remark 2.7 is also inaccurate: non-commutation of the projection with delta does not change the rank of (I - P)delta.
[§2.4, Proposition 2.9] The conclusion that R_c(t) has all positive eigenvalues because it is 'the summations and produces of positive semi-definite matrices' is not a valid inference. Products of positive semi-definite matrices are not generally symmetric positive semi-definite, and even when each term has nonnegative eigenvalues, the eigenvalues of a sum of non-normal matrices cannot be inferred from the eigenvalues of the summands. A separate argument is needed. Since this proposition underpins the finite steady state of b(t) and hence the constrained long-term growth claim, the constrained stability analysis is not established even if the issues with Proposition 2.8 were resolved.
[§2.3, Proposition 2.4] A similar gap appears in the unconstrained analysis. The proof that R_u(t) has all positive eigenvalues relies on the assertion that the terms are sums and products of positive semi-definite matrices and on Lemma 2.1. This is not sufficient: -C(t) Q_u is a product of two positive semi-definite matrices and is not symmetric, and Lemma 2.1(a) concerns the field of values rather than eigenvalue positivity of a sum. Since Proposition 2.5 uses the finite steady state of b(t) to obtain the long-term growth rate, the unconstrained long-term stability claim also needs a corrected proof.

minor comments (3)

[§2.4, Eq. (2.37)] The myopic control for the constrained portfolio is stated as pi_m^* = (1/(1-gamma))(Sigma_1^{-1} - Sigma_c)(mu + delta z). Substituting C = b = 0 into Eq. (2.36) gives the same expression with mu - delta z, matching the unconstrained myopic control in Eq. (2.21). The plus sign appears to be a typo; please correct it and verify which version was used in the backtests.
[§2.4, Eq. (2.29)] In the wealth dynamics display, the risk-free contribution is written with an extra factor W_t after dividing by W_t; it should be r(1 - pi^T 1) dt, not r(1 - pi^T 1) W_t dt.
[§2.4, Proof of Proposition 2.8] The proof contains notational slips: after defining A_c, the text refers to Gamma(A_u^T, E_u^T) and to (A_u^T, E_u^T) instead of A_c in the controllability/observability discussion. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivations are self-contained or externally supported; the flagged rank-condition issue is a correctness problem, not circularity.

full rationale

The derivation chain is not circular. Starting from the assumed SDEs (2.1)-(2.4), the HJB equation (2.11), the power-utility ansatz (2.12), the exponential ansatz (2.16), and the resulting ODE system (2.17)-(2.19) are all derived in the paper. The unconstrained Riccati stability proof uses Proposition 2.2 (Q_u positive definite, P_u negative definite) and Wonham's theorem (1968), an external mathematical result. The constrained case similarly derives its ODEs (2.38)-(2.40) and invokes Wonham's theorem with a stated rank condition; this is a mathematical hypothesis, not a renaming or fitting of the desired boundedness conclusion. The empirical claims about parameter sensitivity and high-volatility profitability come directly from the sliding-window backtests in Section 3, not from the model assumptions. The self-citations (Yeo and Papanicolaou 2017; Lee and Papanicolaou 2016) provide empirical context and a conceptual 'Nirvana' interpretation, but the central stability derivations do not reduce to those papers. Separate from circularity, there is a serious correctness concern: the first condition in (2.42) is algebraically impossible for m>0 when delta is invertible, since (I-P)delta has rank d-m with P=beta(beta^T beta)^{-1}beta^T a rank-m projection. This would make the constrained stability theorem vacuous as stated, but that is a mathematical error rather than a circular derivation.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central theory rests on standard stochastic control assumptions plus a negative risk-aversion condition; the empirical results depend on several hand-chosen hyperparameters and ignore transaction costs. No new physical or economic entities are introduced.

free parameters (7)

risk aversion gamma = -70 (backtests), -100 (Figures 3.4 and 3.5)
Chosen by the authors; no data-driven calibration. All stability results require gamma<0.
number of factors m = 6
Selected as the largest six eigenvalues of the correlation matrix; a hyperparameter.
number of stocks d = <=15
Co-integrated stocks sorted by mean-reversion speed delta, with the top 15 chosen when available. A hyperparameter that affects portfolio composition.
training window length = 190, 200, 210, 220, 230, 240, 250 days
Sliding-window grid; results vary substantially with this choice, as the paper documents.
testing window length = 10, 12, 14, 16 days
Sliding-window grid; results vary with this choice.
interest rate r = 0.01
Assumed constant; not estimated from data.
survivorship bias parameter alpha_b = estimated via regression of principal eigenportfolio returns on SPY
Used to adjust stock returns for survivorship bias; fitted to data in Section 3.1.

assumptions (6)

domain assumption Spreads Z_i follow stationary OU processes with delta_i > 0 (equation 2.4)
The model assumes co-integration with factors produces stationary mean-reverting residuals; central to the state dynamics.
domain assumption Diffusion matrix Sigma_1 is symmetric positive definite and invertible
Required for the optimal control formulas to be well-defined (Section 2.2).
domain assumption gamma < 0 for all stability and no-arbitrage results
Propositions 2.2, 2.3, 2.4, 2.7, 2.8, and 2.9 assume a negative risk-aversion parameter; results are not established for gamma > 0.
ad hoc to paper Full-rank conditions (2.42) for constrained stability
Proposition 2.8 requires rank(delta - beta(beta^T beta)^{-1} beta^T delta) = d and rank(beta^T beta) = m; no economic justification is given, only that it implies delta not proportional to identity and d >= m.
domain assumption Market neutrality is equivalent to pi^T beta = 0
The paper defines market-neutral portfolios as those with zero factor loading; this is a modeling choice in Section 2.4.
ad hoc to paper No transaction costs, leverage limits, or short-sale constraints in backtests
The empirical experiments ignore costs and liquidity; the authors acknowledge this in the final paragraph of Section 3.3.

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Cite this review

Pith. "Pith review of Statistical Arbitrage for Multiple Co-Integrated Stocks." pith.science (2026). https://pith.science/paper/QAZE5RQP

@misc{pith2026190802164,
  author       = {Pith},
  title        = {Pith review of: Statistical Arbitrage for Multiple Co-Integrated Stocks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QAZE5RQP}},
  note         = {Machine review of arXiv:1908.02164}
}

read the original abstract

In this article, we analyse optimal statistical arbitrage strategies from stochastic control and optimisation problems for multiple co-integrated stocks with eigenportfolios being factors. Optimal portfolio weights are found by solving a Hamilton-Jacobi-Bellman (HJB) partial differential equation, which we solve for both an unconstrained portfolio and a portfolio constrained to be market neutral. Our analyses demonstrate sufficient conditions on the model parameters to ensure long-term stability of the HJB solutions and stable growth rates for the optimal portfolios. To gauge how these optimal portfolios behave in practice, we perform backtests on historical stock prices of the S&P 500 constituents from year 2000 through year 2021. These backtests suggest three key conclusions: that the proposed co-integrated model with eigenportfolios being factors can generate a large number of co-integrated stocks over a long time horizon, that the optimal portfolios are sensitive to parameter estimation, and that the statistical arbitrage strategies are more profitable in periods when overall market volatilities are high.

Figures

Figures reproduced from arXiv: 1908.02164 by the authors.

**Figure 3.1.** Survivorship bias of the principal eigenportfo [PITH_FULL_IMAGE:figures/full_fig_p019_3_1.png] view at source ↗

**Figure 3.2.** Co-integrated processes Zt of the in-sample training period 2000-01-03 to 2021-04-15. Their mean-reversion speeds δ are the fifteen largest among all the OU processes Zt whose augmented Dickey-Fuller tests reject the unit-root hypothesis with p-value ≤ 0.01. 3.3 Portfolio Performance The final step before evaluating the performances of optimal portfolios is to solve the system of ordinary differential equations for … view at source ↗

**Figure 3.3.** Sliding window in-sample training and out-of-s [PITH_FULL_IMAGE:figures/full_fig_p022_3_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 3.4.** Figure 3.4: Sliding window out-of-sample testing for uncon [PITH_FULL_IMAGE:figures/full_fig_p023_3_4.png]

**Figure 3.5.** Figure 3.5: Sliding window out-of-sample testing for const [PITH_FULL_IMAGE:figures/full_fig_p024_3_5.png]

**Figure 3.6.** Figure 3.6: Optimal wealth trajectories for γ = −70 of the unconstrained portfolio. The vertical axis is in logarithmic scale with base 10. Interest rate is r = 1%. Factor number is 6. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_3_6.png]

**Figure 3.7.** Figure 3.7: Optimal wealth trajectories for γ = −70 of the constrained portfolio to be market neutral. The vertical axis is in logarithmic scale with base 10. Interest rate is r = 1%. Factor number is 6. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_3_7.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Outperformance and tracking: Dynamic asset allocation for active and passive portfolio management

Ali Al-Aradi and Sebastian Jaimungal. Outperformance and tracking: Dynamic asset allocation for active and passive portfolio management. Applied Mathematical Finance, 25 0 (3): 0 268--294, 2018. ISSN 1350-486X

work page 2018

[2] [2]

Stochastic modeling and methods for portfolio management in cointegrated markets

Bahman Angoshtari. Stochastic modeling and methods for portfolio management in cointegrated markets. PhD thesis, University of Oxford, 2014

work page 2014

[3] [3]

Arnold and Alan J

William F. Arnold and Alan J. Laub. A generalized eigenvalue approach for solving riccati equations. Proceedings of the IEEE, 72 0 (12): 0 1746--1754, 1984. ISSN 0018-9219

work page 1984

[4] [4]

Statistical arbitrage in the US equities market

Marco Avellaneda and Jeong-Hyun Lee. Statistical arbitrage in the US equities market. Quantitative Finance, 10 0 (7): 0 761--782, August 2010. ISSN 1469-7688

work page 2010

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