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REVIEW 3 major objections 6 minor 26 references

Strategic Insider Trading Equilibrium with a Non-fiduciary Market Maker

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A non-fiduciary market maker who sets price by adding a fee proportional to cumulative order flow can earn expected profits comparable to or larger than a perfectly informed insider, while making prices more volatile than fundamentals…

desk verdict A genuine but conditional continuous-time extension of Kyle with a fee-charging market maker; the headline profit comparison is selected by the regulator's volatility cap rather than derived as an equilibrium outcome, yet the integral-equation derivation deserves refereeing. read the letter →

arxiv 1908.08777 v1 pith:CDVSQLI5 submitted 2019-08-23 q-fin.TR math.OC

Knut Aase , Bernt {O}ksendal This is my paper · ORCID
classification q-fin.TRmath.OC MSC 60G3562M2093E1094Axx
keywords insidertradingmarketmakingorder-flowfeeasymmetricinformationpricevolatilitynon-fiduciarymakercontinuous-timeauctionlinearfiltering
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 argues that the standard assumption of a fiduciary, zero-profit market maker is the wrong baseline. In a continuous-time version of the classic single-insider auction model, the market maker sets the price equal to the fair conditional expectation plus a fee proportional to cumulative order flow. With that fee schedule, the market maker's expected profit can match or exceed the profit of a perfectly informed insider, even when the fee is modest, and the resulting price is more volatile than fundamentals alone would produce. The paper derives the insider's optimal trading intensity from an integral equation and shows numerically how a regulator can cap the fee by monitoring relative price volatility.

What carries the argument

The central object is the fee schedule $k_t=(T-t)\kappa$ embedded in the market maker's price rule $p_t=E[\tilde v \mid \mathcal{F}^y_t]+k_t y_t$. Because $k_t$ is positive and linear in cumulative order flow, the price is pushed above the fair value after buying pressure and below it after selling pressure; since the order flow $y_t$ mean-reverts around zero, the market maker systematically buys low and sells high. The insider's trading intensity $\beta_t$, defined by $dx_t=(\tilde v-p_t)\beta_t\,dt$, is the other half of the mechanism: the insider's optimal $\beta_t$ solves the integral equation in Theorem 1, and it declines as $\kappa$ grows. The paper's main analytical tool is the linear-filtering representation of the conditional expectation $m_t$, which gives the variance $V(t)=E[y_t^2]$ and the mean-square error $\gamma_t(\beta)$ in closed form; these feed directly into both profit functions.

What would settle it

Let a second market maker enter and undercut the fee by any positive amount; if the only subgame-perfect outcome is $\kappa=0$, then the profit-dominance result depends entirely on the assumed absence of fee competition and would not survive entry.

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Extended reading notes

Core claim

The paper's central claim is that replacing the fiduciary price rule with a non-fiduciary one changes the economics of the single-insider auction model without destroying equilibrium. The market maker announces the price rule $p_t=E[\tilde v \mid \mathcal{F}^y_t]+k_t y_t$, with $k_t=(T-t)\kappa$; the insider still trades with intensity $\beta_t$ against his information advantage $\tilde v-p_t$. The expected profits are $J^M(\kappa,\beta)=w^M_0+\int_0^T(k_s^2 V(s)\beta_s+\kappa V(s))\,ds$ for the market maker and $J^I(\kappa,\beta)=w^I_0+\int_0^T\beta_s(\gamma_s(\beta)+k_s^2V(s))\,ds$ for the insider, so the fee adds a positive, order-flow-driven income for the market maker. The insider optimally trades more softly when $\kappa$ rises, but the market maker's fee income still dominates beyond a moderate threshold, which the paper's baseline calibration places around $\kappa=0.06$. The paper also derives the price variance $\operatorname{var}(p_t)=\operatorname{var}(m_t)+k_t^2 V(t)+2k_t\operatorname{cov}(m_t,y_t)$, which exceeds the fundamental-based variance, and constructs a regulator-observable relative-volatility measure $rv(t,\kappa)$ that ties $\kappa$ to a cap on price distortion.

Load-bearing premise

The whole result depends on the assumed price rule $p_t=E[\tilde v \mid \mathcal{F}^y_t]+(T-t)\kappa y_t$, with a single scalar $\kappa$ chosen by the market maker under a regulatory volatility cap; if market makers could undercut the fee, if the fee were nonlinear in order flow, or if the regulator's cap were tighter, the stated profit comparison and volatility result could fail.

Editorial extensions

If this is right

  • A modest order-flow fee can make the market maker's expected profit exceed that of a perfectly informed insider; in the paper's baseline calibration the crossover occurs around $\kappa=0.06$.
  • Speculative price volatility contains a fee-driven component $k_t^2 V(t)+2k_t\operatorname{cov}(m_t,y_t)$, so prices can be more volatile than fundamentals imply even without changes in dividend information.
  • The insider trades less aggressively when the fee increases, but intensity rises again near the horizon as $k_t\to 0$, so the fee mainly shifts the timing and size of informed trading.
  • A regulator can control the distortion by monitoring the relative volatility $rv(t,\kappa)$; a 15% cap in the baseline example keeps the market open for $\kappa$ up to about 0.07, and a 21% cap would allow $\kappa=0.09$.
  • Price informativeness $\iota(t,\kappa)$ falls as $\kappa$ rises, so fee-driven price distortion makes prices less informative about fundamentals at every horizon.

Reading between the lines

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

  • If the market maker's fee-setting power were subject to competition from a second market maker who could undercut $\kappa$ by a tiny amount, the profit-dominance result would likely shrink or vanish; the paper assumes a single market maker with regulatory protection, and testing that entry margin is a natural extension.
  • The linear fee $k_t y_t$ is what makes the profit formulas tractable; a nonlinear fee such as a fee on absolute order flow would change the mean-reversion argument, and the ranking of market-maker versus insider profits might not survive.
  • The model's volatility prediction is testable at the level of price impact: if the fee component is real, the coefficient of signed order flow on subsequent price changes should be positive and increasing in the market maker's market power, and the residual variance should exceed the variance implied by a fiduciary price.
  • Because the fee is paid by both insider and noise traders, the model implies that noise traders bear the cost of intermediation; in an overlapping-generations or entry version, this could generate persistent wealth transfer to intermediaries, connecting to the paper's motivation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper extends Kyle's (1985) continuous-time insider trading model by replacing the fiduciary market maker with a non-fiduciary market maker who sets prices according to pt = E[v|F_y] + (T−t)κ y_t. The insider's trading intensity β_t is characterized in Theorem 1 as a solution of an integral equation, and the paper derives closed-form expressions for the expected profits of the insider, the market maker, and the noise traders, as well as for the order-flow variance, relative volatility, and a price-informativeness measure. Numerical sections show that, as the fee parameter κ increases, the market maker's expected profit rises and eventually exceeds the insider's expected profit; the authors connect κ to a regulator-imposed cap on relative volatility rv*. The paper claims that setting a modest order-flow fee allows the market maker to earn profits of the same order as, or larger than, a perfectly informed insider, and that the model explains excess price volatility relative to fundamentals.

Significance. If the central claims were established, the paper would be a useful contribution to the literature on market power and intermediation in continuous-time insider trading: it provides explicit formulas for profits and price dynamics under a non-fiduciary price rule, and it frames a regulator's volatility cap as a constraint on the market maker's fee. The derivations in Section 3 and Appendix 1 are nontrivial, and the paper is transparent about several limitations, including the absence of an existence proof for the integral equation and the non-strategic treatment of the market maker. However, the headline profit comparison is currently presented as an unconditional result even though it is a consequence of an exogenously chosen regulatory threshold, and the technical optimality claim in Theorem 1 is not fully verified. With appropriate restatement and additional analysis, the continuous-time extension could be publishable.

major comments (3)
  1. [Section 4 and Appendix 1, Eq. (4.3)] Theorem 1 is derived as a first-order necessary condition, but the paper does not prove existence or uniqueness of a solution to the integral equation (4.3), nor does it show that the resulting β is a global maximizer of the insider's problem (4.2). The iterative procedure described in Section 7.2 is numerical, and no convergence argument is supplied, so the subsequent profit comparisons cannot be read as verified equilibrium outcomes. In addition, the displayed algebra is not internally consistent: the k^2 term appears with a plus sign in Eq. (10.3) and with a negative sign in Eq. (10.4), and the exponential factors in (10.4) do not obviously match the definition of V(t) used in the final expression (4.3). These points need to be fixed before the main theorem can be checked.
  2. [Appendix 2 and Section 3.3, Eq. (3.21)] Theorem 11.2 states that any Nash equilibrium has κ*=0 or κ*=K, never an interior 'modest' value. Combined with Eq. (3.21), which shows that JM is increasing in κ and grows without bound as κ→∞, an optimizing market maker would always choose the regulator's upper bound. The crossover near κ=0.06 reported in Section 7.4 is therefore not an equilibrium prediction of the model; it is an exogenous selection of the volatility cap. Table 2 confirms this cap-selection property: at rv*=1.08 the insider still earns pI=0.100 versus pM=0.060, while at rv*=1.15 the market maker earns pM=0.087 versus pI=0.055. The abstract and conclusion present the profit comparison unconditionally; this overstates what the model actually establishes.
  3. [Sections 6 and 9] The claim that the model 'indicates why speculative prices are more volatile than predicted by fundamentals' is not supported by the analysis as written. The model compares the market price p_t to the fair conditional price m_t; it does not compare prices to any fundamental process such as dividends or earnings. The numerical finding in Table 1 that var(p_t) exceeds var(m_t) is a statement about the distortion caused by the fee schedule relative to a fiduciary benchmark, not about excess volatility relative to fundamentals. The conclusion in Section 9 should be reformulated as a conditional statement about fee-induced price distortion.
minor comments (6)
  1. [Section 1] The reference to 'Campbell and Schiller (1988)' should be 'Campbell and Shiller (1988)'.
  2. [Section 3.2] The phrase 'by by the above observation' contains a duplicated 'by'.
  3. [Section 3.3, Eq. (3.23)] In the second exponential in Eq. (3.23), 'ke' appears where 'kr' is intended.
  4. [Section 7.1] The phrase 'Vi have' should be 'We have'.
  5. [Appendix 3] The reference to 'Theorem 3.2' appears to be an error; the intended reference is likely Theorem 11.2 in Appendix 2.
  6. [Section 5, Table 2] Table 2 reports profit values at t=9, while the text describes rv* as a supremum over t; the exact time convention used for the reported crossover should be stated explicitly.

Circularity Check

1 steps flagged · score 4.0 of 10

The headline profit comparison is cap-selection rather than equilibrium: JM grows in κ and Appendix 2 allows only corner κ, so the claim that the market maker can beat the insider is imposed by the regulator's 15% volatility cap, not derived as a prediction.

  1. fitted input called prediction [Abstract; Section 2 Eq. (2.2); Section 3.3 Eq. (3.21); Section 5 Table 2; Appendix 2 Theorem 11.2]
    "by setting a modest fee conditional of the order flow, the market maker is able to obtain a profit of the order of magnitude, and even better than, a perfectly informed insider. ... (2.2) p_t =E[˜v +u_t|F^y_t ] :=m_t +E[u_t|F^y_t ], ... u_t =k_ty_t for k_t≥ 0 ... We assume that k_t = (T−t)κ ... (3.21) J_M(k,β ) =w^M_0 + ∫_0^T (k_s^2V(s)β_s +κV(s)) ds. ... rv∗ 1.15 κ∗ .070 pM(9,κ∗) .087 pI(9,κ∗) .055"

    Equation (3.21) is monotonically increasing in κ, and the text states that the profit 'increases without bounds as ... κ→∞.' Hence the comparison 'market maker profit > insider profit' is not an equilibrium prediction but a choice of the free parameter κ. The paper's own Appendix 2, Theorem 11.2, says any Nash equilibrium has κ*=0 or κ*=K, never the interior 'modest' value 0.07 used for the headline. Table 2 makes the selection explicit: at rv*=1.08 the insider still wins (pI=0.100 vs pM=0.060), while at rv*=1.15 the market maker dominates (pM=0.087 vs pI=0.055). The claimed crossover near κ≈0.06 is therefore equivalent to assuming a 15% relative-volatility cap; if the regulator imposed a 10% cap the central conclusion would fail.

full rationale

The model's mathematical derivation is largely self-contained: the insider's integral equation (Theorem 1), the filtering equations, and the profit formulas are derived from stated assumptions, and the numerical examples are explicit about base-case parameters. The genuinely circular feature is in the presentation of the headline: because JM(k,β) grows without bound in κ and Appendix 2 shows equilibrium occurs only at a corner, the 'modest fee' comparison is selected by the exogenous regulator cap (rv*=1.15), not predicted by the theory. The self-citations to Aase-Gjesdal (2017/2018) and Aase et al. (2012) are used for standard filtering and order-form facts and are not load-bearing uniqueness claims. Thus the conditional contribution of the model remains valid, but the abstract's unconditional claim overstates what is established; the core profit comparison reduces by construction to the chosen cap.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The model postulates a non-fiduciary market maker with an exogenous fee rule and a regulator cap, plus standard normality assumptions. No new physical entities are introduced. The central free parameter is κ, whose level determines the headline profit comparison.

free parameters (5)
  • κ (market maker fee intensity) = 0.025, 0.035, 0.045, 0.070, 0.090 in Table 2; κ around 0.06 to 0.07 for MM dominance in Section 7.4
    The decision variable k_t=(T-t)κ controls the fee. The conclusion that MM profit exceeds the insider's is obtained for κ values above roughly 0.06; the paper does not derive κ endogenously from primitives.
  • σ (noise trader volatility) = 0.20
    Base case constant volatility chosen for all numerical illustrations; it scales profits but does not affect the qualitative comparison in the examples.
  • σ_v (prior standard deviation of asset value) = 0.30
    Base case prior standard deviation used in all figures and tables; chosen for illustration.
  • T (trading horizon) = 10
    Illustrative horizon in every figure and table; the time profiles depend on this choice.
  • rv* (regulator volatility threshold) = 1.15 (15% increase) in the main comparison; also 1.03, 1.05, 1.08, 1.21 in Table 2
    The 'modest fee' claim is tied to a 15% relative volatility increase; other thresholds are tabulated. This is an arbitrary policy parameter, not derived from economic primitives.
assumptions (6)
  • domain assumption The asset value v and the noise process B are Gaussian and independent; noise orders follow a Brownian motion with variance rate σ_t^2.
    Used throughout the paper; this gives the Kalman filter and Gaussian covariance computations. Stated in Section 2.
  • domain assumption The insider restricts to order flow of the form dx_t=(v-p_t)β_t dt with deterministic β_t.
    Equation (2.3); motivated by first-order conditions in prior work, not re-derived here. It restricts the insider's strategy space and is essential for the integral equation solution.
  • domain assumption The market maker sets the price as p_t=E[v|F_y]+k_t y_t with k_t=(T-t)κ, κ≥0.
    Equation (2.2), the paper's deviation from Kyle's model. The entire profit and volatility analysis depends on this fee schedule.
  • domain assumption The terminal condition p_t converges to v as t approaches T holds, so the cross term in the insider's profit vanishes.
    Section 2 states 'we find it natural to simply assume this'. It is load-bearing for the insider's expected profit formula JI. It was derived for the k=0 case in Aase et al. (2012a) but is asserted here for k not equal to 0.
  • domain assumption A regulator caps κ by imposing an upper bound on the relative volatility measure rv.
    Section 5. Without the cap, the market maker's profit is unbounded (Section 3.3 and Figure 2), so the paper's profit comparison requires an exogenous regulatory limit.
  • standard math Kalman-Bucy filtering equations and the Riccati equation for the conditional error variance γ_t are applicable.
    Used to derive m_t and γ_t in Sections 3 and 4; standard filtering theory.

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Cite this review

Pith. "Pith review of Strategic Insider Trading Equilibrium with a Non-fiduciary Market Maker." pith.science (2026). https://pith.science/paper/CDVSQLI5

@misc{pith2026190808777,
  author       = {Pith},
  title        = {Pith review of: Strategic Insider Trading Equilibrium with a Non-fiduciary Market Maker},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CDVSQLI5}},
  note         = {Machine review of arXiv:1908.08777}
}
read the original abstract

The continuous-time version of Kyle's (1985) model is studied, in which market makers are not fiduciaries. They have some market power which they utilize to set the price to their advantage, resulting in positive expected profits. This has several implications for the equilibrium, the most important being that by setting a modest fee conditional of the order flow, the market maker is able to obtain a profit of the order of magnitude, and even better than, a perfectly informed insider. Our model also indicates why speculative prices are more volatile than predicted by fundamentals.

Figures

Figures reproduced from arXiv: 1908.08777 by the authors.

Figure 1
Figure 1. illustrates a graph of the covariance function C(s, t; κ) := E[ysyt ] when κ = 0.045 for s, t ∈ [0, 10] [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. The profit functions of the market maker as a function of [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. The insider’s trading intensities as functions of [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The insider’s trading intensities as functions of [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]
Figure 5
Figure 5. Figure 5: The variance function V (t, κ) as a function of t. The variance of the total order flow yt is seen to increase sharply in the 28 [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]
Figure 6
Figure 6. Figure 6: The variance function V (t, κ) as a function of κ. The upper curve in [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]
Figure 7
Figure 7. Figure 7: The two net profits as functions of time. [PITH_FULL_IMAGE:figures/full_fig_p030_7.png]
Figure 8
Figure 8. Figure 8: The two net profits as functions of time. [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]
Figure 9
Figure 9. Figure 9: rv(t, κ) as a function of κ when t = 0.01, 3.6 and 9.0. The lowest curve in [PITH_FULL_IMAGE:figures/full_fig_p031_9.png]
Figure 10
Figure 10. Figure 10: rv(t, κ) as a function of t when κ = 0.07 [PITH_FULL_IMAGE:figures/full_fig_p031_10.png]
Figure 11
Figure 11. Figure 11: rv(t, κ) as a function of t when κ = 0.09 [PITH_FULL_IMAGE:figures/full_fig_p032_11.png]

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    Aase, K. K. and F. Gjesdal (2017). ”Insider trading with non-fiduciary market makers.”. (Last version 2018) Discussion Paper FOR 8, Norwe- gian School of Economics

    work page 2017

  2. [2]

    Aase, K. K. and B. Øksendal (2018). ”Strategic Insider Trading in Con- tinuous Time: A New Approach.” Discussion Paper FOR, Norwegian School of Economics. 9The result by Allinger and Mitter proved a long-standing conjecture by Kailath. 43

    work page 2018

  3. [3]

    and Øksendal, B

    Aase, K., Bjuland, T. and Øksendal, B. (2012a). ”Partially informed liquidity traders.” Mathematics and Financial Economics 6, 93-104

    work page 2012

  4. [4]

    and Øksendal, B

    Aase, K., Bjuland, T. and Øksendal, B. (2012b). ”Strategic Insider Trading Equilibrium: A Filter Theory Approach.” Afrika Matematika 23(2),145-162

    work page 2012

  5. [5]

    and Øksendal, B

    Aase, K., Bjuland, T. and Øksendal, B. (2011). ”An anticipative linear filtering equation.” Systems & Control Letters 1-4

    work page 2011

  6. [6]

    Admati, A. R. and Pfleiderer, P. (1988). ”A Theory of Intraday Patterns: Volume and Price Variability.”The Review of Financial Studies 1, 1, 3– 40

    work page 1988

  7. [7]

    Allinger, D. F. and Mitter, S. K. (1981). ”New Results on the Innova- tions Problem for Non-Linear Filtering”. Stochastics 4, 339–348

    work page 1981

  8. [8]

    Back, K. (1992). ”Insider Trading in Continuous Time”. The Review of Financial Studies Vol. 5, No 3, 387-409

    work page 1992

Show all 26 references
  1. [9]

    Biagini, F., and Øksendal, B. (2005). ”A general stochastic calculus approach to insider trading”. Appl. Math. Optim. 52, 167–181

    2005

  2. [10]

    Davis, M. H. A. (1977). Linear Estimation and Stochastic Control. Chapman and Hall

    1977

  3. [11]

    Davis, M. H. A. (1984). Lectures on Stochastic Control and Nonlinear Filtering.. Tata Institute of Fundamental Research, Bombay

    1984

  4. [12]

    Eide, I. B. (2007). ”An equilibrium model for gradually revealed asym- metric information”. Preprint, University of Oslo 6/2007

    2007

  5. [13]

    R., and Milgrom, P

    Glosten, L. R., and Milgrom, P. R. (1985). ”Bid, Ask and Transaction Prices in a Specialist Market with Hetrogeneously Informed Traders”. Journal of Financial Economics , 14, 71–100

    1985

  6. [14]

    J., and Stiglitz, J

    Grossman, S. J., and Stiglitz, J. E. (1980). ”On the Impossibility of Informationally Efficient Markets”. American Economic Review , 70 , 393–408. 44

    1980

  7. [15]

    Holden, C. W. and Subrahmanyam, A. (1992). ”Long-Lived Private In- formation and Imperfect Competition”. The Journal of Finance, XLVII, 1 247–270

    1992

  8. [16]

    Kallianpur, G. (1980). Stochastic Filtering Theory. Springer

    1980

  9. [17]

    Kalman, R. E. (1960). ”A new approach to linear filtering and prediction problems”. J. Basic Engineering D 82, 35-45

    1960

  10. [18]

    Schreve (1988)

    Karatzas I., and S. Schreve (1988). Brownian motion and stochastic calculus. New York: Springer-Verlag

    1988

  11. [19]

    Kyle, A. S. (1985). ”Continuous Auctions and Insider Trading”. Econo- metrica Vol.53, No. 6, 1315–1336

    1985

  12. [20]

    Liptser, R. S. and Shiryaev, A. N.: Statistics of Random Processes II. Springer 1978

    1978

  13. [21]

    Øksendal, B. (2013). ”Stochastic Differential Equations.” 6th Edition. Springer

    2013

  14. [22]

    Russo, F., and Vallois, P. (1993). ”Forward, backward and symmetric stochastic integration”. Probab. Theory Related Fields 97, 403–421

    1993

  15. [23]

    and Vallois, P

    Russo, F. and Vallois, P. (1995). ”The generalized covariation process and Itˆ o formula”.Stoch. Process. Appl. 59, 81–104

    1995

  16. [24]

    and Vallois, P

    Russo, F. and Vallois, P. (2000). ”Stochastic calculus with respect to continuous finite quadratic variation processes”. Stoch. Stoch. Rep. 70, 1–40

    2000

  17. [25]

    Tucker, H. G. (1967). A graduate course in probability. Academic Press, New York and London

    1967

  18. [26]

    Yong, J. and X. Y. Zhou (1999). ”Hamiltonian Systems and the HJB Equations”. Springer-Verlag, New York, Inc. 45

    1999

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