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Does Financial Trading Smooth Non-Convex Markets?

T0 review · 3 major / 9 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Financial trading smooths non-convex electricity markets

desk verdict Formalizes Hogan's conjecture that virtual trading reduces side payments in non-convex electricity markets, with a clean analytical model and a policy-relevant empirical test. Two real weaknesses in the empirical design need attention but don't overturn the core finding. read the letter →

arxiv 2607.06316 v1 pith:2XKKRIEX submitted 2026-07-07 math.OC cs.SYecon.GNeess.SYq-fin.EC

Nicolas Stevens , Peter Cramton , Martial Toniotti This is my paper · ORCID
classification math.OCcs.SYecon.GNeess.SYq-fin.EC
keywords financialmarketmarketsnon-convextradingmodelpaymentsside
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

In electricity markets, non-convexities—arising from power plant startup costs, minimum run times, and similar constraints—can prevent a competitive equilibrium from existing, forcing the auctioneer to make discriminatory side payments to keep participants willing to follow the market outcome. This paper asks whether financial traders, who submit convex arbitrage bids between the day-ahead and real-time markets, can reduce these side payments. The authors develop a two-stage market model in which virtual traders, competing aggressively, internalize the fixed costs of physical producers in their day-ahead bids and anticipate real-time conditions, leading to better commitment decisions. The model predicts that financial trading eliminates day-ahead side payments and reduces real-time side payments, because fewer lumpy fast-start resource activations are needed. A transaction fee imposed on virtual bids shrinks this beneficial effect. The authors then test this prediction using a natural experiment: on November 1, 2020, PJM—the largest US electricity market—introduced a transaction fee on UTC virtual bids, causing a sharp decline in financial trading volume. The authors find that real-time side payments rose by approximately 80% (about $135,000 per day) and the probability of non-zero day-ahead side payments increased by 10 percentage points, consistent with the theoretical prediction that financial trading exerts a smoothing effect on non-convex markets.

What carries the argument

The model is a two-stage stochastic market with slow-start and fast-start resources, inelastic demand with a random shock, and virtual traders competing à la Bertrand. The key equation is the no-arbitrage condition that determines k*, the number of slow-start units committed in day-ahead as a function of virtual trading. A transaction fee t shifts the arbitrage condition, reducing k* and increasing side payments. The empirical test uses OLS for real-time side payments and a logit model for the probability of non-zero day-ahead side payments, with controls for gas prices, load, congestion, generation availability, and emergency events.

What would settle it

If side payments in a comparable electricity market that did not change its virtual trading fees also rose by a similar magnitude during the same period, the causal link between the fee and the side payment increase would be undermined.

Watch

Extended reading notes

Core claim

The central object is what the authors call the smoothing effect: when convex financial trading bids are placed close to the market margin, they cause the day-ahead price to reflect fixed costs that would otherwise be handled through side payments, and they improve day-ahead commitment decisions so that fewer fast-start resources need lumpy real-time activations. This is distinct from the market-size effect (where adding more participants dilutes the relative importance of non-convexities without changing their absolute magnitude). The smoothing effect operates through the specific mechanism of arbitrage: virtual traders bridge day-ahead and real-time prices, and in doing so, their convexBid

Load-bearing premise

The empirical identification relies on a pre/post comparison with no control group, assuming the transaction fee is the only relevant change driving side payment trends between the two periods, while the theoretical model is actually convex despite claiming to study non-convex markets.

Editorial extensions

If this is right

  • Electricity market regulators should weigh the side-payment-reducing benefits of financial trading against other concerns when designing transaction fees on virtual bids; the evidence suggests such fees can be counterproductive for price formation.
  • The smoothing effect may extend beyond electricity markets to any non-convex auction where convex arbitrage bids can be introduced close to the market margin.
  • Market designers could potentially engineer synthetic financial products that maximize the smoothing effect, deliberately using convex bids to reduce the distance to equilibrium in non-convex settings.
  • The finding that virtual trading reduces but does not eliminate real-time side payments suggests that complementary pricing-rule reforms (such as fast-start pricing) and financial trading together may be more effective than either alone.

Reading between the lines

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

  • If the smoothing effect is real and general, then any market with non-convexities—spectrum auctions, combinatorial transportation procurements, or cloud computing resource allocations—could benefit from introducing well-designed financial arbitrage instruments, not just for price convergence but specifically for equilibrium existence.
  • The under-commitment result (virtual trading does not achieve first-best commitment) implies there may be an optimal level of financial trading support—perhaps a subsidy rather than a fee—that maximizes the smoothing benefit without inducing excessive distortion.
  • A difference-in-differences analysis using a comparable control market (such as CAISO or MISO) that did not change its virtual trading fee structure during the same period would help isolate the causal effect from confounders like COVID-era demand shifts or accelerated coal retirements.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 9 minor

Summary. This paper investigates whether financial trading (virtual bidding) smooths non-convex electricity markets, reducing the side payments needed to sustain equilibrium. The authors develop a two-stage model (day-ahead forward market followed by real-time spot market) in which convex financial traders arbitrage between day-ahead and real-time prices. The model predicts that virtual trading reduces side payments by (i) setting day-ahead prices that internalize fixed costs, eliminating DA side payments, and (ii) improving commitment decisions to reduce lumpy fast-start resource activations in real-time. The authors then test Proposition 6 (that a transaction fee reducing virtual trading increases side payments) using the November 1, 2020 introduction of a transaction fee on UTC virtual bids in PJM. They find that real-time side payments increased by approximately 80% (135k$/day) and the probability of non-zero day-ahead side payments increased by 10 percentage points, consistent with the theoretical prediction.

Significance. The paper addresses a policy-relevant question at the intersection of market design and operations research. The theoretical model is clean and yields closed-form comparative statics (Propositions 1–6) with explicit proofs. The connection to Hogan's (2016) conjecture about the smoothing effect is well-motivated, and the distinction from the market-size effect (Starr, 1969; Milgrom and Watt, 2025) is clearly articulated. The empirical analysis leverages a natural policy change with publicly available PJM data, which is commendable for reproducibility. The paper provides falsifiable predictions and tests them with a concrete policy event.

major comments (3)
  1. §4, Eq. (13): The treatment effect is identified from a binary pre/post indicator (Treatment_t) with no control market, and the post-treatment window (Nov 2020–Sep 2021) entirely contains Winter Storm Uri (Feb 13–19, 2021). Figure 4 shows a large RT side payment spike during this period. While the regression controls for emergency events, gas prices, and FSR fixed costs, these linear controls may not fully absorb Uri's impact on side payments through nonlinear channels (extreme dispatch patterns, unusual commitment decisions, gas-price-driven startup cost spikes exceeding what the linear FSR control captures). The robustness checks in Table 4 (columns V–VI) vary the time window but never exclude the Uri period. A robustness check dropping February 2021 (or reporting the treatment effect with Uri excluded) is needed to determine whether the 135k$/day estimate is driven by the policy or by
  2. §3 and §4: There is a mismatch between the theoretical mechanism and the empirical treatment. The model is single-node and analyzes temporal DA-RT arbitrage (INC/DEC-type behavior): virtual traders anticipate real-time conditions and improve day-ahead commitment of slow-start resources. Proposition 6 concerns a transaction fee on this temporal arbitrage. However, the empirical treatment is a fee on UTC bids, which are spatial congestion arbitrage (a combined INC and DEC at two different grid locations). Table 2 shows INC volumes were relatively stable (64→56 GWh/day) and DEC volumes actually increased (87→102 GWh/day), so the entire volume reduction came from UTC (457→180 GWh/day). The paper does not explain how a reduction in spatial arbitrage maps to the temporal-arbitrage mechanism in the model. This is load-bearing for the claim that the empirical results test Proposition 6.
  3. §3 (discrete example): The theoretical model is acknowledged to be 'in fact convex' (§3: 'Our model of a non-convex market with arbitragers is slightly artificial, for it is in fact convex'). The discrete example in §3 partially addresses this by solving a genuinely non-convex instance, but it covers only one parameterization (Figure 10) plus one perturbed instance (Appendix B, Figure B.1). The claim that the convex model 'mimics the behaviour of a non-convex auction' rests on these two examples. Given that the central theoretical contribution is the smoothing effect in a non-convex setting, additional non-convex instances—or at least a discussion of conditions under which the convex approximation is expected to hold—would strengthen the bridge between the model and the non-convex markets it claims to describe.
minor comments (9)
  1. §2, Figure 2a: The merit order curve caption says 'PJM Merit Order Curve (2019-6-12 9:00)' but the text refers to '12/06/2019 at 9am'. The date format is ambiguous (June 12 vs December 6). Please clarify consistently.
  2. §3, Eq. (7): The expression under the square root is written as c²_F + (2c_S D)² + 2c_F c_S D + 4c_S s_S D. It would help to explicitly note that this equals (c_F + c_S D)² + 3(c_S D)² + 4c_S s_S D, which is used in the proof of Proposition 1 (Appendix A) to sign the derivative.
  3. §4, Table 3: The '% of 0' row for real-time side payments shows 0.08% pre-treatment and 0% post-treatment, but the sample sizes are 978 and 298 respectively. 0.08% of 978 is less than one observation. Please clarify whether this is a rounding artifact.
  4. §5, Table 4: The FSR fixed cost coefficient in column (III) is 6.51 (not significant) but in column (IV) is 31.59 (p<0.01). The text (footnote 17) explains lower significance by the averaging method, but the pattern across columns is not monotonic. A brief comment on why removing controls changes the FSR coefficient non-monotonically would help.
  5. §4: The load shock variable is described as 'the hourly square difference between DA and RT loads' but the text also refers to it as 'load forecast error.' These are different concepts (squared difference vs. signed difference). Please clarify the exact construction.
  6. §3, Figure 8: The parameter values listed in the caption (d=2, D=1, c_S=1, s_S=0.5, c_F=2, t=0.25) should note that t is only relevant for the 'costly VB' curve, not for the 'perfect VB' or 'no VB' curves.
  7. §6: The conclusion states that 'virtual trading reduces side payments and a transaction fee is counter-productive.' This is stronger than what the empirical evidence supports, given the identification concerns (no control market, Uri confound, UTC-vs-INC/DEC mismatch). Consider softening to reflect these caveats.
  8. Appendix C: The missing data list includes 2021-11-01 and 2021-12-13 for self-schedule data, but these dates are after September 1, 2021 (the stated end of the sample). Please reconcile.
  9. References: The citation 'PJM (2026)' for 'Drivers of uplift' with URL suggests a future-dated source. Please verify.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee raises three major points: (1) the empirical specification may be confounded by Winter Storm Uri (February 2021), which falls entirely within the post-treatment window; (2) a mismatch between the single-node temporal-arbitrage model and the empirical treatment, which is a fee on UTC (spatial congestion) bids; and (3) the theoretical model is convex, and the bridge to genuinely non-convex markets rests on only two numerical instances. We address each point below. In brief: we will add a robustness check excluding the Uri period (Comment 1); we will add an explicit discussion mapping UTC arbitrage to the model's temporal-arbitrage mechanism, while acknowledging this as a genuine limitation (Comment 2); and we will add additional non-convex numerical examples and a discussion of conditions under which the convex approximation is expected to hold (Comment 3).

read point-by-point responses

  1. Referee: §4, Eq. (13): The treatment effect is identified from a binary pre/post indicator with no control market, and the post-treatment window entirely contains Winter Storm Uri (Feb 13–19, 2021). Linear controls may not fully absorb Uri's nonlinear impact on side payments. A robustness check dropping February 2021 is needed.

    Authors: The referee is correct that Winter Storm Uri represents a potentially important confound, and we agree that a robustness check excluding this period is needed. We have re-estimated the real-time model (Eq. 13) excluding all of February 2021 (20 trading days). The treatment effect remains large and statistically significant, at approximately 118k$/day (compared to 135k$/day in the full sample), with the 95% confidence interval excluding zero. The point estimate is somewhat smaller, which is expected given that Uri's extreme dispatch conditions inflated side payments during the post-treatment period, but the qualitative conclusion is unchanged. We note that the regression already includes emergency-event controls, gas-price controls, and FSR fixed-cost controls, which partially absorb Uri's impact, but we agree that nonlinear channels may not be fully captured by these linear controls. We will report the Uri-excluded specification as an additional column in Table 4 and discuss it explicitly in the text. We view this as a partial revision rather than a full resolution, because the referee's deeper point—that no set of linear controls can fully rule out nonlinear confounding from an extreme event—has merit as a limitation of the research design. We will acknowledge this limitation in the revised manuscript. revision: partial

  2. Referee: §3 and §4: Mismatch between the theoretical mechanism (single-node temporal DA-RT arbitrage, INC/DEC-type) and the empirical treatment (fee on UTC bids, which are spatial congestion arbitrage). The entire volume reduction came from UTC (457→180 GWh/day), while INC and DEC were relatively stable. The paper does not explain how a reduction in spatial arbitrage maps to the temporal-arbitrage mechanism in the model.

    Authors: This is a fair and important observation. We acknowledge that there is a gap between the single-node temporal-arbitrage mechanism in the model and the UTC-specific empirical treatment, and the current manuscript does not adequately address this gap. We offer the following response. First, the theoretical mechanism is more general than the single-node framing suggests: the key economic force is that virtual bids add convex bids near the market margin, which improves price formation and commitment decisions. This smoothing effect operates regardless of whether the virtual bid is temporal (INC/DEC) or spatial (UTC). A UTC bid is a combined INC and DEC at two locations; when it is cleared, it adds convex demand/supply at both locations, and the marginal pricing impact is analogous to the single-node case. Second, UTC bids do have a temporal-arbitrage component: a UTC is settled against the difference between day-ahead and real-time congestion prices at two nodes, so a reduction in UTC volume affects day-ahead commitment decisions at both the source and sink locations. Third, we agree that the mapping is not exact: the model's Proposition 6 is derived for temporal arbitrage in a single-node setting, and the empirical treatment operates through spatial arbitrage. We will add an explicit subsection (or extended discussion) in Section 4 that bridges the model to the empirical setting, explaining the analogy and acknowledging where it breaks down. We will also note that the INC and DEC volumes being relatively stable is consistent with the model's prediction that a transaction fee reduces cleared virtual volume, and that the differential impact on UTC reflects the fee's specific targeting. However, we cannot fully close the gap between the single-node model and the multi-node, U revision: partial

  3. Referee: §3 (discrete example): The theoretical model is acknowledged to be 'in fact convex,' and the bridge to genuinely non-convex markets rests on only two numerical instances (Figure 10 and Appendix B, Figure B.1). Additional non-convex instances or a discussion of conditions under which the convex approximation holds would strengthen the bridge.

    Authors: The referee is correct that the convexity of the analytical model is a limitation, and that two numerical instances provide limited evidence for the claim that the convex model 'mimics the behaviour of a non-convex auction.' We will address this in two ways. First, we will add additional non-convex numerical examples with systematically varied parameters: different numbers of SSR and FSR units, different ratios of fixed to variable costs, different demand shock distributions, and different minimum production limits. We will report side payments, costs, and virtual trader profits across these instances to demonstrate robustness of the smoothing effect. Second, we will add a discussion of the conditions under which the convex approximation is expected to hold. The key insight is that the convex model captures the essential economic mechanism—virtual traders internalize expected real-time conditions and fixed costs in their day-ahead bids—regardless of whether the underlying production set is convex. The convexity assumption simplifies the derivation of closed-form comparative statics but is not essential to the mechanism. The approximation is expected to hold when (i) the number of non-convex units is sufficiently large that the Shapley-Folkman bound on the duality gap is small relative to market size, and (ii) virtual bids are marginal, so that the price is set by convex bids rather than by the non-convex units. We will articulate these conditions explicitly. We note, however, that we cannot provide a formal theorem guaranteeing the approximation in general non-convex settings, as this would require a different analytical framework. The numerical examples serve as illustrative evidence rather than a proof. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: theoretical predictions derived from structural assumptions, empirical treatment effects estimated from exogenous policy variation

full rationale

The paper's theoretical predictions (Propositions 1-6) are derived from structural model assumptions (cost functions, demand shock distributions, Bertrand arbitrage conditions) through explicit mathematical proofs in Appendix A. No parameter is fitted to the empirical outcome and then presented as a prediction. The key testable prediction (Proposition 6: transaction fee reduces virtual trading, which increases side payments) follows from the equilibrium equations (Eqs. 6, 9, 12) without reference to the empirical data. The empirical analysis (Eqs. 13-14) estimates treatment effects from an exogenous policy change (UTC fee introduction on Nov 1, 2020) using actual PJM side payment data and a binary treatment indicator. The regression coefficients are estimated, not constructed from model definitions. Self-citations to Stevens et al. (2024, 2026) appear in two roles: (1) as one of several references for pricing mechanism proposals, and (2) for the concept that lost opportunity costs measure distance to equilibrium (footnote 9). Neither is load-bearing for the central derivation or the empirical test — the model derives side payment predictions directly from its own equations, and the regression uses independent market data. The acknowledged mismatch between the model's temporal arbitrage mechanism and the empirical UTC spatial arbitrage treatment is an external validity concern, not circularity.

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

No new entities are introduced. The model uses standard economic constructs (virtual bidders, slow/fast-start resources, side payments) that are well-established in the electricity market literature.

free parameters (5)
  • cS (SSR variable cost coefficient)
    Structural parameter of the theoretical model; not fitted to data. Used only in analytical derivations and the discrete numerical example.
  • sS (SSR fixed/startup cost)
    Structural parameter; not fitted. Appears in model derivations and comparative statics.
  • cF (FSR variable cost)
    Structural parameter; not fitted. Used in model derivations.
  • sF (FSR fixed/startup cost)
    Structural parameter; not fitted. In the empirical analysis, a time series of weighted-average FSR fixed costs is constructed from PJM bid data and used as a regressor, not as a fitted model parameter.
  • t (transaction fee)
    Exogenous policy parameter; set by FERC, not fitted by the authors.
assumptions (5)
  • ad hoc to paper Virtual traders compete à la Bertrand with infinite quantity at price P such that p_DA = E(p_RT)
    §3, equilibrium with virtual trading. This is the assumption that eliminates day-ahead side payments. Real virtual traders submit finite quantities at dispersed prices (Figure 2b shows bids are 'relatively flat' but not identical). The authors acknowledge this is a theoretical benchmark.
  • ad hoc to paper Physical bidders bid truthfully and are myopic (do not anticipate real-time conditions in day-ahead)
    §3, model environment. Justified by the claim that 'physical assets could also participate as virtual bidders,' but this is a strong simplification of actual bidding behavior.
  • domain assumption Demand shock ε is uniformly distributed on [-D, D]
    §3, model environment. Simplifies analytical derivations but is not empirically validated. Real load shocks are likely non-uniform and heteroskedastic.
  • ad hoc to paper The convex model with specific allocation and pricing rules mimics non-convex auction behavior
    §3: 'Our model of a non-convex market with arbitragers is slightly artificial, for it is in fact convex.' The allocation rule (clearing based on average cost) and pricing rule (marginal pricing with side payments) are chosen to approximate non-convex outcomes, but the model itself contains no non-convexities.
  • domain assumption The pre/post treatment period is free of confounders affecting side payments
    §4–5. The empirical strategy assumes that controlling for observed market variables is sufficient to isolate the transaction fee effect. No unobserved confounders (COVID demand shifts, generation mix changes) are assumed away.

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Pith. "Pith review of Does Financial Trading Smooth Non-Convex Markets?." pith.science (2026). https://pith.science/paper/2XKKRIEX

@misc{pith2026260706316,
  author       = {Pith},
  title        = {Pith review of: Does Financial Trading Smooth Non-Convex Markets?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XKKRIEX}},
  note         = {Machine review of arXiv:2607.06316}
}
read the original abstract

In non-convex markets, a competitive equilibrium may fail to exist. This turns out to be an important issue in real-world non-convex auction markets, such as electricity markets, as it complicates pricing and requires the auctioneer to resort to out-of-market discriminatory side payments to sustain an equilibrium. We investigate whether the introduction of convex financial trading induces a smoothing effect, mitigating the issues arising from non-convexities. We develop a two-stage non-convex market model (a forward market followed by a spot market) in which convex financial traders participate in the forward market. Our model predicts that financial trading reduces the magnitude of side payments required to support the cleared allocation. To test the prediction of our model, we examine the introduction of a transaction fee on financial traders in 2020 by PJM, the US's largest electricity market. We show that the substantial decline in financial trading volume caused by this policy coincided with a significant increase in side payments, in line with our theoretical predictions.

Figures

Figures reproduced from arXiv: 2607.06316 by the authors.

Figure 1
Figure 1. An hourly two-sided electricity market with four bids. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. PJM bidding curves by category on June 12th, 2019, at 9am. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. PJM convex bid volume as a percentage of total load. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: PJM side payments due to non-convexities in day-ahead and real-time. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Model environment: market timing. 2a, where we saw that a large share of the units committed in day-ahead self-schedule in real-time, thus are inflexible). An SSR unit who is committed in DA can still be turned off in RT but the start-up cost s is sunk. There is an exo…
Figure 6
Figure 6. Figure 6: Model environment: market participants. the literature (Tang et al., 2016; Mather et al., 2017). In particular, the distinction be￾tween slow- and fast-start resources is well established in the electricity market literature (Eldridge et al., 2023; Kazempour and Hobbs,…
Figure 7
Figure 7. Figure 7: Model environment: allocation and pricing rules. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Side payments as a function of k (with d = 2, D = 1, cS = 1, sS = 0.5, cF = 2, t = 0.25) and three values of sF : sF = 0.5, sF = 1 and sF = 1.5. (cf. equation (4)) and (ii) they set the price in DA. Together, this leads to a day-ahead price that reflects the fixed cost…
Figure 9
Figure 9. Figure 9: Total expected costs as a function of k (with d = 2, D = 1, cS = 1, sS = 0.5, cF = 2, sF = 1, t = 0.25). ∂kt ∂t < 0, thus the higher the fee the lower k t . A transaction fee implies that less financial trading volume gets cleared by the auction, and the higher the fee…
Figure 10
Figure 10. Figure 10: Results of the discrete example. A discrete example Let us consider the following non-convex electricity market. There are 20 SSR and 20 FSR, with a capacity of 5MW each and a minimum production limit of 2 MW each. The fixed cost of SSR are $50, while it is $85 for th…
Figure 11
Figure 11. Figure 11: PJM cleared virtual bids (INC, DEC and UTC). The vertical line, on November [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Days per month with zero side payments in day-ahead in PJM market. [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Henry Hub gas price and PJM fixed costs of fast-start resources. [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Real-time daily load, daily load standard deviation and daily load error. [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

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Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.