ReLOBGen: Replayable Limit Order Book Message Generation

Junoh Kang; Kiseop Lee; Bohyung Han

cs.LGSep 26, 2026

ReLOBGen: Replayable Limit Order Book Message Generation

Authors: Junoh Kang, Kiseop Lee, Bohyung Han

Organizations: ECE & 3IPAI, Seoul National University · Department of Statistics, Purdue University · 3IPAI, Seoul National University

Abstract

We propose ReLOBGen, a method for generating limit order book (LOB) messages that are replayable by construction. Replayability is required for closed-loop market simulation, yet existing LOB message generators may produce non-replayable raw messages, i.e., messages inconsistent with the current market state. These generators therefore rely on post-hoc correction or rejection followed by resampling, which may alter the replayed message distribution or increase inference cost. ReLOBGen instead ensures replayability during generation: it selects the referenced order from the resting orders in the current LOB and then generates the remaining message fields to be consistent with that order and the market state. For realistic reference selection, ReLOBGen samples from a learned distribution over eligible resting orders, efficiently computed from cached order representations and a context-dependent query. It then enforces the consistency of the remaining fields by masking out invalid tokens. Together, these components enable efficient generation of realistic messages without post-hoc correction or resampling. In 500-message rollouts, ReLOBGen achieves 100% replayability, improves market realism, particularly for top-of-book statistics and the relative prices of LOB messages, and provides a $2.7\text{-}3.6\times$ speedup per replayed message over the LOBS5 baseline.

Figures & tables

Figure 1: Market simulation with an LOB message generator. At each step, the generator produces a raw message $\widehat{M}_{t}$ . When $\widehat{M}_{t}$ is not replayable, existing approaches correct it when possible or otherwise reject it and resample. ReLOBGen, however, generates replayable messages by construction and requires no post-processing. The LOB simulator replays the resulting message $M_{t}$ to update the market state, which conditions the next generation step.

Event type	Eligible resting-order set $\mathcal{C}_{t}$	Event-order constraints
Add	–	Bid: $X_{t}[\mathrm{price}]<\operatorname{BestAsk}$ ; Ask: $X_{t}[\mathrm{price}]>\operatorname{BestBid}$ .
Cancel	$\{o\in\mathcal{O}_{t-1}^{(\mathrm{side})}:o[\mathrm{size}]>1\}$	$X_{t}[\mathrm{price}]=R_{t}[\mathrm{price}]$ and $1\leq X_{t}[\mathrm{size}]<R_{t}[\mathrm{size}]$ .
Delete	$\{o\in\mathcal{O}_{t-1}^{(\mathrm{side})}:o[\mathrm{size}]>0\}$	$X_{t}[\mathrm{price}]=R_{t}[\mathrm{price}]$ and $X_{t}[\mathrm{size}]=R_{t}[\mathrm{size}]$ .
Execute	$\{\operatorname{FrontOfQueue}(\mathcal{O}_{t-1}^{(\mathrm{side})})\}$	$X_{t}[\mathrm{price}]=R_{t}[\mathrm{price}]$ and $1\leq X_{t}[\mathrm{size}]\leq R_{t}[\mathrm{size}]$ .

Table 1: Event-specific replayability conditions. Here,

\mathrm{side}=E_{t}[\mathrm{side}]

\mathcal{O}_{t-1}^{(\mathrm{side})}

contains the resting orders on that side, and

R_{t}

carries the referenced order’s current attributes.

\operatorname{BestAsk}

and

\operatorname{BestBid}

denote the lowest ask and highest bid prices in the current LOB state.

\operatorname{FrontOfQueue}

returns the oldest order resting at the best price on that side. Under the LOB simulator’s price–time priority, an execute event allows only this order as its reference.

Figure 2: ReLOBGen: mechanisms for replayability by construction. (a) The message representation places

R_{t}

before

X_{t}

and encodes the referenced order’s remaining size rather than its initial submitted size. (b) ReLOBGen samples

R_{t}

from a learned distribution over the eligible resting-order set

\mathcal{C}_{t}

. This distribution is efficiently computed by applying a softmax to the scaled dot products between a query derived from the generation context (the hidden state after processing

E_{t}

) and the cached order keys. (c) ReLOBGen masks invalid token values and samples from the remaining valid support during

X_{t}

generation.

Ticker	Method	Correction	Rejection	Reference	Event-order	Per replayed msg.	Per attempt
		Post-processing		Replayability violations		Runtime (ms)
GOOG	LOBS5 (ref.-last)	13.5%	25.7%	39.1%	0.4%	$602\pm 164$	$447\pm 119$
	LOBS5 (ref.-first)	11.7%	27.6%	39.5%	0.1%	$591\pm 145$	$426\pm 100$
	ReLOBGen	0.0%	0.0%	0.0%	0.0%	$\mathbf{227\pm 12}$	$\mathbf{227\pm 12}$
INTC	LOBS5 (ref.-last)	9.7%	16.3%	25.6%	0.5%	$1216\pm 726$	$1018\pm 606$
	LOBS5 (ref.-first)	11.5%	13.2%	24.5%	0.2%	$1058\pm 850$	$921\pm 744$

Table 2: Post-processing rates, replayability violations, and runtime. Runtime is measured on an NVIDIA RTX A6000 GPU and reported as mean

\pm

standard deviation. Aborted rollouts—59 and 17 for LOBS5 (ref.-last) and LOBS5 (ref.-first) on GOOG, respectively, and two for each baseline on INTC—are excluded from evaluation. Neither ReLOBGen nor Ref.-first + uniform requires any restarts on either stock. Lower is better, and bold indicates the lowest value.

			Metric group
Ticker	Method	Overall	State	Times	Volumes	Depths	Levels	Trades
GOOG	LOBS5 (ref.-last)	0.27	0.35	0.16	0.25	0.40	0.21	0.23
	Ref.-first + uniform	0.24	0.31	0.20	0.22	0.23	0.20	0.27
	ReLOBGen	0.17	0.24	0.15	0.18	0.20	0.11	0.18
INTC	LOBS5 (ref.-last)	0.22	0.14	0.14	0.20	0.32	0.17	0.27
	Ref.-first + uniform	0.30	0.10	0.35	0.27	0.36	0.34	0.32

Table 3: LOB-Bench evaluation: metric-group summary of unconditional

L_{1}

distances. Overall averages all metrics, while each metric-group column averages the metrics in its group. Lower is better, and bold indicates the lowest value.

		Conditional ( $L_{1}$ )			Market impact
Ticker	Method	ask vol. $\mid$ spread	spread $\mid$ time	spread $\mid$ volatility	$MO_{0}$	$MO_{1}$	$LO_{0}$	$LO_{1}$	$CA_{0}$	$CA_{1}$
GOOG	LOBS5 (ref.-last)	0.14	0.63	0.46	23.4	16.9	5.5	8.7	4.5	4.7
	Ref.-first + uniform	0.16	0.51	0.57	12.1	13.9	10.5	4.9	7.6	8.8
	ReLOBGen	0.18	0.37	0.42	7.2	10.4	1.7	3.1	3.3	5.3
INTC	LOBS5 (ref.-last)	0.22	0.22	0.16	9.9	5.9	2.3	3.5	0.4	11.8
	Ref.-first + uniform	0.49	0.15	0.09	7.2	5.4	0.8	2.2	1.3	11.6

Table 4: LOB-Bench evaluation: conditional

L_{1}

distances and market-impact discrepancies. Market-impact discrepancies cover market orders (MO), limit orders (LO), and cancellations (CA), with subscripts 0 and 1 for events without and with an immediate mid-price change. Conditional 99% percentile bootstrap confidence intervals have half-widths below 0.004. Lower is better, and bold indicates the lowest value.

Figure 3: LOB-Bench evaluation: market-impact response curves and error accumulation on GOOG. (a) Responses after market orders without (

MO_{0}

) and with (

MO_{1}

) an immediate mid-price change; shading shows 99% bootstrap confidence intervals. (b)

L_{1}

distance over successive 100-message intervals.

Ticker	Token order	Overall	Type	Side	Price	Size	Time (s)	Time (ns)	Price	Size	$\Delta t$ (s)	$\Delta t$ (ns)
			$E_{t}$		$R_{t}$				$X_{t}$
GOOG	Ref.-last	19.01	0.76	0.51	0.00	0.02	0.29	1.33	1.83	1.29	0.00	12.96
GOOG	Ref.-first	18.94	0.75	0.51	0.89	0.55	0.30	1.29	0.96	0.74	0.00	12.97
INTC	Ref.-last	18.50	0.71	0.51	0.00	0.06	0.38	1.65	0.97	1.95	0.01	12.27
INTC	Ref.-first	18.51	0.72	0.50	0.46	0.82	0.37	1.66	0.51	1.20	0.01	12.28

Table 5: Teacher-forced NLLs under reference-last and reference-first token orders. For fields represented by multiple tokens, NLLs are summed across tokens. Shaded columns highlight the price and size fields of the reference and event orders. Lower is better.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

MBO action pattern	LOBSTER output	Side source
Add ( A )	Type 1: Add	side of A
Cancel ( C )	Type 2/3: Cancel/Delete	side of C
Replace ( C $\rightarrow$ A )	Type 2/3 $\rightarrow$ Type 1	side of each C / A
Visible execution ( T $\rightarrow$ F $\rightarrow$ C )	Type 4: Execute	side of F
Cross trade ( F $\rightarrow$ C )	Type 6: Cross trade	side of C
Reset ( R )	No LOBSTER message	—

Appendix

Table 6: Conversion of Databento MBO action patterns to LOBSTER-compatible messages. Reset clears the order state once at session start without emitting a message. Sole trades (single- T groups with side=N ) may correspond to Type 5 but are omitted because their side is unknown.

MBO field	LOBSTER-compatible field	Description
ts_event	Time	Exchange-recorded event timestamp
action	Type	Order-book event category
order_id	Order ID	Individual order identifier
side	Direction	Buy or sell side
price	Price	Order or execution price
size	Size	Order, cancellation, or execution quantity

Appendix

Table 7: Field correspondence between Databento MBO and LOBSTER-compatible messages.

Field	# Tokens	Normalization and encoding
Event type	1	No normalization
Event side	1	Ask: 0 / Bid: 1
Reference-order price	2	$(\operatorname{sign}(\Delta p),\|\Delta p\|)$
Reference-order size	1	$\min(\text{remaining size},9999)$
Reference-order time (s)	2	Seconds component
Reference-order time (ns)	3	Nanoseconds component

Appendix

Table 8: Normalization and encoding in ReLOBGen. Prices are encoded as tick offsets from the tick-aligned mid-price, clipped to

[-999,999]

\Delta p=\operatorname{clip}((p_{\mathrm{order}}-p_{\mathrm{mid}})/\tau,-999,999)

. Here,

p_{\mathrm{mid}}=\tau\lfloor(p_{\mathrm{ask},1}+p_{\mathrm{bid},1})/(2\tau)\rfloor

p_{\mathrm{ask},1}

and

p_{\mathrm{bid},1}

denote the best ask and bid prices, respectively, and

\tau=100

. Each time component is encoded in base 1000, with one token per three-digit group.

Training stage	Dataset	Budget	Batch size	Optimizer	Peak LR	GPU hours
Base model	GOOG	100M	512	Adam	$5\times 10^{-4}$	8 L40S $\times$ 54 h
Base model	INTC	50M	512	Adam	$5\times 10^{-4}$	8 L40S $\times$ 27 h
Reference selection	GOOG	200M	256	AdamW	$1\times 10^{-4}$	1 A6000 $\times$ 33 h
Reference selection	INTC	100M	256	AdamW	$1\times 10^{-4}$	1 A6000 $\times$ 25 h

Appendix

Table 9: Training configuration. In both training stages, learning rates use linear warmup over the first 10% of training steps, followed by cosine decay to 10% of their peak values.

Model	# Params	Details
$\pi_{\theta}$	35.78M	• S5 backbone: $d_{\mathrm{model}}=512$ ; SSM state size 512. • 2 S5 message-encoder layers; 1 pre- and 1 post-book-encoder layer. • 12 S5 layers for message–book fusion.
$f_{\mathrm{query}}$	0.13M	• Layer-normalized 512-d state concatenated with a GELU-projected mid-price displacement ( $1\rightarrow 512$ ). • $1024\rightarrow 128$ projection followed by layer normalization.
$f_{\mathrm{order}}$	2.16M	• Eight 512-d token embeddings flattened into a $4096\rightarrow 512\rightarrow 128$ MLP, with GELU after the first layer and output layer normalization.

Appendix

Table 10: Model architecture.

\pi_{\theta}

is the message generation model;

f_{\mathrm{query}}

and

f_{\mathrm{order}}

are the query and order projection heads, respectively.

Ticker	Event	Method	Attempts	Correction	Rejection	Reference	Event-order
				Post-processing		Replayability violations
GOOG	Add	LOBS5 (ref.-last)	255,720	0.0%	0.2%	0.0%	0.2%
		LOBS5 (ref.-first)	259,199	0.0%	<0.1%	0.0%	<0.1%
		ReLOBGen	243,723	0.0%	0.0%	0.0%	0.0%
	Cancel	LOBS5 (ref.-last)	4,692	9.9%	67.6%	76.1%	5.1%
		LOBS5 (ref.-first)	2,781	9.4%	67.7%	76.7%	1.8%

Appendix

Table 11: Event-wise post-processing rates and replayability violations. Positive rates below

0.1\%

are reported as <0.1%. Lower is better, and bold indicates the lowest value.

Figure 4: Effect of resting-order initialization on replayability. We report rejection and correction rates for LOBS5 (ref.-last) on GOOG under INIT-order and exact resting-order initialization. Rates are computed over generation attempts grouped into 20-message bins by accepted-message position.

			Metric group
Ticker	Method	Overall	State	Times	Volumes	Depths	Levels	Trades
GOOG	LOBS5 (ref.-last)	0.43	0.59	0.31	0.34	0.69	0.37	0.35
	Ref.-first + uniform	0.33	0.39	0.33	0.23	0.28	0.44	0.32
	ReLOBGen	0.22	0.33	0.20	0.19	0.19	0.14	0.28
INTC	LOBS5 (ref.-last)	0.30	0.34	0.26	0.14	0.40	0.34	0.29
	Ref.-first + uniform	0.46	0.23	0.64	0.28	0.51	0.67	0.42

Appendix

Table 12: LOB-Bench evaluation: metric-group summary of unconditional Wasserstein distances. Overall averages all metrics, while each metric-group column averages the metrics in its group. Lower is better, and bold indicates the lowest value.

Figure 5: LOB-Bench evaluation: unconditional

L_{1}

distances. Log time-to-cancel* measures time-to-cancel for all orders canceled during the rollout using exact resting-order information, including cancellations of orders present in the initial LOB that were excluded from the original LOB-Bench implementation. Error bars indicate 99% percentile bootstrap confidence intervals. Lower is better.

Figure 6: LOB-Bench evaluation: unconditional Wasserstein distances. Log time-to-cancel* measures time-to-cancel for all orders canceled during the rollout using exact resting-order information, including cancellations of orders present in the initial LOB that were excluded from the original LOB-Bench implementation. Error bars indicate 99% percentile bootstrap confidence intervals. Lower is better.

		Conditional ( $W_{1}$ )
Ticker	Method	ask vol. $\mid$ spread	spread $\mid$ time	spread $\mid$ volatility
GOOG	LOBS5 (ref.-last)	0.12	1.17	0.91
	Ref.-first + uniform	0.12	0.78	1.03
	ReLOBGen	0.17	0.60	0.83
INTC	LOBS5 (ref.-last)	0.24	0.57	0.50
	Ref.-first + uniform	0.67	0.36	0.23

Appendix

Table 13: LOB-Bench evaluation: conditional Wasserstein distances. Conditional 99% percentile bootstrap confidence intervals have half-widths below 0.012. Lower is better, and bold indicates the lowest value.

Panel of Figure 7 — Figure 7: LOB-Bench evaluation: market-impact response curves. Responses are plotted against event lag for market orders (MO), limit orders (LO), and cancellations (CA), with subscripts 0 and 1 for events without and with an immediate mid-price change. Shaded regions indicate 99% bootstrap confidence intervals.

Panel of Figure 8 — Figure 8: LOB-Bench evaluation: error accumulation in $L_{1}$ distances. Log time-to-cancel* measures time-to-cancel for all orders canceled during the rollout using exact resting-order information, including cancellations of orders present in the initial LOB that were excluded from the original LOB-Bench implementation. Lower is better.

Panel of Figure 9 — Figure 9: LOB-Bench evaluation: error accumulation in Wasserstein distances. Log time-to-cancel* measures time-to-cancel for all orders canceled during the rollout using exact resting-order information, including cancellations of orders present in the initial LOB that were excluded from the original LOB-Bench implementation. Lower is better.

Ticker	Method	Add	Cancel	Delete	Execute	TV
GOOG	Actual	50.2%	0.2%	46.6%	3.0%	–
	LOBS5 (ref.-last)	51.0%	0.3%	36.9%	11.8%	9.6 pp
	Ref.-first + uniform	45.2%	0.2%	38.4%	16.2%	13.2 pp
	ReLOBGen	48.7%	0.2%	41.9%	9.2%	6.2 pp
INTC	Actual	49.7%	0.2%	43.7%	6.4%	–
	LOBS5 (ref.-last)	52.5%	0.1%	43.4%	4.0%	2.8 pp

Appendix

Table 14: Marginal event-type frequencies in actual and generated messages. TV denotes the total variation distance from the actual distribution, reported in percentage points. Lower TV is better, and bold indicates the lowest value.

Figure 10: Mean resting-order coverage by history length on GOOG and INTC. Coverage is evaluated during regular trading hours on the test split.

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