Say, Echo, Do: Strategic Narratives and Revealed Positioning in Financial Markets

Ali Atiah Alzahrani

q-fin.TRSep 29, 2026

Say, Echo, Do: Strategic Narratives and Revealed Positioning in Financial Markets

Authors: Ali Atiah Alzahrani

Organizations: Riyadh, Saudi Arabia

Abstract

Machine-learning signals built from financial text treat what institutions say, and what the media repeat, as evidence about value. But whoever shapes a narrative may be trading against it. We study markets with three observable voices: institutional statements (Say), media repetition (Echo) and revealed positioning (Do). We ask when words should be followed and when they should be faded. In a linear-quadratic model of an informed institution that speaks and trades before a partly credulous crowd, talking an asset down while buying it is optimal exactly when $\varphi^2<2λk<\varphi$ . A distribution-free identity then shows that when the observable Say-Do covariance is negative, words carry negative predictive content and should be faded. For measurement, we derive (i) an exact factorised posterior over which articles are echoes, combining arrival times with embedding similarity; (ii) a return-aligned contrastive objective that attains its bound exactly when squared embedding distances are an increasing affine function of squared outcome distances, with the tightest loss-based certificate of which neighbour rankings survive imperfect training; and (iii) a path-signature statistic for who moved first. In a controlled market with known ground truth, echo sentiment predicts returns with a significantly negative sign in all 29 simulated markets, the rolling Say-Do correlation flags false-alarm events with an AUC of 0.90, and return-aligned embeddings organise headlines by consequence rather than topic. We also report where the tools fail.

Figures & tables

Panel of Figure 1 — Figure 1: Overview. Three observable voices are turned into measurements, and the theory says which sign each measurement should carry. The central object is the Say–Do covariance: when institutions’ words and positions move in opposite directions, their words should be faded.

Panel of Figure 2 — Figure 2: a : regime map of Theorem 3 . False alarms occupy the thin wedge $\varphi^{2}<a<\varphi$ , and virality moves a market to the right. b : bifurcation diagram of adaptive trust ( Theorem 5 ). Dots show the numerically computed long-run attractor, and dashed curves show the closed-form $2$ -cycle. Trust converges to truth only when $a>2$ ; below $a=\frac{5}{3}$ period-doubling continues, and simulations suggest chaos.

Regressor	Source	Prediction	$t$	As predicted
Echo sentiment $s^{\text{echo}}$	Corollary 2	$-$ (fade)	$-8.0$	5/5
News sentiment $s^{\text{new}}$	Corollary 2	$+$ (follow)	$+2.1$	3/5
Say, where rolling $\operatorname{corr}(\text{Say},\text{Do})<-0.2$	Theorem 4	$-$ (fade)	$-3.3$	4/5
Absorption, institution trades	Theorem 14	either	$-11.1$	–
Absorption, institution does not trade	Theorem 14	$-$	$-3.6$	4/5
Signed Lévy area	Theorem 9	$+$	$+1.0$	1/5

Table 1: Theory tests on test events of the main market (five seeds).

t

: firm-clustered, averaged over seeds; last column: seeds with

|t|>2

and the predicted sign. Controls: Say and Do for the first two rows, Do for the third.

Panel of Figure 3 — Figure 3: a : AUC for flagging false-alarm events, for each feature alone (reported as $\max(\text{AUC},1-\text{AUC})$ ) and for a detector on all five trained with true labels (mean $\pm$ s.d., five seeds; dashed line: chance). b : accuracy of the Say–Do switch in labelling words to fade. c : return-aligned training moves headlines’ ten nearest neighbours from sharing a topic to sharing a consequence; the dashed line is the analog IC of TF-IDF neighbours (three seeds).

Table 5

Variant	IC	echo $t$	news $t$	AUC	echo share
Reference	$0.190$	$-7.5$	$+3.6$	$0.941$	$0.425$ ( $0.430$ )
Fat-tailed values ( $t_{3}$ )	$0.141$	$-8.1$	$+2.4$	$0.924$	$0.423$ ( $0.428$ )
Saturating crowd	$0.101$	$-5.9$	$+3.2$	$0.942$	$0.425$ ( $0.430$ )
Power-law echo delays	$0.177$	$-7.9$	$+2.1$	$0.939$	$0.417$ ( $0.429$ )
30% of trade disclosed, noisier	$0.191$	$-7.5$	$+4.1$	$0.857$	$0.425$ ( $0.430$ )
Weak meaning ( $\varkappa=5$ )	$0.188$	$-7.2$	$+1.9$	$0.941$	$0.373$ ( $0.429$ )

Table 4: Robustness (three seeds each; “reference” is the main market on the same seeds). IC: linear model with theory features.

t

: firm-clustered. Echo share: estimated (true).

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Information structure. The echo channel is a downstream garbling of the institutional statement, while news is an independent signal of $v$ . Informed demand reacts to the price (dashed), and the price aggregates every channel.

Panel of Figure 5 — Figure 5: The Say–Do plane. Theorem 13 shows that the optimal forecast is a decreasing function of the signed distance from the line $\hat{x}=\gamma m$ . Top-left is the false-alarm zone; bottom-right is the false-euphoria zone.

Panel of Figure 6 — Figure 6: Proposition 18 with $\varphi_{0}=0.4$ . As the echo share rises, a deep market (small $a$ ) passes through a false-alarm window where the speech slope turns negative. A shallow market (large $a$ ) instead passes through an exaggeration window with $\psi>1$ . Both windows end where no interior optimum exists.

Panel of Figure 7 — Figure 7: Echo versus news. Top: each new article (blue) triggers a cascade of echoes (red). Bottom: the intensity jumps with every arrival and decays to the baseline $\mu$ . Proposition 21 assigns each article its exact posterior probability of being news.

Panel of Figure 8 — Figure 8: Semantic declustering ( Theorem 6 ), illustrative numbers. Timing alone attributes article C to the most recent story A. Adding embedding similarity attributes it to B, of which it is a near-copy. C is thus correctly classified as echo of B, and its sentiment enters $s^{\text{echo}}$ .

Panel of Figure 9 — Figure 9: Theorem 7 in pictures. Before fine-tuning, stories cluster by subject. At the optimum, distances mirror market aftermaths, so “news that got overdone” becomes a region of the map.

Panel of Figure 10 — Figure 10: Theorem 9 : the orientation of the loop in the (positioning, gloom) plane reveals who moved first, and its enclosed area ( Proposition 31 (iv)) measures how strongly.

Panel of Figure 11 — Figure 11: Theorem 33 . Detection error falls exponentially in the length of a strategic episode, at a rate set by the Mahalanobis separation $\Delta$ of the regime fingerprints.

Panel of Figure 12 — Figure 12: Theorem 35 . Expected growth as a function of the Kelly fraction for three signal-to-noise ratios; dots mark $c^{\star}=\mathrm{SNR}/(1+\mathrm{SNR})$ . With noisy edge estimates, full Kelly destroys growth. The conformal interval width of Theorem 34 supplies $s$ .

Three-voice economy ( Section 3.1 )
Result	Quantity	Theory	Check
Corollary 2	Bayes weight on the echo channel	$0$	$-0.0006$
Theorem 1	return loading on the echo channel	$-0.1840$	$-0.1843$
Theorem 13	loading on positioning, $c_{x}$	$0.4012$	$0.4004$
Theorem 14	absorption coefficient, $c_{A}$	$-0.1997$	$-0.2004$
Strategic speech ( Section 3.2 )

Appendix

Table 5: Theory versus independent checks (

2\times 10^{6}

draws, seed 0).

Panel of Figure 13 — Figure 13: Meaning makes echo detection work ( Theorem 6 ). a : accuracy of the most likely parent. b : mean absolute error of each article’s echo probability. Grey: timing only ( $\varkappa=0$ in the posterior). Red: timing and meaning.

Feature	Construction	Source
News, echo	$s^{\text{new}}=\sum_{j}\mathbb{P}(\text{news}_{j})\,s_{j}$ over articles other than the statement; $s^{\text{echo}}=\sum_{j}\mathbb{P}(\text{echo}_{j})\,s_{j}$	Propositions 24 and 6
Say–Do switch	correlation of Say and Do over the firm’s previous $20$ events; words are faded when it is below $-0.2$	Theorem 4
Absorption	price reaction minus the narrative-implied move, fitted linearly on Say, $s^{\text{new}}$ and $s^{\text{echo}}$ (pooled, or per firm; tone is their sum)	Theorem 14
Lead–lag	Lévy area of (positioning, $-$ tone) over the event window, signed by the direction of the positioning change	Theorem 9
Composite	unit weights on the standardised terms above, with the switch applied to Say	Section 4.4

Appendix

Table 6: Features built from the theory. All are computed causally.

Regressor (controls)	Prediction	Coef.	$t$	As predicted
News sentiment $s^{\text{new}}$ (Say, Do)	$+$ ( Corollary 2 )	$+0.35$	$+2.1$	3/5
Echo sentiment $s^{\text{echo}}$ (Say, Do)	$-$ ( Corollary 2 )	$-1.52$	$-8.0$	5/5
Say, fade group (Do)	$-$ ( Theorem 4 )		$-3.3$	4/5
Say, fade group (Do, news, echo)	$-$	$-0.06$	$-0.3$	1/5
Say, follow group (Do, news, echo)	either	$+0.10$	$+0.4$	–
Do (Say by group, news, echo)	$+$	$+0.14$	$+0.9$	1/5

Appendix

Table 7: Theory tests on the test half of the main market (five seeds). Coefficients are in units of fundamental value per training standard deviation of the feature.

t

: firm-clustered, averaged over seeds. The last column counts seeds with

|t|>2

and the predicted sign.

Panel of Figure 14 — Figure 14: Out-of-sample forecasts in the main simulated market. The fitted models use the training half. “Raw voices” are Say, Do, tone, article count and the price reaction; “theory” adds the features of Table 6 .

Variant	fade price	linear	+ theory	echo	news	Lévy	AUC	echo share
	IC			$t$ -statistic
Reference	$0.184$	$0.187$	$0.190$	$-7.5$	$+3.6$	$+3.6$	$0.941$	$0.425$ ( $0.430$ )
Fat tails ( $t_{3}$ values)	$0.134$	$0.137$	$0.141$	$-8.1$	$+2.4$	$+0.5$	$0.924$	$0.423$ ( $0.428$ )
Saturating crowd	$0.090$	$0.099$	$0.101$	$-5.9$	$+3.2$	$-0.0$	$0.942$	$0.425$ ( $0.430$ )
Power-law echo delays	$0.171$	$0.176$	$0.177$	$-7.9$	$+2.1$	$+1.9$	$0.939$	$0.417$ ( $0.429$ )
30% of trade disclosed	$0.184$	$0.183$	$0.191$	$-7.5$	$+4.1$	$+2.9$	$0.857$	$0.425$ ( $0.430$ )

Appendix

Table 8: Robustness: one change at a time, three seeds each. “Reference” is the main market on the same seeds. IC: out-of-sample information coefficient. Echo share: estimated (true).

Method	Week-ahead IC	Same consequence	Same topic
TF-IDF neighbours (generic)	$0.192\pm 0.003$	$0.506$	$0.846$
Random 16-d projection (generic)	$0.039\pm 0.016$	$0.327$	$0.622$
Ridge regression on words (supervised)	$-0.019\pm 0.016$	–	–
Gradient boosting on words (supervised)	$0.837\pm 0.008$	–	–
Return-aligned embedding, 16-d	$\mathbf{0.892\pm 0.009}$	$\mathbf{1.000}$	$0.344$

Appendix

Table 9: Text experiment:

1{,}000

test headlines, mean

\pm

s.d. over three seeds. Neighbour columns: share of the ten nearest training headlines with the same consequence (tone and cue) or the same topic; chance is

0.25

for both. The ceiling for the IC is

0.936

Panel of Figure 15 — Figure 15: Return-aligned training. a : the ten nearest neighbours of test headlines switch from sharing a topic to sharing a consequence, and the analog forecast improves with them (mean of three seeds). b : share of ordered outcome comparisons certified by Theorem 8 on $150$ training anchors. With noise-free outcomes and a bandwidth of $0.3$ times the median, the exact form certifies about half of them. The Pinsker form certifies none, and with realistic outcome noise neither form certifies any. No certified comparison was ever wrong.

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