Cross-Asset Speculation in Stock MarketsÃ¢ÂˆÂ— - Econometrics at Illinois ...

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3, speculative trading intensities on intrinsic information are characterized byb ∼ (AΣ e A ′ ) −1/2 Σ 1/2u .In sharp contrast, with unobservable prices, Corollary 1 immediately reveals that trading strategiesare not proportional to (AΣ e A ′ ) −1/2 .BCross-asset correlationsWe now derive a result of the unraveling by market makers of the filtering by traders of order flow.Proposition 5 In the fully symmetric, two-speculator observable price environment of Proposition3, the covariance matrix of price changes, λΓΨΓ ′ λ ′ , is independent of Σ u , i.e., of the distributionalproperties of liquidity trade. Only the variance-covariance structure of the innovations to assetvalues, Σ e , affects the correlation structure of price changes.The economic force driving this result is that if the correlation structure of noise trade affected thecorrelation structure of prices, then prices would be systematically driven by processes other thanvalue—market makers would be systematically mispricing stocks—and speculators would exploitthis. Formally, using b ∼ Σ 1/2u , we have that the inner product of net order flow, Ψ = bAΣ e A ′ b ′ +Σ u ∼ Σ u ; and λ ∼ Σ −1/2uunwinds the correlation structure of liquidity trade, leaving only theimpact of the variance-covariance structure of intrinsic information. Our numerical investigationsindicate that this result extends to arbitrary numbers of speculators, and to asymmetric settings.We next prove that in fully symmetric environments, the strategic behavior by speculatorscauses price changes to be more correlated than the underlying fundamentals; and our numericalinvestigations strongly suggest that this result extends more generally to asymmetric environments.Proposition 6 In the fully symmetric, two-speculator observable price environment of Proposition3, price changes (returns) are more correlated than the underlying fundamentals. For smallcorrelations in fundamentals, θ, the price change correlation is approximately 5 3 θ.Corollary 1 revealed that in symmetric settings, unobservable prices cause speculators to tradeonly on their direct signals; whereas with observable prices, speculators weight positively their signal15
for one asset when trading the other asset. Proposition 6 just showed that observable prices causeprice changes to be more correlated than the underlying fundamentals; and we find numericallythat price changes are more correlated when prices are observable than when they are not. Thesefindings would seem to suggest that the cross-asset correlations in net order flow should be higherwhen speculators see prices than when they do not. We now show that this intuition is completelymisplaced. The variance-covariance matrix of net order flows is Γ(bAΣ e A ′ b ′ + Σ u )Γ ′ . The nextresult shows that this matrix takes a simple form.Proposition 7 The variance-covariance matrix of net order flows is Σ u Γ ′ . In the fully symmetric,two-speculator observable price environment of Proposition 3, only the variance-covariance structureof liquidity trade, and not the correlation structure of the innovations to asset values, Σ e , affectsthe correlation structure of order flows.Numerical analysis suggests that this result extends more generally to asymmetric environments.The economic force driving this result is analogous to that for the correlation structure of pricing—were net order flows to reflect the correlation structure of speculators’ private information, marketmakers would exploit this in their pricing, reducing speculator profits. The upshot of Propositions5 and 7 is that the correlation structures of order flow and prices are dual to each other in thefollowing sense: the covariance structure of prices is driven only by the covariance structure ofvalue, while the covariance structure of order flows is driven only by the covariance structure ofliquidity trade.The next proposition derives the surprising consequences of Proposition 7 for the correlationstructure of net order flows when liquidity trade is uncorrelated so that only the intrinsic correlationin fundamentals and strategic speculative behavior affect the order flow correlation.Proposition 8 Consider the fully symmetric, two-speculator environment of Proposition 3, andsuppose that liquidity trade is uncorrelated across assets and asset values are positively correlated,i.e., θ > 0. Then, with observable prices, cross-asset net order flows are negatively correlated. If,instead, prices are unobservable, cross-asset net order flows are positively correlated.With observable prices, the negative correlation in cross-asset net order flows is immediate:with uncorrelated liquidity trade, Σ u is diagonal, so the cross-asset correlation takes the sign of16
Page 1 and 2: Cross-Asset Speculation in Stock Ma
Page 3: ship model of Kyle (1985) and Admat
Page 6 and 7: We next set out the model and analy
Page 8 and 9: of net order flows. Specifically, w
Page 10 and 11: and defining γ i ≡ B i (I −
Page 12 and 13: (from Proposition 1) speculator i
Page 14 and 15: Unobservable prices. This theoretic
Page 18 and 19: the off-diagonal term in Γ = Γ
Page 20 and 21: Substituting for λ 11 − c = λ 1
Page 22 and 23: isolates how the division of privat
Page 24 and 25: mation is highly correlated. Profit
Page 26 and 27: The increased competition between s
Page 28 and 29: of the less volatile asset’s valu
Page 30 and 31: V AppendixProof of Proposition 1. U
Page 32 and 33: under the trace yields{0 = tr= tr
Page 34 and 35: where once more we commute under th
Page 36 and 37: We now establish the contraction pr
Page 38 and 39: We substitute (24) for bAΣ e A ′
Page 40 and 41: We have⎛ ⎞V Σ e A ′ b ′
Page 42 and 43: ⎛( ) ⎛ ⎞⎛⎞⎞⎜ ⎜Σ e
Page 44 and 45: Figure 1: Expected profit for asymm
Page 46 and 47: Figure 3: Trading intensities for a
Page 48 and 49: Figure 5: µ coefficients for asymm
Page 50 and 51: Figure 7: Trading intensity for asy
Page 52: VI ReferencesAdmati, Anat, 1985, A

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