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

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isolates how the division of private information between speculators affects equilibrium outcomes.The normalization that preserves total expected speculator profits when a market maker only observesown order flow holds constant the total variance of each asset and preserves the cross-assetvalue correlation structure. Within this setting, we characterize the impact of increased competitionby varying the number of speculators who receive the same set of signals.Division of Liquidity Trade Between Assets. We impose symmetry across asset values and speculatorsignals, and vary the share of liquidity trade between assets. This isolates the effects of relativeliquidity levels between assets. With independently-distributed liquidity trade, the appropriatenormalization holds constant the sum of the standard deviations of liquidity trade across assets.Division of Uncertainty Between Assets. We impose symmetry across speculators and liquiditytrade and vary the variance share of asset values between assets. This isolates the effects of relativelevels of private information between assets on equilibrium outcomes. The normalization holdsconstant the sum of the standard deviations of asset values and preserves the correlation structure.ADivision of Information Between SpeculatorsOur numerical analysis focuses on the economic environment of Example 2, in which speculatorssee one signal for each asset. In our base two-speculator, two-asset setting, liquidity trade is independentlyand normally distributed with u i ∼ N(0,1). Using the notation of Proposition 3, thevalue of asset j is v j = e 1 j + e2 j . Speculator i sees signal si 1 = ei 1 for asset 1 and signal si 2 = ei 2 forasset 2. The signals e 1 j and e2 jthat the two speculators receive for asset j are independently andnormally distributed with mean zero. However, even though speculator 1 does not see e 2 1 , we allowcross asset signal correlations to be positive, corr(e j 1 ,ei 2 ) = θ, for i ≠ j, so that speculator 1’s signale 1 2 about asset 2 also provides him information about the innovation e2 1to asset 1 that he does not21
see. The variance-covariance matrix for asset value innovations ise 1 1e 2 1e 1 2e 2 2⎛⎜⎝e 1 1 e 2 1 e 1 2 e 2 2ν 0 0 θξ0 (1 − ν) θξ 00 θξ ν 0θξ 0 0 (1 − ν)⎞,⎟⎠where ξ = √ ν(1 − ν) is the normalization factor that preserves the correlation structure as wevary the information that speculator 1 has relative to speculator 2. We vary ν from 0.1 (so thatspeculator 1 sees 10% of the variance of each asset value) to ν = 0.9. Because of the cross-assetcorrelation structure, speculator 1’s share of the private information isvar(e 1 1 ) + var(E[e2 1 |e1 2 ])var(e 1 1 ) + var(E[e2 1 |e1 2 ]) + var(E[e1 1 |e2 2 ]) + var(e2 1 ) = ν + θ2 (1 − ν)1 + θ 2 .We focus on a high correlation of θ = 0.75, as higher correlations largely serve to magnify thequalitative impacts. 9The top line in Figure 1 represents expected total speculator profits when market makers onlysee ‘own’ net order flow and speculators do not see prices. Their constancy reflects that we havenormalized informational parameters to leave profits in this setting unaffected by the division of informationbetween speculators. The middle line plots profits in the Caballe-Krishnan environmentin which market makers see stock prices, but speculators do not. Finally, the bottom line plots profitsin the integrated market that we study, where both market makers and speculators see all prices.[Figure 1 about here]Figure 1 shows how the observability of prices amplifies competition between speculators. Theadded channel of prices through which speculators gain and release information intensifies competitionbecause each speculator internalizes the fact that as he increases an order, other speculatorssee the price impacts and reduce their orders. This “order reduction” raises the speculator’s profit.In turn, speculators raise trading intensities, competing away more of the potential profits.Figure 1 also highlights that the observability of prices amplifies the sensitivity of total expectedspeculator profits to the division of information between speculators, especially when their infor-22
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 16 and 17: 3, speculative trading intensities
Page 18 and 19: the off-diagonal term in Γ = Γ
Page 20 and 21: Substituting for λ 11 − c = λ 1
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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