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

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V AppendixProof of Proposition 1. Use the law of recursive projections to expand the first term in thefirst-order condition (5),E [ v ∣ ∣ s i ,X + u ] = E [ v ∣ ∣ X + u]+ E[v − E [ v ∣ ∣ X + u] ∣ ∣ ∣∣ s i − E [ s i∣ ∣ X + u] ][= p + E v − E [ v ∣ ] ∣ X + u ∣∣ s i − E [ e ∣ ] ] X + u ,where the last equality reflects that E [ v ∣ ∣(X + u) ] = p. By construction, the vector of net orderflows X + u is orthogonal to the vector of forecast errors s i − E [ s i∣ ∣ X + u], so thatE[E [ v ∣ ∣X + u ] ∣ ∣ ∣∣ s i − E [ s i∣ ∣X + u ]] = 0.Hence, we can remove the conditioning on X+u from the first-order condition (5), which reduces to[ (p [ ∣0 = E + E v∣s i − E [ ∣s i∣ ]]] )∣ X + u − p − qi ∣∣∣ ′λ ′ x i s i= E[E [ v ∣ ∣s i − E [ s i∣ ∣X + u ]] ∣ ] ∣∣∣− q i′ λ ′ x i s i .Solving for x i yieldsx i = (λ ′ ) −1 (q i′ ) −1 E [ E [ v ∣ ∣ s i − E [ s i∣ ∣ X + u]]].Proof of Lemma 1. Substitute for γ i and Γ into the first-order condition (5). Recognizing thatwe do not need to condition on net order flow, u drops out, and the first-order condition becomes[()∣N∑∑N ∣∣∣∣0 = E v − λΓ b k s k − q i′ λ ′ b i s i + γ i b k s k s].ik=1k=1The goal is to remove q i = (I − ∑ k≠i Bk ) −1 from the first-order condition. To do this, we useΓ = I + ∑ iγ i= I + ∑ iB i (I − ∑ kB k ) −1= (I − ∑ kB k + ∑ kB k )(I − ∑ kB k ) −1= I(I − ∑ kB k ) −1 (15)29
andI + γ i = I + B i (I − ∑ kB k ) −1= (I − ∑ kB k + B i )(I − ∑ kB k ) −1= (I − ∑ k≠iB k )(I − ∑ kB k ) −1= (q i ) −1 Γ. (16)Hence, if we multiply the first-order condition by I + γ i′ = Γ ′ (q i′ ) −1 , the q i terms cancel.Proof of Proposition 2. We first rewrite the speculator’s objective, exploiting the fact that hisexpected profit is a scalar, so it is equal to its trace. Commuting under the trace yields[ {(max E tr V − λΓb i ,γ iN∑b k A kk=1− λΓ) ′ (( N)∑b i A i + γ i b k A kk=1γ i )(eu) }](e ′ u ′ ) .Taking expectations, we restate the objective as⎧⎛⎪⎨max tr ⎜⎝ V ′ − ∑ ⎞( (Nk=1 Ak′ b k′ Γ ′ λ ′N)⎟∑⎠ b i A i + γ i b k A kb i ,γ i ⎪ ⎩ Γ ′ λ ′k=1) ⎛ ⎞⎫γ i ⎜ Σ e 0⎪⎬⎟ ⎝ ⎠0 Σ u⎪⎭ . (17)First-order conditions for b i .{ ⎛ ⎜ V ′ − ∑ Nk=10 = trAk′ b k′ Γ ′ λ ′⎝−Γ ′ λ ′⎞⎛ ⎞⎟⎠ (A i + γ i A i ⎜0) ⎝ Σ e 0 ⎟⎠0 Σ u⎛ ⎞⎛ ⎞⎜− ⎝ Ai′ Γ ′ λ ′⎟∑N⎠(b i A i + γ i A k b k γ i ⎜) ⎝ Σ }e 0 ⎟⎠0k=1 0 Σ u{ ⎛ ⎜ V ′ − ∑ ⎞⎛ ⎞Nk=1= trAk′ b k′ Γ ′ λ ′ ( )⎟⎝ ⎠ A i +γ i A i ⎜0 Σ e −⎝ Ai′ Γ ′ λ ′⎟ ∑⎠((b N i A i +γ i A k b k) ) }Σ e γ i Σ u .−Γ ′ λ ′0k=1The off-diagonals drop out due to the trace operation, leaving{N∑∑N0 = tr (V ′ − A k′ b k′ Γ ′ λ ′ )(A i + γ i A i )Σ e − (A i′ Γ ′ λ ′ )(b i A i + γ k A k b k )Σ e}.k=1We now manipulate the first-order condition to isolate the {b k } terms on the left-hand side. Thetrace is preserved under transposition: taking the transpose of the second term and then commutingk=130
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 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 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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