Cross-Asset Speculation in Stock Markets∗ - Econometrics at Illinois ...

We have⎛ ⎞V Σ e A ′ b ′ ⎜= 2⎝ ⎟⎠ b i = 2(I + p)b i , V Σ e V = 2(I + p) and bAΣ e A ′ b ′ = 2b i (I + p)b i′ .θ 1Exploiting the symmetric-countersymmetric structure, we commute and write (29) as(V Σ e V ′ )(bAΣ e A ′ b ′ )Ψ −1 , (30)where V Σ e V ′ is the value covariance matrix. Now substitutebAΣ e A ′ b ′ = Ψ − Σ u = (f(C) − I)Σ u and Ψ = f(C)Σ u (31)from the proof of Proposition 4 into (30) to obtain(V Σ e V ′ )(f(C) − I)Σ u (f(C)Σ u ) −1 = (V Σ e V ′ )(f(C) − I)f(C) −1 . (32)To calculate the correlation, we divide by the square root of the product of the diagonal terms: inthis symmetric environment, to prove that the correlation in price changes exceeds that in values,we must prove that the ratio of an off-diagonal element to a diagonal element exceeds θ. To dothis, we use the structure of the solution f(C) for Ψ −1 Σ u = Γ from equation (27). Writing (27) asC(I − Γ) = Γ 2 , (33)we complete the square,and take the square root to solve forΓ 2 + CΓ + 1 4 C2 = C + 1 4 C2Γ = − 1 2 C + C1/2 (I + 1 4 C)1/2 . (34)We next useand substitute into (32) to obtainC(I − Γ) = Γ 2 ⇒ −(Γ − I) = C −1 Γ 2(V Σ e V ′ )(f(C) − I)f(C) −1 = 2(I + p)(Γ − I)Γ −1 = −2(I + p)C −1 Γ= 2(I + p)C −1 (− 1 2 C + C1/2 (I + 1 4 C)1/2 )= (I + p)(−I + C −1/2 (4I + C) 1/2 ), (35)39