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

and defining γ i ≡ B i (I − ∑ k Bk ) −1 , we write speculator i’s vector of trades asx i = b i s i + γ i( N ∑k=1)b k s k + u . (6)We next add across traders, and defining Γ ≡ I + ∑ k γk , we write the vector of net order flows asX + u =(I +N∑γ k)( ∑N )b k s k + u = Γk=1k=1( N ∑k=1)b k s k + u . (7)Lemma 1 exploits this structure to write i’s first-order conditions solely as functions of γ i and Γ.Lemma 1 The first-order condition characterizing speculator i’s orders is given by:[N∑0 = E (I + γ i′ )v − (I + γ i′ )λΓ b k s k − Γ ′ λ ′( ∑Nb i s i + γ i b k s k)∣ ∣ ∣∣∣s]. i (8)k=1k=1The crucial point to note in this first-order condition is that the “same” b i sets the expectationto zero for each s i . As a result, we can integrate out over s i : the b i that maximizes expected tradingprofits given any s i also maximizes expected profits unconditionally. The unconditional maximizationproblem is easier to solve because it exploits the fact that the same b i works for every s i .AThe Unconditional ProblemInformed speculator i’s trading intensities maximize unconditional expected trading profits, solving[( ( N)) ′ ( (max E ∑N))]∑V e − λΓ b k A k e + u b i A i e + γ i b k A k e + u ,b i ,γ ik=1k=1where we recall that v = V e and s i = A i e.Turning to the market maker’s objective, competitiondrives expected market maker profit to zero. This implies that pricing minimizes the variance ofthe forecast error between prices and values—prices equal the projection of the vector of equityvalues onto net order flows. Hence, pricing solves the following least-squares prediction problem:[( ( N)) ′ ( (∑N))]∑min E V e − λΓ b k A k e + u V e − λΓ b k A k e + u .λk=1Proposition 2 details the first-order conditions for b i ,γ i and λ. To ease presentation, we define⎛ ⎞A 1bA ≡ (b 1 ... b N )⎜ .⎟⎝ ⎠ .A Nk=19