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

(from Proposition 1) speculator i’s vector of orders is orthogonal to the vector of net order flows.Even so, the associated first-order conditions characterizing optimization are non-linear becausethe {b k } enter {γ k }, which, in turn, multiply the {b k }. This reflects the strategic use by traders ofthe information in prices. The practical consequence is that we cannot just invert a matrix to solvefor equilibrium trading strategies. Rather, we must iterate, first solving for the best responses toconjectured trading strategies and pricing, and then iteratively substituting these best responsesin as the basis for a new round of conjectures.Stacking the first-order conditions for b ′ , we can write the recursion as:⎛⎞ ⎛⎞∑k≠1 A1 Σ e A k′ b k′ µ(µ ′ (I + γ 1 ) + (I + γ 1′ )µ) −1 A 1 Σ e V ′ (I + γ 1 )(µ ′ (I + γ 1 ) + (I + γ 1′ )µ) −1(aΣ e A ′ )b ′ = ⎜.⎟⎝⎠ + ⎜.⎟⎝⎠ .∑k≠N AN Σ e A k′ b k′ µ(µ ′ (I + γ N ) + (I + γ N ′ )µ) −1 A N Σ e V ′ (I + γ N )(µ ′ (I + γ N ) + (I + γ N ′ )µ) −1(13)The left-hand side represents the vector of best-responses by speculators to conjectured strategiesof other speculators and the associated pricing. The conjectured b ′ shows up on the right-handside both directly and in the {γ k } projections. That is, a conjectured b ′ implies values of {γ k } andhence λΓ on the right-hand side; and the left-hand side is the best-response to those conjectures.We use an iterative algorithm to solve for the equilibrium values of b. We first conjecture a valueof b ′ , use this value to compute {γ k } and µ, and then substitute these values into the right-handside of (13). We then compute the best response to these conjectures. Finally, we use this vectorof best responses as the new vector of conjectured strategies, and iterate.II Theoretical Characterizations under SymmetryThe theoretical analysis is more complicated than the discussion above suggests. The iterative mappingmight not be stable: each speculator responds to an entire matrix of trading intensities, andalso to a pricing matrix. Because his response influences the pricing matrix, a speculator’s best responsemight be to overreact to the trading intensities of rivals. Iteration on this overreaction wouldvitiate the equilibrium. We now demonstrate that the iterated best responses are in fact stable.Specifically, we prove in symmetric settings that for sufficiently small correlations in asset values,this best-response mapping is a contraction on the appropriate space of matrices; and our numerical11