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

Unobservable prices. This theoretical analysis also provides an easy way to derive equilibriumoutcomes if speculators cannot see prices. With unobservable prices, we obtain equilibrium tradingintensities by setting γ i = 0 for all i. In symmetric settings such as Proposition 3, this allows us toextend Caballe and Krishnan to obtain closed-form solutions: substituting γ i = 0 into (14) yieldsQ = Q(I + p) −1 p + 1 N Σ u.Re-arranging, and substituting back for Q yieldsb i (I + p)b i′ (I − (I + p) −1 p) = 1 N Σ u.Expanding the left-hand side yieldsb i (I + p)b i′ (I − (I + p) −1 p) = b i (I + p)b i′ (I + p) −1 (I + p − p)= b i (I + p)b i′ (I + p) −1 = b i b i′ .Hence, b i b i′ = 1 N Σ u. When liquidity trade is uncorrelated across assets, the following is immediate.Corollary 1 If speculators do not see prices, then in symmetric settings in which liquidity trade isuncorrelated across assets, speculators only trade directly on their signals—b is a diagonal matrixwith identical entries σu √N.Concretely, Corollary 1 says that in symmetric settings where speculators do not see prices, eachspeculator uses his Merck signal only to trade Merck independently of the correlation of the Mercksignal with the value of Glaxo. This remarkable outcome arises because speculators understandthat a market maker for Merck will use order flow for Glaxo to price Merck, estimating the valueof Merck via a projection onto the order flows for both stocks.In contrast, when speculators see prices, they can account for the market maker’s projections.The γ’s in trading strategies are (minus) the coefficients of the projections of the speculators’ directtrades onto order flows (equivalently, prices). Speculators subtract these projections, so that theirtotal net trades are effectively the market makers’ forecast errors of their direct trades on signals.Hence, the same direct trading intensities convey less information to market makers, mitigating thetotal price impacts of their trades. In turn, this encourages speculators to trade more aggressivelyon the remaining forecast error components of private information. In particular, in contrast to the13