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

of net order flows. Specifically, we conjecture that speculator i’s order for asset j takes the form,x i j = b i j1s i 1 + b i j2s i 2 + ... + b i jL is i jL i+ B i j1(X 1 + u 1 ) + B i j2(X 2 + u 2 ) + ... + B i jM(X M + u M ).Hence, the vector of speculator i’s orders is x i = b i s i + B i (X + u), i.e.,⎛ ⎞ ⎛⎞ ⎛ ⎞ ⎛x i 1 b i 11 b i 12 ... b i 1L is 1 B 11 i B12 i ... B1Mi x i 2b i 21 b i 22 ... b i 2L =is 2B21 i B22 i ... B2Mi . +⎜ . ⎟ ⎜ . . .. . . ⎟ ⎜ . ⎟ ⎜ . . .. .⎝ ⎠ ⎝⎠ ⎝ ⎠ ⎝x i Mb i M1b i M2... b i ML is Li BM1 i BM2 i ... BMMi⎞⎛⎟⎜⎠⎝⎞X 1 + u 1X 2 + u 2.. ⎟⎠X M + u MThus, b i represents i’s trading intensities on his signals, and B i represents i’s trading intensitieson the net order flows that he sees. The conjecture that pricing is a linear function of net orderflows implies that p = λ(X + u), where the pricing matrix λ is an M × M invertible matrix thataggregates information from the order flows of all assets to determine prices, i.e.,⎛ ⎞ ⎛⎞⎛⎞p 1 λ 11 λ 12 ... λ 1M X 1 + u 1p 2λ 21 λ 22 ... λ 2MX 2 + u 2=..⎜ . ⎟ ⎜ . . .. . ⎟⎜. ⎟⎝ ⎠ ⎝⎠⎝⎠p M λ M1 λ M2 ... λ MM X M + u MThe extent to which a market maker uses information from other stocks to price a particular stock,and not just a stock’s own order flow, is captured by the non-diagonal elements of λ. Because thepricing matrix λ is invertible, observing prices is equivalent to observing net order flows.Equilibrium. In a linear equilibrium, speculators maximize expected trading profits given correctconjectures, net order flows are consistent with speculator optimization, and market makers earnzero expected profits from filling each order given the vector of net order flows.Speculator’s Optimization Problem. Speculator i seeks to maximize expected trading profitsgiven his signals s i and the net order flows X + u, solving⎡max E [ (v − p)) ′ x i |s i ,X + u ] ( ( N ′⎤∑= max E ⎣ v − λ x k + u))x i |s i ,X + u⎦ . (1)x ix iTo solve for speculator i’s best response to the conjectured linear trading strategies of the otherspeculators and the linear pricing, we first rewrite the vector of net order flows from i’s perspective,X + u = x i + ∑ k≠ik=1()b k s k + B k (X + u) + u. (2)7