Cross-Asset Speculation in Stock MarketsÃ¢ÂˆÂ— - Econometrics at Illinois ...

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We next set out the model and analyze a speculator’s optimization problem. We then specializeto a symmetric setting to obtain analytical results. Section 4 offers numerical characterizations,showing how asymmetries affect outcomes. A conclusion follows. All proofs are in the appendix.IThe modelIn our multi-asset model of speculative trade in a stock market, speculators are strategic and internalizehow their trades affect prices, and hence the inferences and trades of other speculators.N risk-neutral informed speculators and exogenous noise traders trade claims to M assets. Pricesare set by risk-neutral, competitive, uninformed market makers.We denote the vector of asset values by v ′ = (v 1 ,...,v M ). We consider a very general formulationin which asset values are linear functions of K underlying fundamentals, e ′ = (e 1 ,...,e K ).These K fundamentals are jointly normally distributed with means that we normalize to zero andan arbitrary variance-covariance matrix, Σ e . The value of asset j is given byv j = v j1 e 1 + v j2 e 2 + ... + v jK e K .We can write the vector of asset values as v = V e, where V is the M × K matrix,⎛⎞v 11 v 12 ... v 1Kv 21 v 22 ... v 2K..⎜ . . .. . ⎟⎝⎠v M1 v M2 ... v MKWe allow for the possibility that each speculator has access to many sources of private informationabout asset values. Specifically, we let speculator i see a vector of signals about the valuefundamentals,s i = A i e, where A i is an L i × K matrix,⎛ ⎞ ⎛s i 1 A i 11 A i 12 ... A i 1Ks i 2A i 21 A i 22 ... A i 2K=.⎜ . ⎟ ⎜ . . .. .⎝ ⎠ ⎝s i L iA i L i 1 A i L i 2 ... A i L i K⎞⎛⎟⎜⎠⎝⎞e 1e 2.. ⎟⎠e KExample: One speculator sees an asset’s value. There are M assets, N = M speculators,and K = M fundamentals. The value of asset j is v j = e j , i.e., V is the M × M identity matrix.5
Only speculator i sees v i = e i , i.e., s i i = e i and s i j = 0, j ≠ i. The Ai matrix capturing speculatori’s information is an M × M matrix, which has only zero entries save for A i ii , which is one.Example: Each speculator receives one signal for each asset. There are M assets, N speculators,and K = MN fundamentals. The value of asset j is v j = e (j−1)N+1 +...+e (j−1)N+i +...+e jN .Thus, V is an M × MN matrix, where the j th row of V captures the value of asset j, so it hasv j,(j−1)N+i = 1, for i = 1,... ,N, and zeros elsewhere. Speculator i sees s i j = e (j−1)N+i for assetj = 1,... M. Then A i is an M × MN matrix, where j th row of the A i matrix captures speculatori’s information about asset j: it has A i j,(j−1)N+i= 1, and zeros elsewhere.In addition to his signals s i , speculator i knows the prices at which orders for each asset willbe executed. That is, our economy is a noisy rational expectations economy in which agents arestrategic and internalize how their trades affect prices and the information content of prices. Itis in this sense that our model extends and melds the models of Admati (1985) and Caballe andKrishnan (1993), allowing us to capture key aspects of a dynamic market in a simpler static setting.Let x i j be speculator i’s order for asset j, let xi′ = (x i 1 ,xi 2 ,... ,xi M) be the vector of his orders,and let X ′ = (X 1 ,X 2 ,... ,X M ) ′ = ( ∑ Ni=1 xi 1 ,∑ Ni=1 xi 2 ,...,∑ Ni=1 xi M) be the vector of net order flowsacross assets from speculators. In addition to trade from speculators, there is exogenous liquiditytrade of asset j of u j . We let u ′ = (u 1 ,u 2 ,... ,u M ) be the vector of liquidity trades. Liquiditytrade is distributed independently of the value fundamentals, and hence of speculator signals. Weallow for a general correlation structure in liquidity trade across assets, assuming only that liquiditytrade is jointly normally distributed with zero mean and variance-covariance matrix Σ u .Markets are cleared by competitive market makers who use the trading activity throughout theentire stock market when setting prices. In particular, a market maker observes the net order flowfor his asset, and the prices of other assets. Equivalently, with linear pricing, market makers seethe vector of net order flows for all assets, (X + u) ′ = (X 1 + u 1 ,X 2 + u 2 ,... X m + u m ). Marketmakers set the price of each asset equal to its expected value given the vector of net order flows,i.e., p j = E[v j |(X + u) ′ ]: competition drives market-maker expected profits from filling each orderdown to zero. At the period’s end, assets are liquidated and trading profits are realized.We focus on equilibria in which speculators adopt trading strategies that are linear functions ofboth their private signals and net order flows, and market makers set prices that are linear functions6
Page 1 and 2: Cross-Asset Speculation in Stock Ma
Page 3: ship model of Kyle (1985) and Admat
Page 8 and 9: of net order flows. Specifically, w
Page 10 and 11: and defining γ i ≡ B i (I −
Page 12 and 13: (from Proposition 1) speculator i
Page 14 and 15: Unobservable prices. This theoretic
Page 16 and 17: 3, speculative trading intensities
Page 18 and 19: the off-diagonal term in Γ = Γ
Page 20 and 21: Substituting for λ 11 − c = λ 1
Page 22 and 23: isolates how the division of privat
Page 24 and 25: mation is highly correlated. Profit
Page 26 and 27: The increased competition between s
Page 28 and 29: of the less volatile asset’s valu
Page 30 and 31: V AppendixProof of Proposition 1. U
Page 32 and 33: under the trace yields{0 = tr= tr
Page 34 and 35: where once more we commute under th
Page 36 and 37: We now establish the contraction pr
Page 38 and 39: We substitute (24) for bAΣ e A ′
Page 40 and 41: We have⎛ ⎞V Σ e A ′ b ′
Page 42 and 43: ⎛( ) ⎛ ⎞⎛⎞⎞⎜ ⎜Σ e
Page 44 and 45: Figure 1: Expected profit for asymm
Page 46 and 47: Figure 3: Trading intensities for a
Page 48 and 49: Figure 5: µ coefficients for asymm
Page 50 and 51: Figure 7: Trading intensity for asy
Page 52: VI ReferencesAdmati, Anat, 1985, A

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