Asset Pricing under Asym Information Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Asset Pricing under Asymmetric Information Modeling Information Solution Concepts Classification of Models Markus K Brunnermeier Princeton University August 21 2007 Asset Pricing under Asym Information References Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Books Brunnermeier 2001 Asset Pricing under Asym Info Vives 2006 Information and Learning in Markets O Hara 1995 Market Microstructure Theory Classification of Models Artciles many see syllabus Some parts of these slides rely on Princeton lecture notes by No ldeke 1993 Asset Pricing under Asym Information Modeling Info Equilibria Two Interpretations of Asymmetric Information Modeling Information Partitions Distribution Solution Concepts Classification of Models different information different interpretation of the same information different background information Asset Pricing under Asym Information Modeling information I Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Classification of Models State space state full description of reality fundamentals signals state space is common knowledge and fully agreed among agents Asset Pricing under Asym Information Modeling information II Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Classification of Models Partition 1 2 3 4 5 6 7 8 P1i P2i P3i partition cells later more about knowledge operators etc Field Sigma Algebra F i Probability measure distribution P Asset Pricing under Asym Information Modeling information III Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Classification of Models Prior distribution Common prior assumption CPA Harsanyi doctrine any difference in beliefs is due to differences in info has strong implications Rational Expectations priori objective distribution i implies CPA Non common priors Problem almost everything goes Way out Optimal Expectations structure model of endogenous priors Updating Signal Extraction Asset Pricing under Asym Information Modeling information III Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Classification of Models Updating general Bayes Rule P i En D P i D En P i En P i D if events E1 E2 EN are a partition P i D En P i En P i En D PN i i n 1 P D En P En Asset Pricing under Asym Information Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Classification of Models Updating Signal Extraction general case Updating Signal Extraction v S desired property signal realization S H is always more favorable than S L formally G v S H FOSD G v S L Milgrom 1981 shows that this is equivalent to fS S v satisfies monotone likelihood ratio property MLRP fS S v fS S v is increasing decreasing in S if v v fS S v fS S 0 v 0 v v and S 0 S fS S v 0 fS S 0 v 0 another property hazard rate fS S v 1 F S v is declining in v Asset Pricing under Asym Information Modeling Info Equilibria Modeling Information Updating Signal Extraction Normal distributions updating normal variable X after receiving signal S s Partitions Distribution Solution Concepts Classification of Models E X S s Cov X S Var S s X S 2 Cov Var S E X Var X S s Var X n multidimensional random variable X S n 1 X X S X E S S N X X S S S n n Projection Theorem X S s 1 N X X S 1 s S X X X S S S S X S S Asset Pricing under Asym Information Modeling Info Equilibria Modeling Information Special Signal Structures N Signals of form Sn X n Let X be a scalar and y 1 Var y Partitions Distribution Solution Concepts E X s1 sN X Classification of Models Var X s1 sN X X 1 PN N X 1 PN n 1 n n 1 n 1 n n sn X 1 X s1 sN If in addition all n i i d then 1 E X s1 sN X N n X N n z N X 1 sn X N n 1 Var X s1 sN where s PN n 1 1 N sn is a sufficient statistic Asset Pricing under Asym Information Special Signal Structures Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Classification of Models N Signals of form X S E X S s s Var X S s Var Binary Signal Updating with binary state space signal q precision prob X H S S H Truncating signals v S S v is Laplace double exponentially distributed or uniform posterior is a truncated exponential or uniform see e g Abreu Brunnermeier 2002 2003 Asset Pricing under Asym Information Solution Equilibrium Concepts Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Classification of Models Rational Expectations Equilibrium Competitive environment agents take prices as given price takers Rational Expectations RE CPA General Equilibrium Theory Bayesian Nash Equilibrium Strategic environment agents take strategies of others as given CPA RE is typically assumed Game Theory distinction between normal and extensive form games simultaneous move versus sequential move Asset Pricing under Asym Information The 5 Step Approach Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Step 1 Classification of Models Step Step Step Step 2 3 4 5 REE Specify joint priors Conject price mappings P S 1 S I u RJ Derive posteriors Derive individual demand Impose market clearing Impose Rationality Equate undet coeff BNE sim moves Specify joint priors Conjecture strategy profiles Derive posteriors Derive best response Impose Rationality No one deviates Asset Pricing under Asym Information A little more abstract Modeling Info Equilibria Modeling Information Partitions Distribution Solution Concepts Classification of Models REE Fixed Point of Mapping MP P 7 P BNE simultaneous moves Fixed Point of Mapping strategy profiles 7 strategy profiles What s different for sequential move games late movers react to deviation equilibrium might rely on strange out of equilibrium moves refinement subgame perfection Extensive form move games with asymmetric information Sequential equilibrium agents act sequentially rational Perfect BNE for certain games nature makes a move in the beginning chooses type action of agents are observable Asset Pricing under Asym Information Modeling Info Equilibria A Classification of Market Microstructure Models Modeling Information Partitions Distribution Solution Concepts Classification of Models simultaneous submission of demand schedules competitive rational expectation models strategic share auctions sequential move models