Market environments, stability and equlibria Gordan Zitkovi c - PowerPoint PPT Presentation

Market environments, stability and equlibria Gordan ˇ Zitkovi´ c Department of Mathematics University of Texas at Austin Linz, October 22, 2008

A Toy Model The Information Flow • two states of the world: Ω = { ω 1 , ω 2 } • one period t ∈ { 0 , 1 } • nothing is known at t = 0, everything is known at t = 1: F 0 = {∅ , Ω } , F 1 = 2 Ω . agents two economic agents characterized by • random endowments (stochastic income)  3 ff  1 ff E 1 = , E 2 = 1 4 • utility functions  x 1 ff U 1 ( ) = 1 2 log( x 1 ) + 1 2 log( x 2 ) x 2  x 1 ff 7 x 1 / 3 7 x 1 / 3 U 2 ( ) = 1 + 6 1 2 x 2

A Toy Model The Information Flow • two states of the world: Ω = { ω 1 , ω 2 } • one period t ∈ { 0 , 1 } • nothing is known at t = 0, everything is known at t = 1: F 0 = {∅ , Ω } , F 1 = 2 Ω . agents two economic agents characterized by • random endowments (stochastic income)  3 ff  1 ff E 1 = , E 2 = 1 4 • utility functions  x 1 ff U 1 ( ) = 1 2 log( x 1 ) + 1 2 log( x 2 ) x 2  x 1 ff 7 x 1 / 3 7 x 1 / 3 U 2 ( ) = 1 + 6 1 2 x 2

A toy example the financial instrument  1 ff S 0 = p , S 1 = , B 0 = B 1 = 1: 10 0 1 p 5 0 Market clearing 0.2 0.4 0.6 0.8 • The demand functions: ∆ i ( p ) = argmax U i ( E i + q ( S 1 − p )) q � 5 • Equilibrium conditions: ∆ 1 ( p ) + ∆ 2 ( p ) = 0

A toy example the financial instrument  1 ff S 0 = p , S 1 = , B 0 = B 1 = 1: 10 0 1 p 5 0 Market clearing 0.2 0.4 0.6 0.8 • The demand functions: ∆ i ( p ) = argmax U i ( E i + q ( S 1 − p )) q � 5 • Equilibrium conditions: ∆ 1 ( p ) + ∆ 2 ( p ) = 0

What happens when markets are incomplete and trading is dynamic? • In the IC&mp case, the S 11 p 1 equilibrium conditions determine S 12 both prices and the geometry (degree of incompleteness) of the S 21 market. S 22 p 2 • Another complication : no p 0 representative-agent analysis. S 23 The First Welfare Theorem does S 31 not hold anymore. S 32 p 3 S 33 • Instead of one price p ∗ , we need to determine the whole price process ( p 0 , ( p 1 , p 2 , p 3 )). • C IC 1p * * mp * +

What happens when markets are incomplete and trading is dynamic? • In the IC&mp case, the S 11 p 1 equilibrium conditions determine S 12 both prices and the geometry (degree of incompleteness) of the S 21 market. S 22 p 2 • Another complication : no p 0 representative-agent analysis. S 23 The First Welfare Theorem does S 31 not hold anymore. S 32 p 3 S 33 • Instead of one price p ∗ , we need to determine the whole price process ( p 0 , ( p 1 , p 2 , p 3 )). • C IC 1p * * mp * +

What happens when markets are incomplete and trading is dynamic? • In the IC&mp case, the S 11 p 1 equilibrium conditions determine S 12 both prices and the geometry (degree of incompleteness) of the S 21 market. S 22 p 2 • Another complication : no p 0 representative-agent analysis. S 23 The First Welfare Theorem does S 31 not hold anymore. S 32 p 3 S 33 • Instead of one price p ∗ , we need to determine the whole price process ( p 0 , ( p 1 , p 2 , p 3 )). • C IC 1p * * mp * +

What happens when markets are incomplete and trading is dynamic? • In the IC&mp case, the S 11 p 1 equilibrium conditions determine S 12 both prices and the geometry (degree of incompleteness) of the S 21 market. S 22 p 2 • Another complication : no p 0 representative-agent analysis. S 23 The First Welfare Theorem does S 31 not hold anymore. S 32 p 3 S 33 • Instead of one price p ∗ , we need to determine the whole price process ( p 0 , ( p 1 , p 2 , p 3 )). • C IC 1p * * mp * +

Financial frameworks Information A filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ), where P is used only to determine the null-sets. Agents A number I (finite or infinite) of economic agents, each of which is characterized by • a random endowment E i ∈ F T , • a utility function U : Dom( U ) → R , • a subjective probability measure P i ∼ P . Completeness constraints A set S of ( F t ) t ∈ [0 ,T ] -semimartingales (possibly several-dimensional) - the allowed asset-price dynamics.

Financial frameworks Information A filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ), where P is used only to determine the null-sets. Agents A number I (finite or infinite) of economic agents, each of which is characterized by • a random endowment E i ∈ F T , • a utility function U : Dom( U ) → R , • a subjective probability measure P i ∼ P . Completeness constraints A set S of ( F t ) t ∈ [0 ,T ] -semimartingales (possibly several-dimensional) - the allowed asset-price dynamics.

Financial frameworks Information A filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ), where P is used only to determine the null-sets. Agents A number I (finite or infinite) of economic agents, each of which is characterized by • a random endowment E i ∈ F T , • a utility function U : Dom( U ) → R , • a subjective probability measure P i ∼ P . Completeness constraints A set S of ( F t ) t ∈ [0 ,T ] -semimartingales (possibly several-dimensional) - the allowed asset-price dynamics.

The equilibrium problem Problem Does there exist S ∈ S such that X π i ˆ t ( S ) = 0 , for all t ∈ [0 , T ] , a.s, i ∈ I where Z T E P i [ U i ( E i + π i ( S ) = argmax ˆ π u dS u )] π 0 denotes the optimal trading strategy for the agent i when the market dynamics is given by S . Problem If such an S exists, is it unique? Problem If such an S exists, can we characterize it analytically or numerically?

The equilibrium problem Problem Does there exist S ∈ S such that X π i ˆ t ( S ) = 0 , for all t ∈ [0 , T ] , a.s, i ∈ I where Z T E P i [ U i ( E i + π i ( S ) = argmax ˆ π u dS u )] π 0 denotes the optimal trading strategy for the agent i when the market dynamics is given by S . Problem If such an S exists, is it unique? Problem If such an S exists, can we characterize it analytically or numerically?

The equilibrium problem Problem Does there exist S ∈ S such that X π i ˆ t ( S ) = 0 , for all t ∈ [0 , T ] , a.s, i ∈ I where Z T E P i [ U i ( E i + π i ( S ) = argmax ˆ π u dS u )] π 0 denotes the optimal trading strategy for the agent i when the market dynamics is given by S . Problem If such an S exists, is it unique? Problem If such an S exists, can we characterize it analytically or numerically?

Examples of Completeness Constraints • Complete markets. S contains all ( F t ) t ∈ [0 ,T ] -semimartingales. (If an equilibrium exists, a complete one will exist). • Constraints on the number of assets. S is the set of all d -dimensional ( F t ) t ∈ [0 ,T ] -semimartingales. If d < n , where n is the spanning number of the filtration, no complete markets are allowed. • Information-constrained markets. Let ( G t ) t ∈ [0 ,T ] be a sub-filtration of ( F t ) t ∈ [0 ,T ] , and let S be the class of all ( G t ) t ∈ [0 ,T ] -semimartingales. • Partial-equilibrium models. Let ( S 0 t ) t ∈ [0 ,T ] be a d -dimensional semimartingale. S is the collection of all m -dimensional ( F t ) t ∈ [0 ,T ] -semimartingales such that its first d < m components coincide with S 0 . • “Marketed-Set Constrained” markets Let V be a subspace of L ∞ ( F T ), satisfying an appropriate set of regularity conditions. Let S be the collection of all finite dimensional semimartingales ( S t ) t ∈ [0 ,T ] such that Z T π t dS t : x ∈ R , π ∈ A} ∩ L ∞ ( F T ) = V, { x + 0

Examples of Completeness Constraints • Complete markets. S contains all ( F t ) t ∈ [0 ,T ] -semimartingales. (If an equilibrium exists, a complete one will exist). • Constraints on the number of assets. S is the set of all d -dimensional ( F t ) t ∈ [0 ,T ] -semimartingales. If d < n , where n is the spanning number of the filtration, no complete markets are allowed. • Information-constrained markets. Let ( G t ) t ∈ [0 ,T ] be a sub-filtration of ( F t ) t ∈ [0 ,T ] , and let S be the class of all ( G t ) t ∈ [0 ,T ] -semimartingales. • Partial-equilibrium models. Let ( S 0 t ) t ∈ [0 ,T ] be a d -dimensional semimartingale. S is the collection of all m -dimensional ( F t ) t ∈ [0 ,T ] -semimartingales such that its first d < m components coincide with S 0 . • “Marketed-Set Constrained” markets Let V be a subspace of L ∞ ( F T ), satisfying an appropriate set of regularity conditions. Let S be the collection of all finite dimensional semimartingales ( S t ) t ∈ [0 ,T ] such that Z T π t dS t : x ∈ R , π ∈ A} ∩ L ∞ ( F T ) = V, { x + 0