On the Wealth Dynamics of Self-financing Portfolios under - PowerPoint PPT Presentation

On the Wealth Dynamics of Self-financing Portfolios under Endogeneous Prices Jan Palczewski Faculty of Mathematics University of Warsaw and School of Mathematics University of Leeds Vienna, September 2007 Joint work with Jesper Pedersen and Klaus Schenk-Hoppé. On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 1 / 16

Motivation Mathematical Finance Economics Classical continuous Supply and demand time theory Prices by market Price process given clearing Option pricing Interaction of investors Optimal investment Evolution of investors’ wealth Price formation Optimal strategies On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 2 / 16

Classical continuous-time finance Investors are price-takers Trades have no impact on the market Dynamics of asset prices are given by a stochastic process, e.g. S t = S 0 exp ( µ t + σ B t ) . There is infinite supply of financial assets There is infinite divisibility of financial assets Standing assumption Small investors!!! Infinite divisibility of financial assets = ⇒ big investors On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 3 / 16

Large trader and large trades Option hedging has significant impact on stock prices 1 Empirical “proofs” Large trader models: Frey (1998), Platen and Schweizer (1998), Bank and Baum (2004) Large trades cannot be performed without being noticed 2 splitting large trades into smaller to lower market impact – algorithmic trading using strategies based on econometric and mathematical reasoning: Keym and Madhavan (1996), He and Mamaysky (2005) strategies based on analysis of limit order books Limitations only one large trader trader’s impact on the market is ad-hoc specified On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 4 / 16

Equilibrium with heterogeneous agents many investors, heterogeneous beliefs dividends investors are utility maximizers prices determined to clear the market one-period models and overlapping generations (De Long, Shleifer, Summers, Waldmann) dynamic models are very complicated and often unsolvable (Hommes) On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 5 / 16

The Market Asset k k = 1 , 2 Price S k ( t ) Cumulative dividends D k ( t ) Assets in net supply of 1. � t D k ( t ) = δ k ( s ) ds 0 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 6 / 16

The Market Asset k k = 1 , 2 Price S k ( t ) Cumulative dividends D k ( t ) Assets in net supply of 1. � t D k ( t ) = δ k ( s ) ds 0 Investor i i = 1 , 2 Portfolio Wealth V i ( t ) number of shares of asset k : Consumption rate cV i ( t ) λ i k V i ( t ) S k ( t ) Constant proportions trading strategy ( λ i 1 , λ i 2 ) On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 6 / 16

Wealth dynamics dV i ( t ) = capital gains + dividends − consumption On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 7 / 16

Wealth dynamics dV i ( t ) = capital gains + dividends − consumption Capital gains λ i k V i ( t ) 2 S k ( t ) dS k ( t ) � k = 1 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 7 / 16

Wealth dynamics dV i ( t ) = capital gains + dividends − consumption Capital gains Dividends λ i k V i ( t ) λ i k V i ( t ) 2 2 � S k ( t ) dD k ( t ) S k ( t ) dS k ( t ) � k = 1 k = 1 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 7 / 16

Wealth dynamics dV i ( t ) = capital gains + dividends − consumption Capital gains Dividends Consumption λ i k V i ( t ) λ i k V i ( t ) 2 2 � S k ( t ) dD k ( t ) S k ( t ) dS k ( t ) cV i ( t ) dt � k = 1 k = 1 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 7 / 16

Wealth dynamics dV i ( t ) = capital gains + dividends − consumption 2 λ i k V i ( t ) dV i ( t ) = � � − cV i ( t ) dt dS k ( t ) + dD k ( t ) � S k ( t ) k = 1 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 7 / 16

Market clearing Market clearing condition k V i ( t ) k V i ( t ) λ 1 + λ 2 k = 1 , 2 . = 1 , S k ( t ) S k ( t ) Equivalent to the net clearing condition: d θ 1 k ( t ) + d θ 2 k ( t ) = 0 , k = 1 , 2 . On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 8 / 16

Price formation Dividend intensities δ k ( t ) + Investment strategies ( λ i 1 , λ i 2 ) + Investor’s wealth dynamics + Market clearing condition ⇓ Asset prices S k ( t ) , k = 1 , 2 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 9 / 16

Price formation Theorem For any feasible V 1 ( 0 ) , V 2 ( 0 ) there exists a unique � � 1 V 1 ( t ) , V 2 ( t ) satisfying wealth dynamics and market � � clearing condition. Asset price dynamics are given by 2 S k ( t ) = λ 1 k V 1 ( t ) + λ 2 k V 2 ( t ) , k = 1 , 2 . On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 10 / 16

Markovian dividend intensities δ 1 ( t ) ρ ( t ) = Relative dividend intensity δ 1 ( t ) + δ 2 ( t ) ∈ [ 0 , 1 ] Assumptions ρ ( t ) is a positively recurrent Markov process 1 its state space is countable 2 its initial distribution is stationary (stationary economy) 3 Theorem Relative dividend intensity process is ergodic: � t 1 ρ ( s ) ds = E ρ ( 0 ) . lim t t →∞ 0 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 11 / 16

Selection dynamics Theorem If investor 1 follows strategy Π ∗ = ( λ 1 1 , λ 1 2 ) = ( E ρ ( 0 ) , 1 − E ρ ( 0 )) 2 ) � = Π ∗ then and investor 2 follows a strategy ( λ 2 1 , λ 2 � t V 1 ( s ) 1 V 1 ( s ) + V 2 ( s ) ds = 1 . lim t t →∞ 0 Remarks Π ∗ is based on fundamental valuation. 1 Relative wealth of investor 2 converges to zero. 2 At odds with findings in discrete-time evolutionary models 3 (Evstigneev, Hens, Schenk-Hoppé). On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 12 / 16

Price dynamics If one of the investors follows trading strategy Π ∗ then asset prices converge: � t S 1 ( s ) 1 S 1 ( s ) + S 2 ( s ) ds = E ρ ( 0 ) . lim t t →∞ 0 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 13 / 16

Price dynamics If one of the investors follows trading strategy Π ∗ then asset prices converge: � t S 1 ( s ) 1 S 1 ( s ) + S 2 ( s ) ds = E ρ ( 0 ) . lim t t →∞ 0 Fundamental valuation Our valuation E δ 1 ( 0 ) � δ 1 ( 0 ) � E E δ 1 ( 0 ) + E δ 2 ( 0 ) δ 1 ( 0 ) + δ 2 ( 0 ) Remarks Fundamental valuation comes as a result of computing 1 average historical payoffs. Our valuation is a fundamentally different benchmark. 2 On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 13 / 16

Almost sure convergence Assumption For every state x E x ( τ x ) 2 < ∞ . Theorem If investor 1 follows strategy Π ∗ and investor 2 follows a 1 2 ) � = Π ∗ then strategy ( λ 2 1 , λ 2 V 1 ( t ) a . s . lim V 1 ( t ) + V 2 ( t ) = 1 t →∞ If one of the investors follows strategy Π ∗ then asset prices 2 converge to our benchmark value: S 1 ( t ) a . s . lim S 1 ( t ) + S 2 ( t ) = E ρ ( 0 ) t →∞ On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 14 / 16

Proof What we hoped to do Linearization and Lagrange multipliers Multiplicative Ergodic Theorem Why? It works fine in discrete-time. Continous-time setting supprised us. Lagrange multiplier at the steady state is zero! What we have done Domination by a Ricatti-type equation with random coefficients. One coefficient depending on the solution of the original problem. Arcsine law. On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 15 / 16

Summary Heterogeneous investors in continuous time model Wealth dynamics Optimal investment strategies Asset pricing - new valuation benchmark Open problems Time varying investment strategies More agents On the Wealth Dynamics under Endogeneous Prices Jan Palczewski AMaMeF 2007, Vienna 16 / 16