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Behavioral Equilibrium and Evolutionary Dynamics

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Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept The Basic Model Main Results Financial Markets: Behavioral Equilibrium and Evolutionary Dynamics Thorsten Hens 1,5 joint work with Rabah Amir 2 Igor Evstigneev 3 Klaus R. Schenk-Hopp´e 4,5 1University of Zurich, 2University of Arizona 3University of Manchester 4University of Leeds 5Norwegian School of Economics, Bergen 31 January 2013/ Maastricht Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept The Basic Model Main Results The Focus of our Work: development of a new theory of market dynamics and equilibrium { a plausible alternative to the classical General Equilibrium theory (Walras, Arrow, Debreu, Radner and others). The characteristic feature of the theory is the systematic application of behavioural approaches combined with the evolutionary modeling of financial markets. The theory addresses from new positions the fundamental questions and problems pertaining to Finance and Financial Economics, especially those related to equilibrium asset pricing and portfolio selection, and is aimed at practical quantitative applications. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept The Basic Model Main Results Outline The Basic Concept Walrasian Equilibrium Behavioral Equilibrium The Basic Model Model Components Equilibrium Market Dynamics Main Results Definition of a Survival Strategy Central Results Relation to Game Theory Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Walrasian Equilibrium The Basic Model Behavioral Equilibrium Main Results Walrasian Equilibrium Conventional models of equilibrium and dynamics of asset markets are based on the principles of Walrasian General Equilibrium theory. This theory typically assumes that market participants are fully rational and act so as to maximize utilities of consumption subject to budget constraints. Walras, Arrow, Debreu. Hicks, Lindahl, Hildenbrand, Grandmont { temporary equilibrium. Radner: equilibrium in (incomplete) asset markets. Text: Magill and Quinzii. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Walrasian Equilibrium The Basic Model Behavioral Equilibrium Main Results "Although academic models often assume that all investors are rational, this assumption is clearly an expository device not to be taken seriously." Mark Rubinstein (Financial Analysts Journal, 05/06 2001, p. 15) Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Walrasian Equilibrium The Basic Model Behavioral Equilibrium Main Results The Fundamental Drawbacks of Conventional GET I the hypothesis of \perfect foresight" I the indeterminacy of temporary equilibrium I coordination of plans of market participants I the use of unobservable agent's characteristics (individual utilities and beliefs) Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Walrasian Equilibrium The Basic Model Behavioral Equilibrium Main Results Behavioral Equilibrium We develop an alternative equilibrium concept { behavioral equilibrium, admitting that market actors may have different patterns of behavior determined by their individual psychology, which are not necessarily describable in terms of utility maximization. Their strategies may involve, for example, mimicking, satisficing, rules of thumb based on experience, spiteful behavior, etc. The objectives of market participants might be of an evolutionary nature: survival (especially in crisis environments), domination in a market segment, capital growth, etc. { this kind of behavioural objectives will be in the main focus of this talk. The strategies and objectives might be interactive { taking into account the behaviour and the performance of the other economic agents. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Walrasian Equilibrium The Basic Model Behavioral Equilibrium Main Results Behavioral economics { studies at the interface of psychology and economics: Tversky, Kahneman, Smith, Shleifer (1990s); the 2002 Nobel Prize in Economics: Kahneman and Smith. Behavioral finance: e.g. Shiller, Thaler (2000s). Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results The Basic Model Randomness. S a measurable space of "states of the world" st 2 S (t = 1; 2; :::) state of the world at date t; s1; s2; ::: an exogenous stochastic process. Assets. There are K assets. Dividends. At each date t, assets k = 1; :::; K pay dividends t Dt;k(s ) ≥ 0; k = 1; :::; K; depending on the history t s := (s1; :::; st) of the states of the world up to date t. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results Assumptions on dividends. K X t t Dt;k(s ) > 0; EDt;k(s ) > 0; k = 1; :::; K; t = 1; 2; :::; k=1 where E is the expectation with respect to the underlying probability P: Asset supply. Total mass (the number of "physical units") of asset k available at each date t is Vk > 0. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results Investors and their portfolios. There are N investors (traders) i 2 f1; :::; Ng. Investor i at date t = 0; 1; 2; ::: selects a portfolio i i i K xt = (xt;1; :::; xt;K ) 2 R+ ; i i where xt;k is the number of units of asset k in the portfolio xt. i The portfolio xt for t ≥ 1 depends, generally, on the current and previous states of the world: i i t t xt = xt(s ); s = (s1; :::; st). Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results K Asset prices. We denote by pt 2 R+ the vector of market prices of the assets. For each k = 1; :::; K, the coordinate pt;k of pt = (pt;1; :::; pt;K ) stands for the price of one unit of asset k at date t. The prices might depend on the current and previous states of the world: t t pt;k = pt;k(s ); s = (s1; :::; st): The scalar product K i X i hpt; xti := pt;kxt;k k=1 i expresses the market value of the investor i's portfolio xt at date t. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results The state of the market at date t: 1 N (pt; xt ; :::; xt ); 1 N where pt is the vector of asset prices and xt ; :::; xt are the portfolios of the investors. Investors' budgets. At date t = 0 investors have initial i endowments w0 > 0 (i = 1; 2; :::; N). Trader i's budget (wealth) at date t ≥ 1 is i i i wt(pt; xt−1) := hDt + pt; xt−1i; where t t t Dt(s ) := (Dt;1(s ); :::; Dt;K (s )): Two components: t i i the dividends hDt(s ); xt−1i paid by the yesterday's portfolio xt−1; i i the market value hpt; xt−1i of the portfolio xt−1 in the today's prices pt. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results Investment rate. A fraction α of the budget is invested into assets. We will assume that the investment rate α 2 (0; 1) is fixed, the same for all the traders. Investment proportions. For each t ≥ 0, each trader i = 1; 2; :::; N selects a vector of investment proportions i i i K λt = (λt;1; :::; λt;K ) 2 ∆ in the unit simplex ∆K , according to which the budget is distributed between assets. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results Game-theoretic framework. We regard the investors i = 1; 2; :::; N as players in an N-person stochastic dynamic i game. The vectors of investment proportions λt are the players' actions or decisions. t Players' decisions might depend on the history s := (s1; :::; st) of states of the world and the market history Ht−1 := (pt−1; xt−1; λt−1); where t−1 p := (p0; :::; pt−1); t−1 1 N x := (x0; x1; :::; xt−1); xl = (xl ; :::; xl ); t−1 1 N λ := (λ0; λ1; :::; λt−1); λl = (λl ; :::; λl ): Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results i K Investment strategies. A vector Λ0 2 ∆ and a sequence of measurable functions with values in ∆K i t t−1 Λt(s ;H ); t = 1; 2; :::; form an investment strategy (portfolio rule) Λi of investor i. i t Basic strategies. Strategies for which Λt depends only on s , and not on the market history Ht−1 = (pt−1; xt−1; λt−1). We will call such portfolio rules basic. Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results Investor i's demand function. Given a vector of investment i i i proportions λt = (λt;1; :::; λt;K ) of investor i, the i's demand function is αλi wi(p ; xi ) i i t;k t t t−1 Xt;k(pt; xt−1) = : pt;k where α is the investment rate. Short-run (temporary) equilibrium. for each t, aggregate demand for every asset is equal to supply: XN i i X (pt; x ) = Vk; k = 1; :::; K: i=1 t;k t−1 Amir, Evstigneev, Hens and Schenk-Hopp´e Behavioral Equilibrium and Evolutionary Dynamics The Basic Concept Model Components The Basic Model Equilibrium Market Dynamics Main Results Equilibrium Market Dynamics Prices: N X i t i pt;kVk = αλt;khDt(s ) + pt; xt−1i; k = 1; :::; K: i=1 Portfolios: αλi hD (st) + p ; xi i i t;k t t t−1 xt;k = ; k = 1; :::; K; i = 1; 2; :::; N: pt;k i i The vectors of investment proportions λt = (λt;k) are recursively determined by the investment strategies i t i t t−1 λt(s ) := Λt(s ;H ); i = 1; 2; :::; N: Under mild "admissibility" assumptions on the strategy profile, the pricing equation has a unique solution pt, pt;k > 0.
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