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FinEconNov8

# FinEconNov8 - LUCAS MODEL BACKGROUND THE PROJECTION METHOD...

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Unformatted text preview: LUCAS MODEL BACKGROUND THE PROJECTION METHOD RESULTS CONCLUSION S OLVING THE L UCAS A SSET P RICING M ODEL U SING A P ROJECTION M ETHOD A PPROACH Andrew Culham [email protected] Department of Mathematics Florida State University November 8, 2005 SOLVING THE LUCAS ASSET PRICING MODEL USING A PROJECTION METHOD APPROACH ANDREW CULHAM LUCAS MODEL BACKGROUND THE PROJECTION METHOD RESULTS CONCLUSION O UTLINE 1 THE LUCAS ASSET PRICING MODEL 2 LEGENDRE POLYNOMIALS AND QUADRATURE RULES 3 THE PROJECTION METHOD 4 RESULTS 5 CONCLUSION SOLVING THE LUCAS ASSET PRICING MODEL USING A PROJECTION METHOD APPROACH ANDREW CULHAM LUCAS MODEL BACKGROUND THE PROJECTION METHOD RESULTS CONCLUSION T HE B ASIC S ETUP Assume: A large number of investors. One stock paying stochastic dividends. One risk-free bond. All agents are identical with utility function u ( c ) = c 1- γ- 1 1- γ , where γ > 0 is the level of risk aversion. SOLVING THE LUCAS ASSET PRICING MODEL USING A PROJECTION METHOD APPROACH ANDREW CULHAM LUCAS MODEL BACKGROUND THE PROJECTION METHOD RESULTS CONCLUSION N OTATION Define the following notation: c t- the agent’s consumption in time t s t , b t- the agent’s holdings of the stock and bond, respectively S t , B t- the market holdings of the stock and bond, respectively ( S t = 1 and B t = 0 at equilibrium) p t , q t- the market price of the stock and bond, respectively d t- the per capita dividend paid by the stock For each t , the agent chooses { c t , s t + 1 , b t + 1 } . The individual states are z t = { s t , b t } and the aggregate states are Z t = { S t , B t , d t } . SOLVING THE LUCAS ASSET PRICING MODEL USING A PROJECTION METHOD APPROACH ANDREW CULHAM LUCAS MODEL BACKGROUND THE PROJECTION METHOD RESULTS CONCLUSION T HE A GENT ’ S P ROBLEM ( CONT ) The agent solves v ( z t , Z t ) = max { ct , s t + 1 , b t + 1 } E ∞ X t = β t u ( c t ) , subject to c t + p t s t + 1 + q t b t + 1 ≤ s t ( p t + d t ) + b t ∀ t , where s and b are known. In addition, c t ≥ , ∀ t . SOLVING THE LUCAS ASSET PRICING MODEL USING A PROJECTION METHOD APPROACH ANDREW CULHAM LUCAS MODEL BACKGROUND THE PROJECTION METHOD RESULTS CONCLUSION T HE E ULER E QUATIONS Stock: p t c- γ t = β E t [ c- γ t + 1 ( p t + 1 + d t + 1 )] Bond: q t c- γ t = β E t [ c- γ t + 1 ] or Stock: pc- γ = β E [( c )- γ ( p + d )] Bond: qc- γ = β E [( c )- γ ] SOLVING THE LUCAS ASSET PRICING MODEL USING A PROJECTION METHOD APPROACH ANDREW CULHAM LUCAS MODEL BACKGROUND THE PROJECTION METHOD RESULTS CONCLUSION A T RANSFORMATION Suppose dividends grow according to d t = e xt d t- 1 , where x t = ( 1- ρ ) μ + ρ x t- 1 + ε t with ε t being i.i.d. N ( ,σ 2 ) and | ρ | < 1. Let v t = p t d t and θ = 1- γ ....
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FinEconNov8 - LUCAS MODEL BACKGROUND THE PROJECTION METHOD...

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