Lecture12.pdf - Lecture 12 Solutions to Mertonu2019s...

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Lecture 12: Solutions to Merton’s Problem 1 Solving Merton’s Problem: The Black–Scholes Case Consider the Black–Scholes market with two assets, one bond and one stock. Assume r, μ, σ are all deterministic constants, and that U 1 ( t, c ) = e - ρt c γ γ , U 2 ( x ) = e - ρT x γ γ where 0 6 = γ < 1. The Merton investment–consumption problem in this case is Maximize J ( π, c ) = E " Z T 0 e - ρt c γ t γ dt + e - ρT x γ T γ # , subject to dx t = ( rx t + t - c t ) dt + σπ t dW t , c t 0 , t [0 , T ] . (1) The corresponding HJB equation is v t ( t, x ) + sup c 0 1 2 σ 2 π 2 v xx ( t, x ) + ( rx + - c ) v x ( t, x ) + e - ρt c γ γ = 0 , ( t, x ) [0 , T ) × R , v ( T, x ) = e - ρT x γ γ , x R . (2) Applying the verification theorem, we need to maximize, for each ( t, x ), what is in the bracket above. The first-order conditions in ( π, c ) yield σ 2 πv xx ( t, x ) + Bv x ( t, x ) = 0 , - v x ( t, x ) + e - ρt c γ - 1 = 0 . Hence the optimal portfolio and consumption policies in feedback forms are π ( t, x ) = - σ - 2 B v x ( t, x ) v xx ( t, x ) , c ( t, x ) = e ρt v x ( t, x ) 1 γ - 1 . (3) Note that we need to check a posteriori that the constraint c ( t, x ) 0 is satisfied, and that v xx ( t, x ) 0 (so that the above first-order condition leads to the maximization). This problem actually admits a closed-form solution. To this end, conjecture that v ( t, x ) = θ ( t ) x γ γ 1
for some function θ of time

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