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Homework 5

FIN 539

Homework 4

FIN 539

1. Use the definition of the dual value function:

(t, t ) = Et  [Zt T e−β(s−t)1 ( s )ds + e−β(T−t)2 ( T )]

and the static budget constraint

Et [ZT Zs s ds + ZT T ] = ZtXt .

(where s  = e−r(s−t)cs , T  = e−r(T−t)XT ) to show that

z (t, t )+ Xt  = 0

2. Under the same assumptions as those covered in class for the one stock case, use the duality

approach to solve for the optimal consumption and optimal trading strategy for the case u1 (c) =  and u2 (x) = 3 . Recover the value function V(t,x) for the dual problem.          Hint : Consider the form V(t,z) = f(t) where b = γ 1 .

Homework 3

FIN 539

1. In the model with no consumption (ct  = 0 for all t) and u(XT ) = ^XT , solve for the optimal trading strategy Tt  using the duality approach.  (recall that the risk free rate is r and the stock price follows St(dS)t   = µdt + σdWt ).

2.  Consider an one-period model with 2 states: u happens with probability 0.6, d happens with probability 0.4.  There is one stock with the following prices:  S0  = 52,S1(u)  = 100,S1(d)  = 40. Suppose that the risk free rate is r = 0. The investor has utility over terminal wealth given by u(X) = log(X).  The investor’s initial wealth is x = 100 and the time discount rate is β = 0. Find the investor’s optimal amount of stock holding using the duality approach.

Homework #2

Fin 539

1.   Under  the  same  assumptions  as  those  covered  in  class,solve  for  the  opti- mal consumption and optimal trading strategy for the case u1 (c) = log(c) + 2 and u2 (x) = 3log(x). Hint: consider the form V(t,x) = g(t)log(x) + f(t)

2.  Under the same assumptions as those covered in class,solve for the optimal consumption and optimal trading strategy for the case u1 (c) = 4cγ /γ and u2 (x) = 2xγ /γ . Hint: consider the form V(t,x) = f(t)xγ /γ

3. Under the same assumptions as those covered in class in the one stock case with constant coefficients,solve for the optimal consumption and optimal trading strategy for the case u1 (c)  =  0 and u2 (x)  =  −e−↵北 .   Hint:  consider the form  V(t,x)  = −f(t)e−g(t)北