Merton Portfolio Optimization with Multiple Assets
We now note how the above results extend to the case where there aren’t many assets. Now suppose that there are assets that can be in invested in. These obey the Stochastic Differential Equation
where is an independent Brownian motion for each .
Wealth now evolves according the SDE
where gives the amount of each asset held in the portfolio at time . Also we define as the wealth in each asset. As given in Def [Merton:Objective], our task is the maximize the objective
We now proceed through exercises that are very similar to the case with a single risky asset.
Ex 1. Show that the HJB equation for the Merton Problem can be written as
Ans 1. The answer follows more-or-less identically to Merton Portfolio Optimization . Note that in this case Ito’s formula applied to gives
This is the drift term applied in the HJB equation, similar to [[cDP:MertonHJB]] and gives the require HJB equation.
Ex 2. Show the optimal asset portfolio in the HJB equation is given by
Ans 2. Considering  we have that
Solving for and substituting back into the maximization gives the answer.
Ex 3. Show that for a CRRA utility the optimal solution to the HJB equation is given by
Ans 3. [Merton:Multi-CRRA] By (Merton Portfolio Optimization )
for some constant . Differentiating twice gives
Rearranging and solving for gives the required solution for .
Def 1. [Merton Portfolio and Market Price Risk Vector] As given in ,
is called the Merton Portfolio and
is called the Market Price Risk Vector.
Dual value function approach
We could solve the CRRA utility case because it had a special structure. We now give a method for solving for general utilities .
Here we assume that is continuous in and , concave in and satisfies
The HJB equation for the Merton problem is
We take the LF transform of ,
Further we define
where is such that .
Ex 4. Show that
Ans 4. We can ignore the dependence of for the first two expressions i.e. take . Now , so
Now reintroducing dependence on ,
Ex 5. Show that the HJB equation becomes
[Noticed that we have got rid of the maximisation are we now have a linear PDE.]
Ans 5. The HJB equation is
Ex 6. Now suppose that , for concave and increasing. Argue that
Ans 6. This the same as (Merton Portfolio Optimization ).
Ex 7. [Continued] Further defining where is such that , show that
Ans 7. This is a straightforward calculation.
Ex 8. [Continued] Show that the HJB equation becomes
[noticed that we have reduced the PDE in  to an ordinary differential equation.]
which implies the stated answer.
Ex 9. Show that the second order ODE given the above has a solution of the form
where and solve the quadratic equation
Ans 9. This is standard ODE theory.