Nov

19

 The Math Department at the University of Michigan has held an annual lecture series called the Ziwet Lectures since 1936. Past speakers include von Neuman, Kac, Thurston, and about a half a dozen Fields medalists. This year, the speaker is I. Karatzas . He is giving a series of three lectures. Today he discussed Stochastic Portfolio Optimization (the next lectures will be on Volatility and Arbitrage respectively). He spent a lot of time introducing the subject, which was good for me. One assumes that there are n risky assets available, S_1, . . .,S_n, and they evolve according to the stochastic differential equation

where dW_i(t) is Brownian motion. X(t) is one's wealth at time t and p_i(t) is the percent of one's wealth invested in asset i at time t. Denote by p=(p_1,…,p_n) our portfolio. U(x) is a utility function, i.e., any increasing function that is concave down. Our goal is to maximize the expected value of utility at the time T. In other words, we let V(x)=sup{E[U(X(T)]} where the supremum is taken over all possible portfolios given that our initial wealth was X(0)=x. Apparently we are guaranteed existence of such a thing in general, but finding the optimal strategy is not very tractable, so as always, one starts with special cases.
 

If we assume that the utility function is U(x)=log(x) or U(x)=x^{a}/a for 0<a<1, then one can find a reasonable solution. However, the solution depends on having reliable values for sigma_{i,j}(t) for all times t as well as for the interest rate.

If one assumes that all coefficients involved are constant, then we can handle the problem of a general utility function. The solution is characterized by a partial differential equation called the Hamilton-Jacobi-Bellman (HJB ) equation. Because we have assumed U is concave down, we can apply the Legendre transform and linearize the partial differential equation. We can then solve the linearized equation.

Karatzas ended the talk with several open problems.

I am not sure whether this lends itself directly to practical application, but perhaps it inspires some more practical ideas.

Jeff Rollert asks:

Why would one assume the coefficients are constant?

Chris Hammond responds:

One answer is that over a reasonably short time horizon, they would be approximately constant. I think the same question could be asked of the Black-Scholes model. It is assumed that if S is the price of an asset, dS=S*r*dt+S*sigma*dW(t), where r is the expected return on the asset, sigma is its volatility, and W(t) is Brownian motion. More sophisticated models assume that the volatility is also a random variable that changes with time sigma=sigma(t). But it makes sense to start with the simpler, constant, case.

In some situations in math, it is insightful to assume very simple behavior to get a model case and view reality as some sort of perturbation of that.

I am not sure it is a good answer, but I'm trying to learn more about these things, so if I find a more satisfying answer, I'll let you know.


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