Joint Probability and Computational Finance Seminar
In the modern mathematical finance the stock prices are modeled by stochastic differential equations, whose solutions produce logarithmic Brownian motions. This is the backbone of what is the classical Black-Scholes option pricing theory and Merton's investment/consumption theory. We consider a dynamical portfolio optimization model in the spirit of the latter. The portfolio consists of several risky assets (Stocks) and one risk-free asset (Bond). The rate of return on Bond is constant while the rate of return of Stocks is governed by the SDE of the logarithmic Brownian motion type. Funds can be transferred from one asset to another, and such transaction involves a penalty (brokerage fees) proportional to the size of the transaction. The objective is to find the policy which maximizes the expected rate of growth of funds. The main mathematical tool in the solution of this problem is singular stochastic control theory. In this theory the control functionals are represented by processes of bounded variation, and the optimal control consists of functionals which reflect the process from an a priori unknown boundary. They are continuous but singular (not absolutely continuous) with respect to time. The analytical part of the solution to singular control is related to a free boundary problem for an elliptic PDE with gradient constraints, similar to the ones encountered in elastic-plastic torsion problems. The existence of the classical solution cannot be proved in general but one can show an existence of viscosity solutions to this equation. The optimal policy is to keep the vector of fractions of funds invested in different assets in an optimal (a priori unknown) boundary. We show how to find these boundaries explicitly in the case of one risky and one risk-free asset when the problem becomes one dimensional. In this case the free boundary problem can be reduced to a Stephan problem for an ODE.