Optimal Portfolios with Bounded Risks
Our focus in this dissertation is on the portfolio selection of a trader subject to a risk limit give in terms of variance or Value-at-Risk (VaR). First, we suppose a Markowitz type portfolio problem founded on a mean-variance analysis. Because of the prohibition of short selling, we employ a simplex based algorithm, namely Wolfe’s method, to evaluate for any feasible level of expected return the portfolio of minimum risk.
Next, we show a generalization of the mean-variance analysis to continuous-time. The method consists of maximizing expected terminal wealth using the martingale approach. All formulas required to implement the method are derived and a comparison with the traditional portfolio selection as introduced by Markowitz is provided.
We then consider a continuous-time model of optimal portfolio choice subject to VaR limits. In a Black-Scholes setting the stochastic control approach is employed to maximize the expected terminal wealth under the constraint of an upper bound for the VaR. As an application, we compare the risk exposure of a trader subject to the VaR limit with that of an unconstraint one.
Finally, we generalize the last optimization problem to a Black-Scholes setting with jumps. All formulas required to implement the method are derived. Since an analytical solution is not available, we give a finite-differences based algorithm to evaluate the quantities of interest.
