Optimal portfolio selection for cashflows with bounded capital at risk

We consider a continuous-time Markowitz type portfolio problem that consists of minimizing the discounted cost of a given cash-fl ow under the constraint of a restricted Capital at Risk. In a Black-Scholes setting, upper and lower bounds are obtained by means of simple analytical expressions that avoid the classical simulation approach for this type of problems. The problem is easily extended to cope with more general discount processes. Keywords: Black-Scholes model, Capital at Risk, portfolio optimization, Value at Risk