CS522-Computational Tools and Methods in Finance

The previous sections described that, given a set of scenarios which fully describe a trading universe, numerical methods from stochastic and linear/nonlinear programming can be used to formally obtain an optimal replication or hedge portfolio for a given benchmark. This section takes a closer look at the scenario generation process, which is the most important support for the optimization framework. The subject is very broad, and this discussion is restricted to provide a “road map ” of implementation difficulties with particular emphasis placed on numerical issues. The spectrum of the approach is fully identified however. 1.1 Formal problem In previous discussions, we assumed the existence of a vector of present values (PV) describing a portfolio, given by. (1.1) The components of this vector can represent PV of arbitrary securities. Computing a single element in the vector is done by means of arbitrage-free models discussed earlier on in the course. These include Black-Scholes-based methods for pricing equity derivatives, and the Heath-Jarrow-Morton (HJM) framework for pricing instruments with a dependence o