Computational Methods Based on the Guaranteed Capture Basin Algorithm

This chapter is devoted to numerical approximation methods for managing replicating portfolios and more complex financial instruments. One aim is to regulate evolutions under uncertainty in order not only to reach a target in finite time but also to fulfill constraints (known as viability constraints) until that time. Considering the portfolio evaluation problem, the target is defined by the payoff function at maturity time, which usually expresses the financial goal to be achieved. Constraints appear when one wants to take into account any limitations on prices, quantities to share, or other restrictive conditions that affect asset prices as well as quantities of time or other characteristics of an agreement. The extension of viability theory to hybrid or impulse systems allows us to evaluate more complex financial instruments. Since reaching a target while remaining in a given set for impulse dynamics can be characterized by a nondeterministic controlled differential equation and a controlled instantaneous reset equation, the set of initial conditions from which a given objective can be reached is computed using the Hybrid Guaranteed Capture Basin Algorithm, which extends the Guaranteed Capture Basin and the Viability Kernel algorithms to more complex dynamic systems. This algorithm can be applied to evaluate options in the presence of barriers or transaction costs, given a lack of observation, or in the frame of Ngarch modeling or other processes of price prevision.