Quasi-Monte Carlo methods in finance

Viac o knihe

Portfolio optimization has been a key focus in finance since Merton's work in the 1960s. While many methods utilize dynamic programming, recent advancements employ martingale techniques to establish optimal portfolio allocations. Following a 2003 journal article, this work demonstrates how optimal portfolio weights can be expressed through conditional expectations of state variables and their Malliavin derivatives. Unlike traditional methods that rely on Monte Carlo simulations, this approach uses Quasi-Monte Carlo methods to enhance efficiency. Although previous studies on Quasi-Monte Carlo simulations of stochastic differential equations suggest they outperform standard Monte Carlo methods, the theoretical optimal order of convergence remains elusive. By leveraging recent findings on Monte Carlo error estimation and conducting computer experiments with a straightforward model featuring an explicit solution, we propose initial strategies to navigate these challenges. The book begins with an overview of Quasi-Monte Carlo methods, illustrating their ability to accelerate standard Monte Carlo sampling through a simple example. The second chapter offers a comprehensive introduction to Malliavin Calculus, establishing essential calculation rules for the subsequent chapter, which centers on portfolio optimization. This section follows the general market model from Detemple, Garcia, and Rindisbacher to derive the optimal portfolio f