Simulation I: Monte Carlo Methods

We introduce the Monte Carlo approach as statistical estimation of the expectation in the martingale characterisation of the value of an option, and illustrate how the approach generalises to exotic options. We implement the Longstaff-Schwarz method for the valuation of American options and illustrate the use of variance reduction strategies such as control variates, antithetic variates, and importance sampling. Finally we look at how these methods combine with finite difference approximation to estimate hedge parameters via the so-called “bump-and-revalue” method, and point out some practical issues and mitigation strategies.