Extensions of the Black-Scholes Theory to Other Types of Options (Futures Options, Currency Options, American Options, Path-Dependent Options, Multi-asset Options)

Abstract

We extend the basic Black-Scholes formula, which was derived in Volume I for European plain vanilla options to more complex types of derivatives like currency options, futures options, American options, path-dependent options, or multi-asset options. We show how to use these extended formulas to price complex types of options with the help of Monte Carlo methods. For some types of path-dependent options (geometric Asian options, barrier options), we also give explicit valuation formulas. We also discuss refinements of Monte Carlo methods, for example, variance reduction methods for Monte Carlo, or quasi-Monte Carlo methods and their application in option pricing.