Exotic Options, Volatility Smile, and Alternative Stochastic Models

This chapter starts with an overview of the zoology of exotic options, i.e., with options that are more complex than plain vanilla ones. Some exotic options can be valued by a modification of the Black-Scholes formula, while for some there are more complicated formulas, developed in the context of the geometric Brownian motion, and the others can be valued only numerically using Monte Carlo simulations, binomial tree techniques, or partial differential equations. For most of the exotic derivatives, it turns out that the geometric Brownian motion model calibrated to value correctly the plain vanilla options might give quite imprecise results. The empirical phenomenon called the volatility smile (or surface) demonstrates that the market does not, in fact, believe in lognormal returns and the volatility constant over time. This fact has led to the development of various alternative stochastic models that try to capture better the behavior of market prices, especially the jumps and stochastic volatilities of the underlying asset returns. We will discuss some of the best-known models in the last section.