Option Valuation and the Volatility Smile

In this chapter, we briefly present the basic concepts of option pricing theory. The readers who are familiar with these topics, can skip this chapter and begin with the next chapter directly. A Brownian motion is an elemental building-block in modeling the dynamics of stock returns, and correspondingly the geometric Brownian motion as an exponential function of Brownian motion is the simplest and most popular process for stock prices, on which the Black-Scholes model is based. Dynamic hedging in the Black-Scholes model is a self-financing trading strategy that ensures no arbitrage, and allows us to derive the Black-Scholes equation and formula. It is shown that no arbitrage implied in the dynamic hedging results in a risk-neutral process with which all financial derivatives may be valued. From the point of view of probability measures, a risk-neutral process can be regarded as a process derived via the change of the historical measure to a measure using the money market account as numeraire. Finally, we can verify that an equivalent martingale measure under certain conditions in turn implies no arbitrage. Therefore, no arbitrage, risk-neutral valuation and equivalent martingale measure are the key concepts in option pricing theory, and are essentially equivalent to each other, but from different points of view. Next, we focus on the practical challenge of option pricing: inconsistent implied volatilities, or the volatility smile, a challenge that arises from the weakness of the Black-Scholes model, and that we will tackle in this book. Implied volatilities display different smile patterns, and are extensively used by markets for quotations of option prices.