Implied Volatility and Volatility Smile › Implied Volatility and Volatility Smile

In the Black-Scholes-Merton framework we developed in Chapter 6, volatility enters as a fixed parameter $\sigma$ representing the constant diffusion of the underlying asset's returns. We treated this parameter as an input, something we might estimate from historical returns. Practitioners reversed this process. Given an observed market price, you can solve for the volatility that produces a Black-Scholes theoretical price matching that market price. This backed-out volatility is called implied volatility.

Implied volatility transformed how traders think about options. Instead of quoting options in dollar terms, market participants began quoting them in volatility terms. When a trader says "the 100-strike call is trading at 25 vol," everyone understands exactly how expensive that option is relative to its theoretical value. More importantly, practitioners discovered systematic patterns when computing implied volatilities across different strikes and maturities. These patterns directly contradicted the Black-Scholes assumption of constant volatility. These patterns are the volatility smile and volatility skew, which reveal the market's collective assessment of tail risks, jump probabilities, and the inadequacy of the lognormal assumption for asset returns.

This chapter develops the concept of implied volatility from first principles, shows you how to compute it numerically, and explores the rich structure of the volatility surface. Understanding these patterns is essential for anyone who trades options, manages option portfolios, or builds derivative pricing models.