CHAPTER 5
Volatility
As I noted in Chapter 3, volatility is a major element of modern option pricing theory. For the first decade and a half after the publication of the Black-Scholes paper, volatility was thought to be a constant, as the model did in fact assume. Then, the 1987 stock market crash occurred. That event convincingly disposed of the belief that volatility is a constant. Market participants were aghast to see quoted volatility on equities and equity-index options rise to staggeringly high levels. They also saw a pronounced excess demand send the quoted volatility of low-delta puts to even higher levels. Some of these same phenomena, now called smile and skew volatility, were subsequently detected in the pricing of currency options.
ALTERNATIVE MEANINGS OF VOLATILITY
The term volatility is ubiquitous in the options world. There are at least five distinct meanings that I will now introduce.
Theoretical Volatility
One of the core assumptions of Black-Scholes option pricing theory is that infinitesimal percentage changes in the spot exchange rate follow the diffusion process
The term dz is a Gaussian white-noise process that has zero mean and standard deviation equal to 
; σ is assumed to be a known constant, which I call theoretical volatility, because it is that volatility ...
