Derman & Miller, The Volatility Smile

Options traders know about, but often don’t know what to do about, the volatility skew/smirk/smile. In The Volatility Smile (the word “practitioners have persisted in using … to describe the relationship between implied volatilities and strikes, irrespective of the actual shape”) Emanuel Derman and Michael B. Miller take us into the weeds, but by their own accounting not too deep into the weeds. The book, over 500 pages long, with the expected math, offers “an accessible, not-too-sophisticated introduction to models of the volatility smile.”

The Black-Scholes-Merton options pricing model “assumes that a stock’s future return volatility is constant, independent of the strike and time to expiration of any option on that stock. Were the model correct, a plot of the implied BSM volatilities for options with the same expiration over a range of strikes would be a flat line.” This is in fact what implied volatilities, at least in equity index option markets, looked like before the stock market crash of 1987. After that fateful day traders realized that they should pay more for low-strike puts than for high-strike calls. The flat line began to slope and curve. And the Black-Scholes model, never in perfect accord with reality, was now seen to be decidedly imperfect.

Researchers thus set out to create a options pricing model that could explain the volatility smile. As might have been expected, they came up with not one but many models, each of which explains some aspects of the volatility smile, but none of which is “perfect” enough to replace BSM as the dominant pricing model. Don’t despair, the authors advise: “Models not only change the pattern of trading in existing markets, but make possible trading in new, previously unimagined markets. Thus, new and improved models lead to new markets which lead to newer models, ad infinitum.” Financial engineers won’t be joining a bread line any time soon.

If you’re a trader, do you care that the BSM model is wrong? In most cases, if you’re trading very liquid options, the answer is no. “The model is merely a quoting convention.” However, “the model becomes critical for vanilla options, even liquid ones, when you want to hedge them, because even if the option price is known, the option’s hedge ratio is model-dependent. … The model is also critical if you want to trade illiquid exotic options, whose prices are not obtainable from a listed market. In that case, you have no choice but to use a model to estimate both the price and a hedge ratio.”

And so the authors delve into models that are consistent with the smile: local volatility models, stochastic volatility models, and jump-diffusion models.

Although there are nuggets in this book that the retail options trader might find useful, the book is really written for professional traders and budding financial engineers. For those using the book as a self-study guide, there are end-of chapter problems with answers.