Ask most options traders which parameter matters more for pricing: volatility or dividends. Almost everyone says volatility. And almost everyone is wrong. Chapter 64 of Wilmott’s book shows that for many common option structures, the sensitivity to dividend yield actually exceeds the sensitivity to volatility. Once you see the numbers, you start treating dividends very differently.
Here is something that should make every options trader stop and think. The “optimal” time to exercise an American option depends on who you are. The textbook answer assumes the holder is delta hedging. But if the holder were delta hedging, why would they buy the option in the first place? Chapter 63 of Wilmott’s book, based on a 1998 paper with Dr. Hyungsok Ahn, digs into this question and reaches a conclusion that is great news for option writers.
Would you rather have a guaranteed $5 million or a 50/50 shot at $10 million? Most people take the sure thing. Mathematically the expected value is the same. But something inside you says the safe option just feels better. That feeling is exactly what utility theory tries to capture, and Chapter 62 of Wilmott’s book lays down the framework for it.
Every derivatives textbook makes the same quiet assumption: option trading does not affect the stock price. The stock does its random walk thing, the option value follows, and hedging is just a passive activity. But think about this. In many markets, the nominal value of options traded exceeds the value of trade in the underlying stock itself. When everyone is delta hedging, they are all buying and selling the stock in predictable amounts at predictable times. Can we really pretend this has no effect? Chapter 61 of Wilmott’s book says no, and the consequences are fascinating.
Delta hedging is wonderful in theory. You adjust your position continuously, and risk vanishes. In practice, it is messy. You have to trade at discrete times. Transaction costs eat your lunch. And for some contracts, like barrier options or anything with a discontinuous payoff, the required hedge ratios become absurd. You end up buying and selling enormous quantities of the underlying at exactly the wrong moments. Chapter 60 of Wilmott’s book introduces static hedging as the cure for many of these headaches.
Almost everything in quantitative finance is built around one assumption: you hedge. You buy the option, you delta hedge, you eliminate risk, and the drift of the stock does not matter. Beautiful theory. But Chapter 59 of Wilmott’s book asks an uncomfortable question: what if you are not hedging?
The jump diffusion models from the previous chapter have a fundamental problem. You have to estimate the probability of a crash, and that is incredibly hard to do. How often does a 15% market drop happen? Once every 5 years? 10 years? 50 years? Nobody really knows. Chapter 58 of Wilmott’s book takes a completely different approach. Instead of guessing crash probabilities, it asks: what if the worst happens?
Here is a thing that bothers every honest quant at some point. The lognormal random walk, the thing Black-Scholes is built on, assumes that stock prices move smoothly. Small steps. Continuous paths. Nice and clean. But if you have ever watched a market during a crisis, you know that prices do not always walk. Sometimes they jump. Chapter 57 of Wilmott’s book tackles this head on and introduces jump diffusion models.
Every few chapters, Wilmott stops talking about theory and shows you a concrete product that exposes why the theory matters so much. Chapter 56 does exactly this. The cliquet option is a structured product that looks innocent on the surface but hides extreme sensitivity to volatility modeling. If you price it with the wrong volatility assumptions, you can be off by a factor of ten in your risk estimate. That is not a rounding error. That is a blowup waiting to happen.
Here is a frustrating reality of stochastic volatility models. You pick a model because it is tractable (Heston, anyone?). You get nice semi-closed-form solutions. But what if the model does not actually describe reality well? You have traded accuracy for mathematical convenience, and in finance, that trade can cost you real money.