From implied to spot volatilities

Given the quote price of a call or put option, the Black-Scholes implied volatility is the unique volatility parameter to be put into Black-Scholes formula to give the same price as the option quote price. This dissertation is concerned with the link between the implied volatility and the actual volatility of the underlying stock. Such a link is of particular practical interest since it relates the fundamental quantity for pricing financial derivatives (the actual volatility of the underlying stock), which is not observable, to directly observable quantities such as implied volatilities. The link that we derive in chapter 2 is a link between the dynamics of the two quantities. So far these quantities were mostly studied at a given time whereas we work at the level of processes. This is the main result of the dissertation. In chapter 1, we shall first review current practical problems in option pricing. Our aim there is twofold. First, we want to show that from a practical point of view, studying dynamics is very natural. Second, we shall identify two practical issues to which we shall propose answers in chapter 3. Although the main motivation of this dissertation comes from contemporary issues in the study of financial markets, chapter 2 also gives a solution to an inverse problem in the mathematical sense. One wishes to recover the structure of a stochastic process from a family of conditional expectations over its distribution. Besides the main result, this dissertation makes the following contributions. It brings new insights about implied volatility dynamics. In particular, it was observed that its motion was extremely ‘rigid’ in the sense that the motion of a specific point determines the entire surface dynamics. This statement will be made more precise. Second, it provides with simple closed form approximations for implied volatilities. These approximations avoid having to compute expectations to get option prices. Third, this dissertation gives qualitative and quantitative understanding of common models used in practice. iii