A Theory of Volatility

Bruno Dupire defined in 1993 an extension of Black & Scholes's model consistent with the market implied volatility smile on equity derivatives. This paper shows how Dupire's theory can be extended to take into account more realistic assumptions, such as interest rates, dividends, stochastic volatility and jumps. Moreover, we describe a similar theory in an interest rate setting, deriving a "caplet smile" formula, similar to Dupire's "local volaitlity" formula, defining an extended Vasicek model consistent with the market implied caplet smile. Our results also lead to an efficient calibration strategy for Markov interest rate models, and they are likely to considerably speed-up the calibration of non Gaussian models, with or without taking into account the volatility smile.The first section introduces the fundamental mathematical concepts and notations, and presents a few applications. It also re-derives Dupire's formula in a different context. The second section shows how Dupire's theory can be extended to take into account realistic market considerations, such as interest rates, dividends, stochastic volatility and jumps. The last section derives similar results in an interest rate framework and shows how the Focker-Planck equation can be applied to design fast calibration algorithms. Sections 1 and 2 are a compilation of existing results presented in the framework of a unified theory. The results in the section 3 are new.Our objective being the mathematical derivation of the results, we leave the discussions on the models themselves and the means to numerically implement them to subsequent papers. We also neglect the regularity conditions in order to focus on the concepts and the core of the calculations.