We consider the problem of pricing derivative securities in an environment of uncertain and changing market volatility. The popular Black-Scholes model relates derivative rices to current stock prices through a constant volatility parameter. The natural extension of this approach is to model the volatility as a stochastic process. In a regime with a multiscale or bursty stochastic volatility we derive an generalized pricing theory that incorporates the main effects of a stochastic volatility. We consider high frequency S&P 500 historical pricing data and analyze these with a view toward identifying important time scales and systematic features. The data shows a periodic behavior that depends on both maturity dates and also the trading hour. We examine the implications of this for modeling and option pricing.

In the 1970s and early 1980s Stark developed a remarkableconjecture aimed at interpreting the first non-vanishing derivative of anArtin L-function $L_{K/k, S}(s, chi)$ at $s=0$ in terms of arithmeticproperties of the Galois extension of global fields K/k. Work of Tate,Chinburg, and Stark himself has revealed far reaching applications ofStark's Conjecture to Hilbert's 12-th Problem and the theory of Galoismodule structure of groups of units and ideal-class groups. In his searchfor new examples of Euler Systems, Rubin has formulated in 1994 a strongversion ("over Z", in Tate's terminology) of Stark's Conjecture forabelian L-functions of arbitrary order of vanishing at s=0. Our study ofthe functorial base-change behavior of Rubin's Conjecture led us toformulating a seemingly more natural Stark-type conjecture "over Z". Wewill discuss and provide evidence for this new statement, as well asbriefly describe the main goals of the conjectural program initiated byStark.