A market model for stochastic smile: a conditional density approach

The purpose of this paper is to introduce a new approach that allows to construct no-arbitrage market models of for implied volatility surfaces (in other words, stochastic smile models). That is to say, the idea presented here allows us to model prices of liquidly traded vanilla options as separate stochastic quantities. The main reason why market models of implied volatilities need to be constructed is that they can capture the stochastic nature of an implied volatility surface. More to the point, market models have a potential of improved pricing of forward volatility depending products, such as compound options. Besides, this framework allows to match the initial vanilla market by construction and hedge with simple call and put options in a natural way. The modelling approach presented in this paper relies on taking a deterministic smile model as a backbone around which a stochastic smile model can be constructed without violating no-arbitrage constraints

Quantum Harmonic Oscillator as a Zariski Geometry

We carry out a model-theoretic analysis of the Heisenberg algebra. To this end, a geometric structure is associated to the Heisenberg algebra and is shown to be a Zariski geometry. Furthermore, this Zariski geometry is shown to be non-classical, in the sense that it is not interpretable in an algebraically closed field. On assuming self-adjointness of the position and momentum operators, one obtains a discrete substructure of which the original Zariski geometry is seen as the complexification.Comment: some typos correcte

The theory of the exponential differential equations of semiabelian varieties

The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.Comment: 53 pages; v3: Substantial changes, including a completely new introductio