Static or Dynamic? Vanilla Market Models

The idea of this paper is to present how Vanilla Market Models, i.e. Models where we directly diffuse the vanillas, can be used in practice. We provide some already known formulae to guarantee that there are neither dynamic nor static arbitrage, extend these results and present them in a financially comprehensive way. More generally, our purpose is to present the interest of having such models and how they can be used in practice. On our way, we demonstrate that a deterministic Sticky Delta Model is necessarily a Black-Scholes model, and we introduce Forward Market Models, which are well suited to represent the stochasticity of dividends. All these analyses are made considering there are no jumps