Tangent Lévy market models

In this paper, we introduce a new class of models for the time evolution of the prices of call options of all strikes and maturities. We capture the information contained in the option prices in the density of some time-inhomogeneous Lévy measure (an alternative to the implied volatility surface), and we set this static code-book in motion by means of stochastic dynamics of Itô's type in a function space, creating what we call a tangent Lévy model. We then provide the consistency conditions, namely, we show that the call prices produced by a given dynamic code-book (dynamic Lévy density) coincide with the conditional expectations of the respective payoffs if and only if certain restrictions on the dynamics of the code-book are satisfied (including a drift condition à la HJM). We then provide an existence result, which allows us to construct a large class of tangent Lévy models, and describe a specific example for the sake of illustration.