The Generalized Local Volatility Modelling Framework

The Local Volatility formula is a precious tool to identify the volatility process that will allow the perfect retrieval of all the market quotes of vanilla options on the market of equity and index derivatives. Since its first inception by Dupire, the local volatility model, applied initially in a modelling framework where only the underlying is stochastic, has been adapted to several more general cases to produce more generalized versions of the initial famous local volatility function. The most encountered one is the case where the interest rates are assumed to be stochastic. Another famous generalization is the case where the underlying has no longer continuous paths but also endures some jumps in its path over time: see for the case of an equity with a jump to ruin only, for a case where jumps are more general, and for the case where the dynamic of the underlying no longer embeds any continuity in its paths at all.Pursuing the idea of generalizing the Dupire's Local Volatility, the question of this short paper is simple: what is the local volatility formula when absolutely all goes stochastic: stochastic rate, stochastic repo, jumps due to proportionnal dividends, jump to ruin as well as more general jumps? The corresponding derivated formula becomes very difficult to handle in practical but it helps to identify what are the missing bricks of the markets and the datas to make this formula usable. In the end, a first application is developed for Jump To Ruin (JTR) models with stochastic rates and local intensity functions in the mood of and but in a more general case than these earlier works