Stochastic Models and Estimation in Mathematical Finance
Vinícius Viana Luiz Albani (IM - UFRJ)

Fitting simultaneously the SPX and VIX smiles: In recent years, the mathematical finance community has been challenged by the problem of finding models that can fit simultaneously observed prices of vanilla (call and put) options on the S&P500 index and the volatility index (VIX). Many of the proposed models are non-Markovian, which may lead to rather involved theoretical analyses of the pricing problem and possibly complex numerical solutions. In contrast, we propose a Markovian approach, based on a stochastic volatility model with functional parameters. The corresponding forward Kolmogorov equation and its adjoint PDE are then used to evaluate option prices on the S&P500 and VIX indices. This approach simplifies the pricing problem, which is important for the estimation. However, there are still some important theoretical gaps that must be filled, which will be discussed. The influence of different variables, like weather and economic activity, is important in commodity prices but difficult to describe mathematically. We will present a model that combines stochastic differential equations and neural networks to address this task. Empirical results with Electricity prices from the Brazilian market will be presented.