Financial models for commodity, energy and equity markets

In this thesis, we propose several financial models to better understand the dramatic price behavior observed in the commodity and energy markets over the past decade. In the first part, we propose a feedback model to account for the "financialization" of commodities, referring to the increased correlations between the commodity and equity markets since the early 2000s. This is conjectured to be due to the influx of external (portfolio optimizing) traders through commodity index funds, for instance. We build a feedback model to capture some of these effects, in which traditional economic demand for a commodity, oil say, is perturbed by the influence of portfolio optimizers. We approach the full utility maximization problem with price impacts through a sequence of problems that can be reduced to linear PDEs, and we find correlation effects proportional to the long or short positions of the investors, along with a lowering of volatility.;The second part of this thesis is motivated by the recent free-fall in oil prices, from around ;In the third part, we apply techniques of optimal control to analyze a class of dynamic portfolio optimization problems that allow for models of return predictability, transaction costs, and stochastic volatility. We propose a multiscale asymptotic expansion when the volatility process is characterized by its time scales of fluctuation. The analyses of the nonlinear Hamilton-Jacobi-Bellman PDEs under fast mean-reverting and slowly fluctuating volatilities can be effectively combined for multifactor multiscale stochastic volatility models. We present formal derivations of asymptotic approximations and demonstrate how the proposed algorithms improve our Profit & Loss using Monte Carlo simulations.