Nils Detering - Local Volatility Models for Commodity Forwards

We present a dynamic model for forward curves in commodity markets, which is defined as the solution to a stochastic partial differential equation (SPDE) with state-dependent coefficients, taking values in a Hilbert space H of real valued functions. The model can be seen as an infinite dimensional counterpart of the classical local volatility model frequently used in equity markets. We first investigate a class of point-wise operators on H, which we then use to define the coefficients of the SPDE. Next, we derive growth and Lipchitz conditions for coefficients resulting from this class of operators to establish existence and uniqueness of solutions.  We also derive conditions that ensure positivity of the entire forward curve. Finally, we study conditions that ensure the existence of an equivalent measure, which is such that related traded, 1-dimensional projections of the forward curve are martingales.

Our approach encompasses a wide range of specifications, including a Hilbert-space valued counterpart of a constant elasticity of variance (CEV) model, an exponential model, and a spline specification which can resemble the S shaped local volatility function that well reproduces the volatility smile in equity markets. A particularly pleasant property of our model class is that the one-dimensional projections of the curve can be expressed as one dimensional stochastic differential equation. This provides a link to models for forwards with a fixed delivery time for which formulas and numerical techniques exist. We then consider a simulation scheme facilitating neural networks for pricing and model calibration. This methodology enables us to overcome some of the inherent numerical complexities associated with this class of models. We observe that a time dependent spline based, S shaped local volatility function well calibrates the volatility surface in electricity markets.