Handbook of Multi-Commodity Markets and Products: Structuring, Trading and Risk Management

CHAPTER 13 Nonparametric Estimation of Energy and Commodity Price Processes

Gianna Figà-TalamancaAndrea Roncoroni

13.1 INTRODUCTION

Random changes in several financial figures such as stock data market indices and commodity prices are commonly represented by a diffusion process X of the following form:

(13.1)

where W is a one-dimensional standard Brownian motion, μ( · ) and σ( · ) are assumed to be regular functions in order to guarantee the existence and uniqueness of a weak solution to equation (13.1). Most model specifications assume parametric forms for both these functions, which are called the drift and the diffusion coefficient of process X, respectively. A possible extension of the above model is obtained by adding a jumping part in the dynamics of process X, that is

(13.2)

Here J is a compensated jump process with intensity λ(x), that is, J(t) − λ(X(t)) is a martingale, and Y is a random jump size whose distribution p_Y is assumed to be independent of that of J and W.

We aim at testing the consistency of the diffusion model described in equation (13.1) with changes in the log-price of several commodities; this is done by applying kernel methods to estimate the infinitesimal conditional moments of the process, as suggested by Stanton (1997), and computing confidence ...