A new technique to estimate the risk-neutral processes in jump–diffusion commodity futures models

Abstract

In order to price commodity derivatives, it is necessary to estimate the market prices of risk as well as the functions of the stochastic processes of the factors in the model. However, the estimation of the market prices of risk is an open question in the jump-diffusion derivative literature when a closed-form solution is not known. In this paper, we propose a novel approach for estimating the functions of the risk-neutral processes directly from market data. Moreover, this new approach avoids the estimation of the physical drift as well as the market prices of risk in order to price commodity futures. More precisely, we obtain some results that relate the risk-neutral drifts, volatilities and parameters of the jump amplitude distributions with market data. Finally, we examine the accuracy of the proposed method with NYMEX (New York Mercantile Exchange) data and we show the benefits of using jump processes for modelling the commodity price dynamics in commodity futures models. JEL classification: G13, G17.

The Jump Size Distribution of the Commodity Spot Price and Its Effect on Futures and Option Prices

“…In this paper, we price natural gas futures assuming that the spot price follows a diffusion process and, then, we also consider a jump-diffusion process with a Normal jump size distribution as in [11] but for a higher prediction period of time. Moreover, we also assume that the jump size follows an Exponential distribution in order to make some comparisons and analyze the role of the jump size distribution.…”

“…Moreover, they are closer to the observed ones. Then, in order to complement [11], we also price futures options when the jump is not taken into account and when Normal as well as an Exponential jump size distributions are considered. In this case, we see that the differences between the prices are higher (in particular for out of money options).…”

“…In this section, we introduce a commodity model with two state variables: the spot price and the convenience yield, for pricing commodity derivatives; see also [11,17]. We assume that the spot price follows a jump-diffusion process, because commodity prices usually suffer from abrupt changes in the markets; see [1].…”

“…In all the cases, we use the approach, the nonparametric techniques and the in-sample data (January 2004-December 2014) as in [11], to estimate the risk-neutral functions. However, we increase the out-of-sample period where we price the natural gas derivatives from January till July 2015.…”

“…If we considered other more realistic functions for the state variables or the market prices of risk or even a nonparametric approach, then, the model would not be affine anymore, a closedform solution could not be obtained, and, therefore, the estimation of the market prices of risk would not be possible. However, [11] shows a new approach to estimate the whole functions of the model although a closed-form solution is not known. They even apply it to a jump-diffusion model where the jump follows a Normal distribution.…”