Abstract and Applied Analysis

Volume 2017, Article ID 3286549, 10 pages

https://doi.org/10.1155/2017/3286549

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

Departamento de Economa Aplicada e IMUVA, Facultad de Ciencias Económicas y Empresariales, Universidad de Valladolid, Avenida del Valle de Esgueva 6, 47011 Valladolid, Spain

Correspondence should be addressed to J. Martínez-Rodríguez; se.avu.oce@ailuj

Received 12 July 2017; Accepted 30 August 2017; Published 18 October 2017

Academic Editor: Lucas Jodar

Copyright © 2017 L. Gómez-Valle et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we analyze the role of the jump size distribution in the US natural gas prices when valuing natural gas futures traded at New York Mercantile Exchange (NYMEX) and we observe that a jump-diffusion model always provides lower errors than a diffusion model. Moreover, we also show that although the Normal distribution offers lower errors for short maturities, the Exponential distribution is quite accurate for long maturities. We also price natural gas options and we see that, in general, the model with the Normal jump size distribution underprices these options with respect to the Exponential distribution. Finally, we obtain the futures risk premia in both cases and we observe that for long maturities the term structure of the risk premia is negative. Moreover, the Exponential distribution provides the highest premia in absolute value.

#### 1. Introduction

In the literature, the commodity price usually follows a diffusion process with continuous paths when pricing commodity derivatives. Although this assumption is very attractive because of its computational, convenience, theoretical derivation and statistical properties, [1–4] others found significant evidence of the presence of jumps in commodity prices.

In traditional jump-diffusion commodity models, the functions of the stochastic processes and the market prices of risk are usually specified as simple parametric functions, for pure tractability and simplicity. Furthermore, the functions of the models are usually chosen to provide an affine model which has a known closed-form solution. For example, [5] considers a three-factor model where the spot price follows a jump-diffusion stochastic process. In [6] existing commodity valuation models were extended to allow for stochastic volatility and simultaneous jumps in the spot price and volatility. The standard geometric Brownian motion augmented by jumps was used by [7] to describe the underlying spot and the mean reverting diffusion processes for the interest rate and convenience yield in gold and copper price models. In [8] a seasonal mean reverting model with jumps and Heston-type stochastic volatility is analyzed.

We consider, in this paper, a two-factor jump-diffusion commodity model, where one of the factors is the commodity spot price and the other is the convenience yield. These factors are often used in the commodity literature. For example, [9, 10] propose affine models with these two factors, though they do not consider jumps. Then, all the functions can be easily estimated and the commodity derivatives priced. However, there is not any empirical evidence or consensus that affine models are the best models to price commodity futures. Furthermore, the market prices of risk are not observed in the markets. 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 closed-form 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. Finally, they estimate the whole functions with a nonparametric technique in order to avoid imposing arbitrary functions on the model.

Other authors have found seasonal patterns in commodity markets and this fact has been taken into account in their models; see [12–15].

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. We find that for short maturities the Normal distribution provides more accurate futures prices. However, the Exponential distribution shows the lowest error for long maturities. Furthermore, for long maturities, the models with both distributions underprice the futures in the market, but the futures prices with the Exponential distribution are higher than with the Normal 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).

Futures prices are potentially a valuable source of information on market expectations of asset prices. In fact, financial investors use futures contracts to hedge against commodity price risk. However, exploiting this information is difficult in practice, because of the presence of a risk premium between the current futures price and the expected spot price of the underlying asset. Moreover, understanding this premium is very important; see [16]. Therefore, in this paper, we also show an out-of-sample analysis of the natural gas futures risk premia. We find that the risk premium with the Exponential distribution is negative more times than with the Normal distribution. In all the cases, we use natural gas data traded at NYMEX and a nonparametric approach to estimate the whole functions of the two-factor model.

The rest of the paper is organized as follows. Section 2 shows a two-factor jump-diffusion model to price commodity derivatives. Section 3 prices futures with a diffusion model and a jump-diffusion model, when the jump size follows a Normal as well as an Exponential distribution. Then a comparison is made. Section 4 compares futures option prices when the jump follows a Normal or an Exponential distribution. Section 5 analyzes the futures risk premium and, finally, Section 6 concludes. All the implementation has been done using MATLAB software.

#### 2. The Valuation Model

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]. However, we assume that the convenience yield is a diffusion process because its behaviour is not affected by extreme changes; see, for example, [6].

Define as a complete filtered probability space which satisfies the usual conditions where is a filtration; see [18–20]. Let be the spot price and the instantaneous convenience yield. We assume that these factors follow this joint jump-diffusion stochastic process: where and are the drifts and and the volatilities. Moreover, and are Wiener processes and the impact of the jump is given by the compound Poisson process, , with jump times , where represents a Poisson process with intensity and is a sequence of identically distributed random variables with a probability distribution . We assume that and are independent of , but the standard Brownian motions are correlated with We also suppose that the jump magnitudes and the jump arrivals time are uncorrelated with the diffusion parts of the processes. We assume that the functions and satisfy suitable regularity conditions: see [20, 21]. Under the above assumptions, a commodity futures price at time with maturity at time , , can be expressed as and at maturity it verifies that .

We assume that the market is arbitrage-free. Then, there exists an equivalent martingale measure, -measure, which is known as the risk-neutral measure; see extended Girsanov-type measure transformation in [22]. The state variables of the model (1) under the risk-neutral measure are as follows: where and are the Wiener processes under the risk-neutral measure and . The market prices of risk associated with and Wiener processes are and , respectively. Finally, is the compensated compound Poisson process under -measure, the intensity of the Poisson process is , and denotes the expectation under the -measure. Then, the futures price can be expressed as

Let be the price of a European call option that matures on on a futures contract that expires at , , and is the strike price. Then, analogously to (4), an European commodity futures option is priced as the expected discounted payoff under the -measure; see [6, 22]: where denotes the instantaneous risk-free interest rate, which is assumed to be constant. Moreover, and are the maturity of the option contract and futures contract, respectively.

#### 3. Valuation of Commodity Futures with NYMEX Data

In this section, by means of an empirical application with natural gas NYMEX data, we illustrate the advantages and disadvantages of modelling the spot price with a jump-diffusion process with an Exponential distribution and a Normal distribution. 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.

In this empirical application, we use the model stated in Section 2, where the factors are the commodity spot price and the convenience yield. For simplicity and tractability and as usual in the literature, we also assume that the distribution of the jump size under -measure is known and equal to the distribution under -measure. This means that all risk premium related to the jump is artificially absorbed by the change in the intensity of the jump from under the physical measure to under the risk-neutral measure; see [8, 11, 23]. Moreover, we assume the jump size follows a Normal distribution (see [11]) or an Exponential distribution (see [6, 24, 25]) among others.

In order to price natural gas futures, we use daily natural gas data from the NYMEX in Quandl platform. Natural gas spot prices were obtained from the U.S. Energy Information Administration (EIA). The sample period covers from January 2004 to July 2015. More precisely, we use data from January 2004 to December 2014 to estimate the risk-neutral functions as in [11] and, then, we keep data from January to July 2015 to make our out-of-sample analysis of the futures prices.

As it is well known in the literature, the convenience yield is not observed in the markets. Then, following [9], we approximate it by the following result where denotes the forward interest rate between and . We obtain this forward interest rate with two daily T-Bill rates with maturities as close as possible to the futures contracts’ ones in order to compute , the one-month ahead annualized convenience yield. The latter is identified with the instantaneous convenience yield ; see [9, 11] for more details.

In order to estimate the risk-neutral functions of the jump-diffusion models, we follow the same approach as [11]. Note that similar techniques have been proposed for interest rate derivatives; see [26, 27].

Firstly, we obtain the compensated risk-neutral drift of the spot price by means of the following equality which relates the futures slope in the origin with the drift of the spot in the stochastic process under -measure; see [11] for more detail:

We approximate the partial derivative by means of numerical differentiation with futures prices with maturities equal to 1, 2, 3, and 4 months. Then, we estimate it by means of the Nadaraya-Watson estimator; see [28] for more details on this estimation technique.

Secondly, for the risk-neutral jump intensity, we use a result proposed in [11] which relates the futures slope in the origin with the spot price, spot price volatility, and parameters of jump size distribution under -measure:

Initially, [11] assumed that the jump size followed a Normal distribution as , then, , and . Furthermore, it is well known that

In this paper we also assume that the jump size follows an Exponential distribution as ; then: This jump size distribution has also been considered by [29] for the volatility and [30] for interest rates. This assumption could be useful for pricing during periods in which positive jumps are expected to dominate negative jumps, for example, coming out of an economic crisis (see [30]) or in certain economic regimes (see [31]).

With both distributions, the parameters of the jump size distribution and the spot price volatility, , are estimated by means of a system of moment equations of a jump-diffusion process (see [11, 32, 33]): More precisely, we use moments , , and for the Normal distribution and moments , , and for the Exponential distribution; see, for example, [34, 35], respectively. Then, Nadaraya-Watson estimator is applied. Once we estimate the parameters of the jump size distribution and the spot volatility and approximate the previous partial derivatives and , we replace them in (9). Then, we estimate the risk-neutral jump intensity of the spot price with the Nadaraya-Watson estimator.

As the convenience yield follows a diffusion process, we estimate its risk-neutral drift by means of see [11]. In order to estimate the correlation, we use the moment and the Nadaraya-Watson estimator; see [36] for more details. Later, we replace the estimated covariance and the approximations of and in (13) and we estimate the risk-neutral drift of the convenience yield by means of the Nadaraya-Watson estimator.

Finally, the volatility of the convenience yield under -measure is equal to the volatility under -measure. Hence, we estimate by means of the second order moment of a diffusion process: and Nadaraya-Watson estimator, with spot and convenience yield data.

Up to this point, we have focused on the estimation of the risk-neutral functions of jump-diffusion processes. If we assume that the spot price follows a diffusion stochastic process, the factors of the model will follow this joint diffusion stochastic process under -measure: with .

The estimation of these functions is made by means of the approach in [37] and the Nadaraya-Watson estimator, with the same natural gas data and numerical differentiation approximation as the jump-diffusion model.

For analyzing the effect of the jumps on the natural gas futures prices, we price natural gas futures with a diffusion model (DM) as well as a jump-diffusion model with a Normal jump size distribution (JDMN) and an Exponential distribution (JMDExp). In order to price natural gas futures it is necessary to solve a partial integrodifferential equation or, equivalently, by means of Feynman-Kac Theorem the expectation in (4). As we use nonparametric methods a closed-form solution cannot be found. Recently, several numerical methods have been developed to solve this kind of problems; see [38, 39].

In this paper, we use the Monte Carlo simulation approach because it is widely used by practitioners in the markets, especially for multiple factor models because of its simplicity and efficiency, [40]. More precisely, we consider 5000 simulations and a daily time step, . We price natural gas futures with maturities from 1 to 44 months and we compare them with those traded at NYMEX along the out-of-sample (January–July 2015). As measures of error, we use the root mean square error (RMSE) and the percentage root mean square error (PRMSE) for the out-of-sample: where is the number of observations, is the futures price traded at NYMEX, and is the predicted futures price with the different models.

Table 1 shows a summary of the RMSE and PRMSE of the different models for the out-of-sample and for several maturities. F1 is the futures price with a maturity of 1 month, F6 with six months, and so on. In this table, we show that for short maturities the RMSE are usually lower than for long maturities. Besides, for very short maturities sometimes the diffusion model prices natural gas futures quite accurately, as for F6. However, for F1 and for maturities higher or equal to 9 months, jump-diffusion models provide lower errors than the diffusion model as in [11]. Moreover, for maturities lower than 18 months the JDMN is more accurate than the JDMExp, but for long maturities (higher or equal to 18 months) the results change and the JDMExp displays lower errors than the JDMN. Therefore, depending on the maturity of the futures to price, some models are more accurate than others. As far as the PRMSE is concerned, we reach the same conclusion but, for maturities longer or equal than 36, the differences between the relative error of the JDMN and JDMExp are higher.