Stochastic volatility models for the Brent oil futures market: forecasting and extracting conditional moments

• Author: Per Bjarte Solibakke
• Date: 01 June 2015
• Published date: 01 June 2015
• DOI: http://doi.org/10.1111/opec.12048

Stochastic volatility models for the Brent oil

futures market: forecasting and extracting

conditional moments

Per Bjarte Solibakke

Professor, Molde University College, Britveien 2, Kvam, 6402 Molde, Norway. Email:

per.b.solibakke@hiMolde.no

Abstract

This paper builds and implements a multifactor stochastic volatility model for the latent (and

observable) volatility from the front month future contracts at the Intercontinental Commodity

Exchange (ICE), London, applying BayesianMarkov chain Monte Carlo simulation methodologies

for estimation, inference and model adequacy assessment. Stochastic volatilityis the main way time-

varying volatility is modelled in financial markets. An appropriate scientific model description,

specifying volatility as having its own stochastic process, broadens the applications into derivative

pricing purposes, risk assessment and asset allocation and portfolio management. From an estimated

optimal and appropriate stochastic volatility model, the paper reports risk and portfolio measures,

extracts conditional one-step-ahead moments (smoothing), forecasts one-step-ahead conditional

volatility (filtering), evaluatesshocks from conditional variance functions, analyses multistep-ahead

dynamics and calculates conditional persistence measures. (Exotic) option prices can be calculated

using the re-projected conditional volatility.Obser vedmarket prices and implied volatilities estab-

lish market risk premiums. The analysis adds insight and enablesforecasts to be made, building up

the methodology for developing validscientific commodity market models.

1. Introduction

This paper builds and assesses scientific stochastic volatility (SV) models for the Brent

oil futures markets and traded at Intercontinental Commodities Exchange (ICE),

London.1Knowledge of the empirical properties of the Brent oil future prices is important

when constructing risk-assessment and management strategies. Energy market partici-

pants who understand the dynamic behaviour of forward prices are more likely to have

realistic expectations about future prices and the risks to which they are exposed. Time-

varying volatility is endemic in financial markets. Such risks may change through time in

complicated ways, and it is natural to build stochastic models for the temporal evolution

Classification: C11, C63, G17, G32

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in volatility. One of the main objectives of the paper is therefore to structure a scientific

model specifying volatility as having its ownstochastic process, appropriately describing

the evolution of the market volatility. The implementation adapts the Markov Chain

Monte Carlo (MCMC) estimator proposed by Chernozhukov and Hong (2003), claimed

to be substantially superior to conventional derivative-based hill climbing optimisers for

this stochastic class of problems. Moreover, under correct specification of the structural

model the normalised value of the objective function is asymptotically χ2distributed (and

the degrees of freedom is well specified).An appropriate and well-specified SV model for

the ICE markets broaden applications into derivative pricing purposes, risk assessment

and management, asset allocation and portfolio management. The main objective of the

paper was therefore to prepare the foundation for methodologies comprising derivative

pricing, implied volatilities for risk premium calculations, asset allocations and risk

management.

Stochastic volatility models have an intuitiveand simple structure and can explain the

major stylised facts of asset, currency and commodity returns. The volatility of future oil

prices are of interest for many reasons. Firstly, energy price volatility could hinder invest-

ments in new and advanced technology and equipment. Secondly, if oil prices and other

energy prices are cointegrated, a volatile market might make it harder for consumers and

businesses to predict their raw-material costs.2In a booming economy, an increase in

prices may make it harder for economic growth to occur. In general, energy prices move

relatively slowly when conditions are calm, while they move faster when there is more

news, uncertainty and trading. One of the most prominent stylised fact of returns on finan-

cial assets is that their volatility changes over time.As the most important deter minant of

the price of an option is the uncertainty associated with the price of the underlying asset,

volatility is of paramount importance in financial analysis. Risk managers are particularly

interested in measuring and predicting volatility,as higher levels imply a higher chance of

large adverse price changes. Forenergy markets like other markets, the motivation for SV

is the observed non-constant and frequently changing volatility.The SV implementation is

an attempt to specify how the volatility changes overtime. Bearing in mind that volatility

is a non-traded instrument, which suggests imperfect estimates, the volatility can be inter-

preted as a latent variable that can be modelled and predicted through its direct influence

on the magnitude of returns. Besides, as the ICE commodity market writes options on the

front month future contracts, SV models are also motivated by the natural pricing of these

options, when over time volatility change. Finally, for energy markets observed returns

and volatility changes seem so frequent that it is appropriate to model both returns and

volatility by random variables.

The paper focuses on the Bayesian MCMC modelling strategy used by Gallant and

Tauchen (2010a, 2010b) and Gallant and McCulloch (2011)3implementing uni- and

multivariate statistical models derived from scientific considerations. The method is a

Stochastic volatility models for the Brent oil futures market 185

OPEC Energy Review June 2015© 2015 Organization of the Petroleum Exporting Countries

systematic approach to generate moment conditions for the generalised method of

moments (GMM) estimator (Hansen, 1982) of the parameters of a structural model.

Moreover, the implemented Chernozhukov and Hong (2003) estimator keeps model

parameters in the region where predicted shares are positive for every observed price/

expenditure vector. For conventional derivative-based hill-climbing algorithms, this is

nearly impossible to achieve. Moreover, the methodology supports restrictions, inequal-

ity restrictions and informative prior information (on model parameters and functionals

of the model). Asset pricing models as the habit persistence model of Campbell and

Cochrane (1999), the long-run risk model of Bansal and Yaron (2004) and the prospect

theory model of Barberis et al. (2001) are all implemented. For the SV model implemen-

tation, the enhanced statistical and scientific stochastic model calibration methodologies

can greatly enhance portfolio management, elaborate and extend the decomposition and

aggregation of overall corporate and institutional risk assessment and management. In

fact, appropriate MCMC-estimated SV model simulations can generate probability dis-

tributions for the calculation of value at risk (VaR/CVaR) and Greek letters for portfolio

rebalancing and model parameters can be the basis for forecasting the mean and volatil-

ity for forward assessment of risk, portfolio management and other derivative pricing

purposes. However,on the downside, as volatility is latent (and unobservable) coinciding

with the fact that the conditional variances are complex functions complicating the

maximum likelihood, estimations will be imperfect and a single optimal estimation tech-

nique is probably not available.

It is simple to make forecasts using the MCMC framework. Hence, both the mean and

volatility are able to be forecasted adding content to future contract prices. Moreover, the

re-projection method (Gallant and Tauchen,1998) based on long simulated data series can

extend projections. The post-estimation analysis can be used as a general-purpose tech-

nique for characterising the dynamic response of the partially observed system to its

observable history. Forecasting the conditional moments and the use of filtered volatility

(with a purelyARCH-type meaning) and multistep-ahead dynamics are some features that

really add strength to the methodology building of scientifically validmodels. Ultimately,

the analysis may therefore contribute to more realistic risk methodologies for energy

markets and for market participants an improved understanding of the general stochastic

behaviour inducing more realistic expectations about future prices and the risks to which

they are exposed. Previousresearch on energy market prices is extensive. Studies adopting

the Heath–Jarrow–Morton (HJM) assuming dynamics for the forward and swapprice evo-

lution have been suggested. In particular, Bjerksund et al. (2000), Keppo et al. (2004),

Benth and Koekebakker (2008) and Kiesel et al. (2009) have used contracts for the

NASDAQ OMX and European Energy Exchange (EEX) electricity markets. The same

approach for energy markets in general can be found in Clewlow and Strickland (2000).

However, modelling the price dynamics, where the contracts deliversover a period, creates

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