Forward Curve Dynamics Research Articles

Stationary covariance regime for affine stochastic covariance models in Hilbert spaces

This paper introduces stochastic covariance models in Hilbert spaces with stationary affine instantaneous covariance processes. We explore the applications of these models in the context of forward curve dynamics within fixed-income and commodity markets. The affine instantaneous covariance process is defined on positive self-adjoint Hilbert–Schmidt operators, and we prove the existence of a unique limit distribution for subcritical affine processes, provide convergence rates of the transition kernels in the Wasserstein distance of order p∈[1,2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p \\in [1,2]$\\end{document}, and give explicit formulas for the first two moments of the limit distribution. Our results allow us to introduce affine stochastic covariance models in the stationary covariance regime and to investigate the behaviour of the implied forward volatility for large forward dates in commodity forward markets.

A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets

We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics.

Regime-switching affine term structures

We consider an HJM model setting for Markov-chain modulated forward rates. The underlying Markov chain is assumed to induce regime switches on the forward curve dynamics. Our primary focus is on the interest rate and energy futures markets. After deriving HJM-drift conditions for the two markets, we prove under the assumption of affine structure for the term structure that the forward curves are solutions to specific systems of ODEs that can be solved explicitly in many cases. This allows for a tractable model setting, and we present an algorithm for obtaining consistent forward curve models within our framework. We conclude by presenting some numerical examples and an application to real data under a one-factor Gaussian model setting.

Gas storage valuation under multifactor Lévy processes

A practical 1 The author's views are his own and do not necessarily reflect those of Gazprom Marketing & Trading Limited or any of its affiliates. problem for energy companies is instituting a consistent framework across its supply and trading activities to deliver on all-important P&L and at-Risk reporting requirements. With a focus on storage assets and wider natural gas market exposures, we present a gas storage valuation methodology, which uniquely uses a flexible multifactor Lévy process setting that allows for consistent valuation and risk management reporting across a general derivative book. Our approach is capable of replicating the complex covariance structure of the natural gas forward curve and capturing time spread volatility, a key driver of extrinsic storage value, while being simultaneously capable of accurately calibrating to market traded options. We begin by extending a single factor Mean Reverting Variance Gamma process to an arbitrary number of dimensions and, by way of specific examples, show how the traditional Principal Component Analysis based view of gas forward curve dynamics can be incorporated into a primarily market based valuation. We develop in the process an innovative implied moments based calibration technique, which allows for efficient calibration of general multifactor forward curve models to delivery period options common in energy and commodity markets. Furthermore, to accommodate the forward curve and traded options market consistency, we propose an appropriate joint market based calibration and historical estimation methodology. Through a formal model specification analysis, we provide evidence that the multifactor Lévy models we propose provide a better joint fit to NBP natural gas options-forward market data, relative to comparative benchmark models. Finally, we develop a novel multidimensional fast Fourier transform based storage valuation algorithm and provide empirical evidence that the multifactor Lévy model suite is better specified to more accurately capture extrinsic value.

A forward dynamic optimization strategy under contango storage arbitrage with frictions

The goal of this paper is to explain and improve the offshore oil storage trade observed in a contango market using a forward dynamic optimization strategy. The strategy is developed using trades in forward contracts and contrasted with the literature. By simulating forward prices based on realistic May 2009 market conditions, our strategy, when compared with selling the oil on the spot in a base case scenario, gains an extra US$8.99/barrel, which is comprised of US$6.19 generated by the initial forward maximization and US$2.80 achieved by the subsequent trades. The impact of the forward curve dynamics is studied by examining the trading decisions based on the realized slope and mean level of the forward curve. The path of the realized slope and stepwise changes in the slope are found to be able to explain most of the trading decisions. The effect of initial conditions, the frequency with which the position is readjusted and the storage cost are also examined. The optimal frequency depends on the storage cost. Friction is introduced into the problem by penalizing the refund of the storage cost when a decision to advance the short position in time is made, where its influence, which depends on the storage cost, is at most US$1.40.

Relaxations of Approximate Linear Programs for the Real Option Management of Commodity Storage

The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs), in part because of the use of high-dimensional models of commodity forward curve evolution, as commonly done in practice. Focusing on commodity storage, we identify a deficiency of approximate linear programming (ALP), which we address by developing a novel approach to derive relaxations of approximate linear programs. We apply our approach to obtain a class of tractable ALP relaxations, also subsuming an existing method. We provide theoretical support for the use of these ALP relaxations rather than their associated approximate linear programs. Applied to existing natural gas storage instances, our ALP relaxations significantly outperform their corresponding approximate linear programs. Our best ALP relaxation is both near optimal and competitive with, albeit slower than, state-of-the-art methods for computing heuristic policies and lower bounds on the value of commodity storage, but is more directly applicable for dual (upper) bound estimation than these methods. Our approach is potentially relevant for the approximate solution of MDPs that arise in the real option management of other commodity conversion assets, as well as the valuation of real and financial options that depend on forward curve dynamics. This paper was accepted by Dimitris Bertsimas, optimization.

Commodity Forward Curve Dynamics with Inventory Information

In this paper we introduce a new two-factor commodity term structure model for which inventories serve as a second state variable. We derive a closed-form formula for futures prices and empirically analyze the model's properties. Besides being economically appealing, our model also outperforms the well-known Gibson and Schwartz (1990) model in terms of hedging abilities.

Gas Storage Valuation Under Multi-Factor Levy Processes

This paper builds upon the work of Kiely et al. [2015] to provide a storage valuation methodology which is capable of utilizing both market and historical information. We begin by extending the Mean Reverting Variance Gamma process to an arbitrary number of dimensions and by way of specific examples show how the traditional Principal Component Analysis based view of gas forward curve dynamics can be incorporated into a primarily market based valuation. We discuss the calibration of such models to the monthly delivery swaption market through a modi cation of the market implied moments technique of Guillaume and Schoutens [2013]. We go on to present details of the Fourier Transform based valuation methodology in a multidimensional setting and fi nally present empirical valuation results for a stylized storage deal under different specif cations of a two factor model of the forward curve.

Relaxations of Approximate Linear Programs for the Real Option Management of Commodity Storage

The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs), in part due to the use of high dimensional models of commodity forward curve evolution, as commonly done in practice. Focusing on commodity storage, we identify a deficiency of approximate linear programming, which we address by developing a novel approach to derive relaxations of approximate linear programs (ALPs). We apply our approach to obtain a class of tractable ALP relaxations, also subsuming an existing method. We provide theoretical support for the use of these ALP relaxations rather than their corresponding ALPs. Applied to existing natural gas storage instances, our ALP relaxations significantly outperform their corresponding ALPs. Our best ALP relaxation is both near optimal and competitive with, albeit slower than, state-of-the-art methods for computing heuristic policies and lower bounds on the value of commodity storage, but is more directly applicable for dual upper bound estimation than these methods. Our approach is potentially relevant for the approximate solution of MDPs that arise in the real option management of other commodity conversion assets, as well as the valuation of real and financial options that depend on forward curve dynamics.

A semiparametric factor model for electricity forward curve dynamics

In this paper we introduce the dynamic semiparametric factor model (DSFM) for electricity forward curves. The biggest advantage of our approach is that it not only leads to smooth, seasonal forward curves extracted from exchange traded futures and forward electricity contracts, but also to a parsimonious factor representation of the curve. Using closing prices from the Nordic power market Nord Pool we provide empirical evidence that the DSFM is an efficient tool for approximating forward curve dynamics.

Detecting market transitions and energy futures risk management using principal components

An empirical approach to analysing the forward curve dynamics of energy futures is presented. For non-seasonal commodities—such as crude oil—the forward curve is well described by the first three principal components: the level, slope and curvature. A principal component indicator is described that detects transitions between the two fundamental market states remarkably well. For seasonal commodities—such as electricity and natural gas—it is shown how to extract the seasonal component from the forward curve. The principal component indicator can then be applied to the de-seasoned forward curve to detect significant price deviations that may support profitable trading strategies. A principal component approach to forward curve modelling is applied to computing portfolio value-at-risk. This approach is combined with a new two-step resampling procedure to improve value-at-risk estimates.

Soybean Inventory and Forward Curve Dynamics

We present two results concerning soybean prices. First, we exhibit a simple relationship between stocks and price volatility. The observation of an increasing price volatility with decreasing inventory is often mentioned in the literature, but has so far been documented using a proxy for inventory (see Fama and French 1987, 1988; Litzenberger and Rabinowitz 1995). Instead, we reconstruct a yearly, quarterly, and monthly database of worldwide soybean inventories using aggregate data from the United States, Brazil, and Argentina. We show that under all time scales, price volatility is an increasing linear function of inverse inventory, which we term “scarcity.” Second, we show how the addition of the factor scarcity in a state-variable approach to the dynamics of the term structure of soybean forward prices improves the quality of the fit. We document this property on a 25-year database of CBOT futures contracts and show that the superior accuracy also affects long-maturity futures contracts, an important property for the valuation of long-term origination contracts between producing countries and the agrifood industry.

Forward curve dynamics in the Nordic electricity market

The forward curve dynamics in the Nordic electricity market is examined. Six years of price data on futures and forward contracts traded in the Nordic electricity market are analysed. For the forward price function of electricity, we specify a multi‐factor term structure models in a Heath‐Jarrow‐Morton framework. Principal component analysis is used to reveal the volatility structure in the market. A two‐factor model explains 75 per cent of the price variation in our data, compared to approximately 95 per cent in most other markets. Further investigations show that correlation between short‐ and long‐term forward prices is lower than in other markets. We briefly discuss possible reasons why these special properties occur, and some consequences for hedging exposures in this market.

Pricing and Hedging Spread Options

We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.