Abstract

Emissions Option Pricing

We give a brief review of the equilibrium theory of cap-and-trade schemes in order to motivate the assumptions of several reduced form models which we present and compare. Pricing and hedging options in these models lead to the solution of forward-backward stochastic differential equations with a non-smooth terminal condition, and we spend the second part of the talk analyzing these equations.

Abstract

Modelling and Estimating the Forward price curve in the Energy Market

The stochastic or random nature of commodity prices plays a central role in models for valuing financial contingent claims on commodities. In this paper, by enhancing a multi factor framework which is consistent not only with the market observable forward price curve but also the volatilities and correlations of forward prices, we propose a two factor stochastic volatility model for the evolution of the gas forward curve. The volatility is stochastic due to a hidden Markov Chain that causes it to switch between “high volatility load” and “low volatility load” states. Based on the structure functional forms for the volatility, we propose and implement the Markov Chain Monte Carlo (MCMC) method to estimate the parameters of the forward curve model. Applications to market gas forward data show that the MCMC approach provides stable estimates.

Abstract

Pricing of Futures on Crude Oil and other Commodities: Some Research Issues.

Pricing commodity futures poses many challenges due to the specific nature of the different contracts. In this talk we review some of the issues we have had to deal with over the years including the number of risk factors, mean reversion, model calibration techniques, multicommodity models, model extrapolation and seasonality, among others.

Abstract

The Electricity Bid Stack: Linking the dynamics of fuel, power and carbon prices.

Evidence from the EEX, PJM and New England electricity markets illustrates the benefit of using observed bid data to better understand the strong relationships between fundamental supply and demand factors and power prices. Exploiting the high correlation between movements of bids and fuel prices, we propose a parametric form for the bid stack function in order to construct a fundamental model for spot power prices. This approach allows for the investigation of possible merit order changes and fuel switching, which are considered key factors impacting prices of carbon emissions allowances. We therefore adapt the model to construct an equilibrium carbon price model, which can begin to capture the important but complicated dependence structure between CO2, electricity and other energy prices.

Abstract

Modeling Energy Derivatives

The presentation addresses special issues and challenges arising in the rapidly expanding markets of energy derivatives, and provides insights into the modeling, hedging and risk management techniques utilized in energy markets. The topics covered in the presentation include the overview of complex energy products and structures, as well as the analysis of energy prices and some of their unique properties required for the development of adequate models. Special attention is given to the recent developments and new trends in energy derivatives.

Abstract

Semi-Lagrangian Methods for Gas Storage

In this talk, the value of leasing a gas storage facility is determined by solving a Hamilton Jacobi Bellman Partial Differential Equation. We propose a one factor, regime switching model for the risk adjusted natural gas spot price and study the implications of the model on the valuation and optimal operation of natural gas storage facilities.

We calibrate the model parameters to both market futures and options on futures. Calibration results indicate that the regime-switching model is a better fit to market data compared to a one-factor mean-reverting model similar to those used by other authors to value gas storage.

We use a semi-Lagrangian timestepping scheme to solve the gas storage pricing problem, essentially a stochastic control problem, and conduct a convergence analysis of the scheme. Numerical results also indicate that the regime-switching model can generate operational strategies for gas storage facilities that reflect the existence of multiple regimes in the market as well as the regime shifts due to various exogenous events.

Abstract

Boom and Bust and Spikes in Commodity and Shipping Markets

The goal of our talk is threefold: i) review some remarkable commodity spot prices and freight rates in the recent past; ii)validate in the case of oil and natural gas the use of the slope of the forward curve as a proxy for inventory (the slope being defined in a way that filters out seasonality for natural gas); iii) analyze directly for these two strategic commodities the relationship between inventory and price volatility.

The role of inventory in explaining the shape of the forward curve and spot price volatility in commodity markets is central in the theory of storage developed by Keynes (1930), Kaldor (1939) and Working. Fama and French (1987) revisit the relationship between inventory and spot price volatility in the case of metals and use as a proxy for inventory the adjusted spread of the forward curve.

In agreement with the theory of storage and using no proxy, we exhibit that a) the negative correlation between price volatility and inventory is globally significant for crude oil; b) this negative correlation prevails only during those periods of scarcity when the inventory is below the historical average and increases importantly during winter periods for natural gas. Our results are illustrated by the analysis of a 15 year-database of US oil and natural gas prices and inventories.

Abstract

Dual methods for the valuation of multiple exercise options

Multiple exercise options are contracts in which the holder can exercise and receive a payout at a number of different times throughout the life of the contract. A typical example is that of a swing option in an electricity market but examples also occur in interest rate markets, real options problems and even some liquidation problems can be formulated in this way. It is generally straightforward to extend the basis function regression approach, initially developed for American options, to provide a lower bound for the price of a multiple exercise option. We will look at some versions of such contracts in discrete time and show how to formulate a dual version in order to provide numerical techniques for upper bounds for prices. We will illustrate the approach with a range examples.

Abstract

Carbon Price Risk Modelling

Tackling climate change is at the top of many agendas. In this context, emission trading schemes are considered as promising tools. A mandatory cap-and-trade system involves its participants in a risky business and creates need for risk management by appropriate financial contracts. In this talk, we address logical principles underlying valuation of emission-linked derivatives. Due to the complexity of emissions markets, risk-neutral dynamics of emission certificates must be addressed in terms of explanatory variables, viewed as proxies of fundamental quantities. Thus, we utilize equilibrium analysis to explain the role of fundamentals in the process of allowance price formation and show tha the t key issue in this context is a feedback relation between allowance prices and market abatement activity.

Abstract

Games with Exhaustible Resources

Two producers own finite resources of a commodity (for example mineral water). It costs nothing to produce, and there is an alternative technology (say, desalination) which allows production at a nonzero cost. The producers compete in a Cournot market and the goal is to determine their optimal production strategies in a continuous-time Markov-perfect setting, including the option of producing using the alternative technology. The resulting differential game leads to a strongly nonlinear pair of coupled partial differential equations with some unexpected properties. I shall describe the model and our work on it (which is very much in progress, with many open issues). Joint work with Chris Harris (Cambridge), Ronnie Sircar (Princeton) and Jeff Dewynne (Oxford).

Abstract

Airline Revenue Management and Pricing in Competitive Markets

We consider a simple one leg revenue management model with two competing airlines and fenceless fare structures. We simulated the situation when both airlines use the same state of the art revenue management system adapted to fenceless fares. We then show that for continuous fares the perfect information game theoretical model has a unique, pure strategy, subgame-perfect equilibrium. Both the results of the simulation and the Nash equilibrium indicate that for larger capacities prices will spiral down to the lowest level and that the revenue is substantially lower compared to a monopoly or cooperative situation. We argue that a more appropriate model for the competitive situation should be a repeated game and propose how current revenue management systems could be adapted.

Abstract

Importance of Commodities in Financial Markets

Global commodities markets (e.g., oil, iron ore, agriculturs) have shown very high volatility in volume and price over the last years. After a decline during the global economic and financial crisis, market conditions have been recovering recently. While institutional and private investors have increased the share of commodities in their total asset allocation to about 1-2%, many corporates are now following more sophisticated hedging strategies. Despite uncertain market outlook, structural drivers like climate change and liberalization will likely continue to drive further growth of commodities markets offering an attractive business opportunity for exchanges, clearing houses, traders and alike. The global revenue pool of commodities amounts to over EUR 30 bn of which banks capture approximately 25%. Banks pursue very different business models ranging from pure financial trading activities to ownership of physical commodities assets. Besides commodities traders like energy companies (RWE, Eon etc), global banks like Goldman and BarCap dominate the competitive landscape in the commodities space. On average they earn about EUR 300+ million in their commodities departments per year. Going forward, especially smaller banks can only participate at the growing commodities business if they leverage their existing client franchise building on improved capabilities along the whole value chain (sales, trading, structuring, research, ...).

Abstract

Systemic risk

I will discuss briefly how systemic risk is currently perceived and then I will consider several models that have played a role in understanding systemic risk and in quantifying it. I will also give analogies with other fields where rapid transitions can occur when small changes in the operating environment accumulate and tend to amplify instabilities.

Abstract

Revenue Management

Major network airlines use published fare structures with limited discrete price points to sell their tickets. The lack of dynamic pricing methods is mainly caused by restriction of the legacy global distribution systems, like Amadeus. Those systems are used by the travel agents and still are the main distribution channel for network airlines. But the trend to sell more and more tickets via internet and airline web sites as new start-up carriers do opens up new flexibility in price optimization.

This presentation gives a high-level overview of the history and the evolution of revenue management methods. It points out some practical constraints and limitations in the airline world and takes a glance at future revenue management projects of Lufthansa.

Abstract

Risk Modeling in Optimization Under Uncertainty

In numerous situations in management and economics, decisions must be made without full knowledge of their consequences because those consequences depend on the outcomes of random variables yet to be realized. There is a desire nonetheless to optimize the decisions from some perspective, but the meaning of this is far from clear. Objectives and constraints must be formulated in terms of additional random variables that are affected by the decisions but only in shaping their distributions rather than pinning them down to specifics. One can look to worst cases, or expectations, or to the probabilities that particular conditions might be violated, but in fact there is now a rich methodology which can provide much broader guidance.

This talk will outline the basics of this methodology, as now emerging, and explain the criteria that supports superior forms of modeling, more amenable to computation.

Abstract

Survey on Modelling Electricity Markets

Electricity markets show a number of characteristics that distinguish this market from other commodity or financial markets, such as price spikes and pronounced seasonalities. The main reasons for this particular behaviour are that electricity is hardly storability and that demand is mostly inelastic. On the other hand, the optionalities embedded in power generation assets or flexible demand contracts are very complex from a modelling perspective. Options related to power generation assets often incorporate complex exercise constraints to represent the technical specification of the generation assets. Furthermore, they involve a multi-commodity setup to take into account fuel price behaviour. Flexible power sales contracts need to deal with volume risk that to some extent is weather dependent.

For the reasons above, a number of mathematical models have been developed particularly for electricity markets. The presentation will outline a selection of different approaches and will also give more background on the main challenges to model electricity markets.

Abstract

Optimization Under Risk in Electricity Production and Trading

In electricity production, mathematical models for optimization under uncertainty were in use already before market deregulation and sustainable resource consumption led to new decision problems where proper handling of uncertainty is even more demanding. As an example take the reliable and cost efficient operation of energy systems with dispersed generation. Here uncertainty is present in power prices, power demand, and infeed from renewables. Difficult optimization problems arise whose critical features include nonlinearity induced by the physics of power grids, combinatorics caused by the commitment of generation units or other grid elements, and nonanticipativity resulting from the interplay over time of making decisions and gaining information. This list is by no means complete, but illustrates a change of paradigms with uncertainty in a pivotal position.

Driven by practical needs and by inner-mathematical aspects, optimization under incomplete information has seen a rapid development within the last decade. Approaches differ considerably, depending on what information is available on the uncertain data. If the latter is just membership in an (explicit) uncertainty set, then robust optimization yields solutions taking into account the full uncertainty set. On the one hand, this makes them worst-case compatible, on the other (overly) conservative, which has motivated research into controlling the level of conservatism. In the talk, robust optimization is not pursued any further, see [BTEGN09] for a recent state-of-the-art.

Stochastic programming addresses optimization under uncertainty if distributional information is available for the unknown data, see [RS03, SDR09] for recent more comprehensible treatments. Depending on when realizations of uncertain data are known and on how this interacts with decision making over time, there are different model frameworks in stochastic programming, e.g., one-stage, two-stage, or multi-stage models. Selection and placement (in the objective or the constraints) of the statistical parameters according to which relevant random variables are to be evaluated is another important issue. This allows to express perceptions such as reliability, risk neutrality, or risk aversion. Finally, the nature of the initial uncertain optimization problem (linear or nonlinear, with or without integer variables) has crucial impact on the resulting stochastic programming model. These aspects lead to a wide variety of models and mathematical techniques for their analysis and algorithmic treatment.

In the talk, we present some stochastic programming models for decision making in power production and trading. Our aim is to explore the borderline of computational tractability for different model setups, all related to two-stage decision schemes. Models live on finite probability spaces which allows for equivalent representation as large-scale, block-structured, mixed-integer linear programs. These problems quickly exceed the sizes that can be handled by off-the-shelf solvers. Tailored decomposition methods then provide practicable algorithmic alternatives.

Motivated by decision problems in cost optimal operation of dispersed energy systems, we consider two-stage mean-risk stochastic integer programs involving different risk measures. Another option for risk aversion is to employ concepts of stochastic dominance to compare relevant random variables with random benchmarks. This is illustrated at an electricity retailer problem, [CGS09].

Two-stage stochastic programs are specific bi-level (leader-follower) optimization problems where the optimal value rather than an optimal solution of the follower problem is passed to the leader, [PW99]. Research on stochastic bi-level programs with optimal follower solutions being passed to the leader still is in its beginning. Following [RC09], we discuss a stochastic bi-level model for the determination of optimal offering strategies for a strategic price-maker power producer trading electric energy in a day-ahead pool market. Inspired by this application, we present a model framework and point to some open questions in algorithm design for bi-level stochastic programs.

References

Abstract

Equilibrium models in the gas and electricity industries

The last twenty years saw the move of the gas and electricity industries from the vertically integrated monopoly structure to a more market-oriented organization. This transition was accompanied by a huge development of models aimed at better understanding the evolutions of these industries in their new environment. We survey these developments, categorize existing models and emphasize some major difficulties remaining with these instruments.

Physical laws drive the flows of gas and electricity. We recall these laws in the introduction of the presentation and note at the outset that they play a fundamental role in electricity market models while they are generally overlooked in gas market models.

The second part of the talk deals with gas models. We identify three different levels of complexity. The conceptually simplest models assume perfectly competitive production and transmission markets. This has led to gas trade models particularly well adapted to the US situation but also to extensions covering world trade. A second class of models, more concentrated on the European gas market, aims at representing the market power of the producers and/or merchant companies while assuming a perfect transmission market. Last we mention some of the problems caused by the introduction of transmission and contractual idiosyncrasies in gas trade models. To the best of our knowledge these are not taken on board in existing models.

The third part of the talk deals with electricity markets. We again begin by distinguishing models that deal with market power and those that assume that agents are price takers. We first analyze this latter class and immediately separate the sophisticated market designs currently implemented in the US and the so-called market coupling that is currently the most advanced realization in Europe. The US market designs offer an integrated set of different submarkets implemented around sophisticated optimization software. These markets are generally considered to be competitive in market reports. In contrast the European electricity market is still organized in a rather primitive way (even though market coupling has been a significant advance). It is generally considered as poorly competitive by EU authorities and large industrial consumers. Market-coupling also reveal bizarre computational and economic properties that we discuss. We next discuss models that aim at representing market power both at the sole energy level or both at the interaction of energy and transmission. We conclude by illustrating the strangeness of applying oligopoly theory to these markets.

Conclusion try to come up with an agenda of what should be done in order to improve the usefulness of these models both in gas and electricity.

Abstract

Management of gas storage

In this talk I will review various scenarios for the operation of gas storage facilities, and the challenges these pose for modelling.

Storage facilities vary widely in their operational characteristics, their connectivity, and the prevailing regulatory environment. Moreover, those holding and operating storage facilities do so from a range of possible motivations. All of these aspects affect both the value (to the owner) and optimal operation of a facility, and have implications for the task of determining these through modelling. I will review of current modelling approaches, and some recent developments, and discuss some of the challenges related to the problem of optimal management of gas storage that future modelling developments may help to meet.

Abstract

Dynamic Pricing

We review the literature and current research on dynamic pricing as part of the growing field of revenue management. The practice of revenue management originated almost 40 years ago in the airline industry but has ever since expanded to many other industries, notably service and retailing industries. In the classical formulation of a dynamic pricing problem, a finite number of a perishable product is sold over a finite selling horizon where price-sensitive customers arrive over time following a stochastic process. A customer makes a purchase as long as the current price is lower than her willingness-to-pay. In such a model, the benefit of dynamic pricing stems from its ability to adjust prices to account for variabilities in the arrival process. The literature on dynamic pricing has been growing along several dimensions. One important extension is the consideration of multiple products with dependent demand. When multiple substitutable products are sold together, the price of one product affects the demand of all related products, resulting in complex dynamics in the model and much harder optimization problems. Much of the existing research focuses on heuristic solution approaches for the problem. Another important aspect is parameter uncertainty and demand learning. Sellers of seasonable, short life-cycle products often face significant uncertainty with respect to demand and price sensitivity parameters. There are at least two reasonable strategies to handle the uncertainty. In a learning approach, a seller can dynamically update the parameters as sales unfold. The complicating factor is that the seller needs to take into account the effect of pricing on demand learning. A related approach is robust pricing that takes into account parameter uncertainty in pricing optimization. Recent years also witness significant growth in consumer behavior modeling in the dynamic pricing literature. Following advances in economics and marketing, this stream of literature devotes to behavioral and psychological considerations in dynamic pricing. Examples include reference price effect where consumer expectation is affected by the price history and product display format effect where consumer purchase decision depends on the available assortment. Finally, we briefly discuss future research directions.