SCIENTIFIC PROGRAMS AND ACTIVITIES

April 23, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

Focus Program on Commodities, Energy and Environmental Finance

August 27-29, 2013 (Tues-Thurs)
Workshop on Stochastic Games, Equilibrium, and Applications to Energy & Commodities Markets
Organizing Committee: René Carmona, Ronnie Sircar

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OVERVIEW
The workshop will address the recent developments in stochastic games in the context of commodity markets. Relevant challenges include understanding of the oligopolistic and game theoretic effects in energy production, carbon emission trading schemes, climate change mitigation and environmental risk transfer. New mathematical tools such as mean- field games, fully nonlinear stochastic differential games and backward-forward stochastic dfferential equations are being developed for these tasks. The workshop will present the latest state-of-the-art and explore outstanding problems.

Preliminary Schedule

August 27, Tuesday
1:15-1:30
Opening Remarks
1:30 - 2:15
Mike Ludkovski, UC Santa Barbara (slides)
Strategic R&D in Cournot Markets
2:15 - 3:00

Mireille Bossy, INRIA (slides)
Game-theory Approach for Electricity and Carbon Allowances: a markets coupling study

3:00 - 3:30
Coffee Break
3:30 - 4:15
Talat Genc, University of Guelph (slides)
Power trade, welfare,and air quality
4:15 - 4:40
Daniel Lacker, Princeton University (slides)
A probabilistic weak formulation of mean field games and applications

4:40 - 5:05

Nina Lange, Copenhagen Business School (slides)
The correlation structure of exchange rates and commodity prices

5:15

Program Reception at the Fields
August 28, Wednesday
9:00 - 9:45
Minyi Huang, Carleton University (slides)
Mean field Consumption-Accumulation Games with Congestion

9:45 - 10:30
François Delarue, Université de Nice Sophia-Antipolis (slides)
Mean Field Games with a Common Noise
11:00 - 11:45
Ronnie Sircar, Princeton University (slides)
Energy Production and Differential Games
11:45 - 12:10
Emmanuel Leclercq, Swiss Finance Institute @ EPFL (slides)
Equilibrium commodity trading
12:10 - 1:30
Lunch Break
1:30 - 2:15
Frank Wolak, Stanford University (slides)
Measuring the Competitiveness Benefits of Transmission Investments in Wholesale Market with Locational Pricing: The Case of the Australian Electricity Market

2:15 - 3:00
Daniel Schwarz, Carnegie Mellon University
Price Modelling in Carbon Emission and Electricity Markets
3:00 - 3:30
Coffee Break
3:30 - 3:55
Xuwei Yang, University of California, Santa Barbara (slides)
Dynamic Cournot Models for Production of Exhaustible Commodities under Stochastic Demand

4:00 - 4:25

Bilkan Erkmen, Two Sigma Investments (slides)
Generalized Multi-Factor Commodity Spot Price Modeling through Dynamic Cournot Resource Extraction Models
4:30 - 5:30
Panel Discussion: Energy Markets and the Environment: could economics and mathematics team up for a better future?
Moderator: Rene Carmona, Princeton University
Panelists:
Matheus Grasselli, Fields Institute
Hans Tuenter, Ontario Power Generation
Frank Wolak, Stanford University

7:00

Program Dinner at Frank
August 29, Thursday
9:00 - 9:45
Sebastian Jaimungal, University of Toronto
Incorporating Cash-Flow Distributions into Real Options: Risk and Ambiguity
9:45 - 10:10
Shilei Niu, University of Waterloo (slides)
An Options Pricing Approach to Ramping Rate Restrictions at Hydro Power Plants
10:10 - 10:35
Nicolas Langrené, Université Paris Diderot (slides)
A numerical algorithm for general HJB equations: a jump-constrained BSDE approach
10:35 - 11:00
Coffee Break
11:00 - 11:.45
Antony Ware, University of Calgary
Modelling shared gas storage facilities

11:45 - 12:30
Ulrich Horst, Humboldt University (slides)
Smooth solutions to portfolio liquidation problems under price-sensitive market impact


Speaker Title and Abstract
Bossy, Mireille
INRIA
Game-theory Approach for Electricity and Carbon Allowances: a markets coupling study

We address the problem of electricity producers interacting between an electricity market and an emissions trading scheme that is fostered by a carbon penalty.

We propose to investigate, through a game-theory approach, the efficiency of such mitigation policies, in terms of electricity production and carbon emission.For this, we will use a (deterministic) two steps game approach, addressing each market, in order to assess the electricity price, the carbon allowance price and the related electricity/carbon production. The description of players strategies imposes to model the structures of the two markets (an equilibrium, if there exits one, will depend on those market designs).

We prove that a Nash equilibrium exists for non cooperative players that strive to maximize their market shares. The players face an elastic electricity demand, coupled to an auction type carbon allowance market.

Delarue, François
Université de Nice Sophia-Antipolis

Mean Field Games with a Common Noise

Motivated by several examples in finance and environmental finance, I will discuss a probabilistic analysis for describing equilibriums in large population of controlled players, when subject to correlated noises. As I will explain, correlation of the noise raises several non trivial difficulties in comparison with simpler models where players are assumed to be driven by independent noises. This is a joint work with René Carmona.
Bilkan Erkmen
Two Sigma Investments

Generalized Multi-Factor Commodity Spot Price Modeling through Dynamic Cournot Resource Extraction Models

We will be presenting an analytically tractable continuous-time model for the optimal extraction of a common property non-renewable resource in a 2-factor stochastic Cournot oligopoly with supply and demand uncertainties. The demand uncertainty is represented as a mean-reverting shock on the inverse demand function while the supply uncertainty is modeled as a jump-diffusion process for the remaining resource stock. We solve the model for a symmetric Markov perfect Nash equilibrium and show that the SDE characterizing the evolution of the spot price of the resource is a generalization of the 2-factor commodity spot price model proposed by Eduardo Schwartz and James Smith in their 2000 paper “Short-Term Variations and Long-Term Dynamics in Commodity Prices”. This is work in progress joint with Michael Coulon (University of Sussex).
Talat Genc
University of Guelph

Power trade, welfare,and air quality

We use detailed data from all generators in a major wholesale electricity market to investigate cross-border electricity trade and its impact on air emissions and social welfare. Using the technical characteristics of the generators and financial data we run a competition model every hour and find that the model generates actual prices and outputs with 94.4% and 96% accuracy, respectively. We show that there is a significant welfare gain from power trade. The air emissions savings are also considerable. For instance, when hourly imports double from current levels CO2 emissions decrease around 13%, and market prices reduce 5.4%. In autarky, CO2, SO2, NOx emissions increase 12%, 22%, 16%, resp., the prices go up 5.8%, and the price volatility rises 12%. However, the impact of negative wholesale prices on market outcomes is small.
Huang, Minyi
Carleton University

Mean field Consumption-Accumulation Games with Congestion

This work presents a neoclassical stochastic growth model with (i) a large number of agents and (ii) a congestion effect in the production dynamics due to the production activity of all agents. We formulate a mean field game with finite horizon and HARA utility. Decentralized strategies are constructed by consistent mean field approximations.

We also discuss the long term behavior of the game which in certain situations gives rise to significant challenges: the mean field generated by self-optimizing individuals exhibits oscillatory or even chaotic behavior. This raises the question whether it is possible at all in such cases to forecast the mean field behavior for the purpose of game theoretic optimization. Some further generalizations of the model will be presented.

Horst, Ulrich
Humboldt University

Smooth solutions to portfolio liquidation problems under price-sensitive market impact

We establish existence and uniqueness of a classical solution to a semilinear parabolic partial differential equation with singular initial condition. This equation describes the value function of the control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price dependent impact functions describing the trading costs in the primary market and price dependent adverse selection costs associated with dark pool trading. We establish the explicit asymptotic behavior of the value function at the terminal time and give the optimal trading strategy in feedback form.

The talk is based on joint work with Paulwin Graewe and Eric Sere.

Sebastian Jaimungal
University of Toronto

Incorporating Cash-Flow Distributions into Real Options: Risk and Ambiguity

The adoption of real options analysis by industry practitioners remains limited. Managers tend to have a very simplified and condensed view of what potential cash-flows might be. Approximating cash-flows and/or project values as GBMs or OUs is standard in most academic works, yet it is difficult to sell such models to practitioners since there is a disconnect to their views. Here, we provide a way to incorporate the manager's cash-flow distributions exactly, within an incomplete market setting, using indifference pricing. We will focus on two prototypical options (i) irreversible investment, and (ii) an entry-exit problem, but our approach is quite general. Moreover, we will outline how managers can account for model ambiguity using robust optimization and explore its implications.
[This is joint work with Yuri Lawryshyn]
Lacker, Daniel
Princeton University

A probabilistic weak formulation of mean field games and applications

Mean field games are studied by means of the weak formulation of stochastic optimal control. This approach allows the mean field interactions to enter through both state and control processes and take a form which is general enough to include rank and nearest-neighbor effects. Moreover, the data may depend discontinuously on the state variable, and more generally its entire history. Existence and uniqueness results are proven, along with a procedure for identifying and constructing distributed strategies which provide approximate Nash equilibria for finite-player games. Our results are applied to a new class of multi-agent price impact models and a class of flocking models. This is a joint work with René Carmona.

Lange, Nina
Copenhagen Business School

The correlation structure of exchange rates and commodity prices

An investor in commodity markets are often faced with both price risk and currency risk, as the commodity is often traded in a different currency than the investor’s own. Often, the news report that the commodity prices and the USD/EUR rate move in opposite directions, indicating that the currency risk and price risk offsets eachother for a Euro denominated investor. In this paper, I investigate if this is in fact the case and to which extent the correlation connects to the volatility of the commodity price and exchange rates. The paper introduces a model which allows for stochastic correlation of both signs and models the futures price curves and option prices in a model. The model is estimated using data on WTI crude oil and EURUSD contracts traded at the Chicago Merchantile Exchange from 1998 to 2013.

Langrené, Nicolas
Université Paris Diderot

A numerical algorithm for general HJB equations: a jump-constrained BSDE approach

In this talk, a probabilistic numerical scheme for solving Backward Stochastic Differential Equations (BSDEs) with constrained jumps will be presented. The class of jump-constrained BSDEs is a generalization of BSDEs that was proposed by Kharroubi and Pham (2012). In particular, the interest of this class is that it is general enough to encompass stochastic control problems such that the volatility of the underlying state variable is controlled. A typical example where such a controlled volatility occurs is the problem of super-replication of a contingent claim under uncertain volatility. Hence, after a partial analysis of the error of the scheme, numerical illustrations of the behavior of our scheme on this problem of pricing under uncertain volatility and/or correlation will be provided, including a comparison with an alternative scheme based on second-order BSDEs proposed by Guyon and Henry-Labordère (2011). The methodology presented here can be naturally extended to stochastic games and HJB-Isaacs equations. This talk is based on a joint work with Huyên Pham (Paris Diderot) and Idris Kharroubi (Paris Dauphine)
Emmanuel Leclercq
Swiss Finance Institute @ EPFL

Equilibrium commodity trading

We develop an equilibrium model of commodity spot and futures markets in which commodity production, consumption, and speculation are endogenously determined. Speculators facilitate hedging by the commodity suppliers. The entry of new speculators thus increases the supply of the commodity and lowers expected spot prices, to the benefits of the end-users. However, this entry may be detrimental to the producers as they do not internalize the price reduction due to greater aggregate supply. In the presence of asymmetric information, speculation on the futures market serves as a learning device. The futures prices and open interests reveal distinct information regarding the supply and demand side of the spot market, respectively. When the accuracy of private information is low, the entry of new speculators makes both production and spot prices more volatile. The entry of new speculators typically increases the correlation between financial and commodity markets.
Ludkovski, Mike
UC Santa Barbara

Strategic R&D in Cournot Markets

We explore optimal investment in Research and Development activities in a competitive market. R&D effort is costly and results in discrete technological advances that gradually lower production costs. Using a Cournot model of competition with substitutable goods (e.g. markets for different energy commodities) we analyze the respective Markov Nash equilibrium strategies for production and R&D effort. Our model represents stages of technological progress by a controlled multi-dimensional counting process. The solution approach is then to study the sequence of the one-step static games arising between R&D successes. We present several numerical examples and some preliminary analysis of the emerging comparative statics. This is work in progress joint with Ronnie Sircar (Princeton).
Niu, Shilei
University of Waterloo

An Options Pricing Approach to Ramping Rate Restrictions at Hydro Power Plants

This paper uses a real options approach to examine the impact of ramping rate restrictions imposed on hydro operations to protect aquatic ecosystems. We consider the effect on profits from electricity generation in order to inform policy decisions about ramping rate restrictions. A novelty of the paper is in examining the optimal operation of a prototype hydro power plant with electricity prices modelled as a regime switching process and comparing with the results from a single regime model. We show that profits are negatively affected by ramping restrictions in both models, however profits are less sensitive in the regime switching model. Interestingly for a large range of restrictions in both models, profit is not sensitive to ramping restrictions. The results point to the importance of accurately modelling electricity prices in gauging the trade offs involved in imposing restrictions on hydro operators which may hinder their ability respond to volatile electricity prices and meet peak demands.
Schwarz, Daniel
Carnige Mellon

Price Modelling in Carbon Emission and Electricity Markets

We present a model to explain the joint dynamics of the prices of electricity and carbon emission allowance certificates as a function of exogenously given fuel prices and power demand. The model for the electricity price consists of an explicit construction of the electricity supply curve; the model for the allowance price takes the form of a coupled forward-backward stochastic differential equation (FBSDE) with random coefficients. Reflecting typical properties of emissions trading schemes the terminal condition of this FBSDE exhibits a gradient singularity. Appealing to compactness arguments we prove the existence of a unique solution to this equation.
We illustrate the relevance of the model at the example of pricing clean spread options, contracts that are frequently used to value power plants in the spirit of real option theory.
Ronnie Sircar
Princeton University

Energy Production and Differential Games

One way to view energy markets is as competition between producers from different fuels and technologies with markedly varied characteristics. For instance, oil is relatively cheap to extract, but in diminishing supply and polluting. Solar power is more expensive to set up, but essentially inexhaustible and clean. We construct dynamic oligopoly models of competition between heterogeneous energy producers to try and understand how the changing landscape may affect energy prices and supply. Key issues are exhaustibility of fossil fuels, exploration and discovery of new sources such as shale oil and technologies such as fracking, and differing costs of production among the various sources. This involves studying non-zero sum (stochastic) differential games of Cournot and Bertrand-type, for which we develop asymptotic approximations and numerical methods, and explicit results in some cases.
Ware, Antony
University of Calgary

Modelling shared gas storage facilities

Natural gas storage facilities are sometimes leased to third parties who are thus given the rights to manage some fraction of the storage capacity. The maximum injection/withdrawal rates for each participant at any given point in time depend on the overall pressure in the facility.

We describe a sub-timestep splitting method for storage valuation [1], and use it to examine the strategic actions of the participants and their impact on the efficient utilization of the storage facility.

[1] A. Ware. Accurate semi-Lagrangian time stepping for stochastic optimal control problems with application to the valuation of natural gas storage. SIAM Journal on Financial Mathematics, 4(1):427–451, 2013.

Wolak, Frank
Stanford University

Measuring the Competitiveness Benefits of Transmission Investments in Wholesale Market with Locational Pricing: The Case of the Australian Electricity Market

Transmission expansions can increase the amount of competition faced by wholesale electricity suppliers with the ability to exercise unilateral market. This perceived increase in competition faced by these strategic suppliers causes them to behave more aggressively and set market-clearing prices closer to competitive benchmark levels.
These lower wholesale prices are the competitiveness benefit of a transmission expansion to electricity consumers. This paper quantifies empirically for an actual wholesale electricity market the competitiveness benefits of a transmission expansion policy that causes strategic suppliers to perceive a smaller frequency and duration of transmission constraints to limit the competition they face for in wholesale market with locational pricing. Using half-hourly generation-unit level offer, output, market-clearing price and congestion data from the Australian Electricity Market from January 1, 2008 to December 31, 2011, this paper builds on the expected profit-maximizing offer model in Wolak (2003 and 2007) and best-reply offer pricing model in McRae and Wolak (2012) to compute two counterfactual reduced congestion half-hourly market prices that are used to compute an estimate of the competitiveness benefits of a proposed transmission expansion. The empirical results find that the competitiveness benefits are a significant fraction, roughly half, of the overall consumer benefits of the transmission expansion. The empirical results argue in favor including competitiveness benefits in all transmission planning processes in wholesale markets in order to ensure that all transmission expansions with positive net benefits to electricity consumers are undertaken.
Xuwei Yang
University of California, Santa Barbara

Dynamic Cournot Models for Production of Exhaustible Commodities under Stochastic Demand

We extend the dynamic Cournot model of Ludkovski and Sircar (2011) by considering stochastic demand which switches between high and low regimes with exogenously given holding rates. We first consider the case of an exhaustible production monopolist who makes decision on the production rates and exploration efforts depending on the reserves amount and demand regime. We study how the two regimes and the corresponding holding rates influence the monopolist’s production rates and exploration efforts. We then consider a stochastic game between such an exhaustible producer and a ”green” producer. We study the two producers’ production rates and exhaustible producers exploration e orts under stochastic demand, and compare with the case of deterministic demand. A novel feature driven by stochasticity of demand is that producers may shut down production during low demand to conserve reserves.


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