
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

Focus
Program on Commodities, Energy and Environmental Finance
August
2729, 2013 (TuesThurs)
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 meanfield games, fully nonlinear
stochastic differential games and backwardforward stochastic dfferential
equations are being developed for these tasks. The workshop will present the
latest stateoftheart and explore outstanding problems.
Preliminary Schedule
August 27, Tuesday

1:151:30

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)
Gametheory 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 ConsumptionAccumulation Games with Congestion

9:45  10:30

François Delarue, Université
de Nice SophiaAntipolis (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 MultiFactor 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 CashFlow 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
jumpconstrained 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 pricesensitive market impact

Speaker 
Title and Abstract 
Bossy, Mireille
INRIA 
Gametheory 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 gametheory 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 SophiaAntipolis 
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 MultiFactor Commodity Spot Price Modeling through
Dynamic Cournot Resource Extraction Models
We will be presenting an analytically tractable continuoustime
model for the optimal extraction of a common property nonrenewable
resource in a 2factor stochastic Cournot oligopoly with supply and
demand uncertainties. The demand uncertainty is represented as a meanreverting
shock on the inverse demand function while the supply uncertainty is
modeled as a jumpdiffusion 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 2factor commodity spot price
model proposed by Eduardo Schwartz and James Smith in their 2000 paper
“ShortTerm Variations and LongTerm 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 crossborder 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 ConsumptionAccumulation 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 selfoptimizing 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 pricesensitive
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 CashFlow 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 cashflows might be. Approximating cashflows
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 cashflow distributions exactly, within an incomplete
market setting, using indifference pricing. We will focus on two prototypical
options (i) irreversible investment, and (ii) an entryexit 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 nearestneighbor 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 finiteplayer
games. Our results are applied to a new class of multiagent 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 jumpconstrained
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 jumpconstrained 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 superreplication 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 secondorder BSDEs proposed by Guyon and HenryLabordère
(2011). The methodology presented here can be naturally extended to
stochastic games and HJBIsaacs 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
endusers. 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
multidimensional counting process. The solution approach is then to
study the sequence of the onestep 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 forwardbackward 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 nonzero
sum (stochastic) differential games of Cournot and Bertrandtype, 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 subtimestep 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 semiLagrangian 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
marketclearing 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 halfhourly generationunit level offer, output, marketclearing
price and congestion data from the Australian Electricity Market from
January 1, 2008 to December 31, 2011, this paper builds on the expected
profitmaximizing offer model in Wolak (2003 and 2007) and bestreply
offer pricing model in McRae and Wolak (2012) to compute two counterfactual
reduced congestion halfhourly 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 eorts 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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