Talk Titles and Abstracts
René Aid (EDF, Université de Paris 9 Dauphine)
A structural risk-neutral model for electricity prices
The objective of this paper is to present a model for electricity
spot prices and the corresponding forward contracts, which relies
on the underlying market of fuels, thus avoiding the electricity
non-storability restriction. The structural aspect of our model
comes from the fact that the electricity spot prices depend on
the dynamics of the electricity demand at the maturity T, and
on the random available capacity of each production means. Our
model explains, in a stylized fact, how the prices of different
fuels together with the demand combine to produce electricity
prices. This modeling methodology allows one to transfer to electricity
prices the risk-neutral probabilities of the market of fuels and
under the hypothesis of independence between demand and outages
on one hand, and prices of fuels on the other hand, it provides
a regression-type relation between electricity forward prices
and forward prices of fuels. Moreover, the model produces, by
nature, the well-known peaks observed on electricity market data.
In our model, spikes occur when the producer has to switch from
one technology to the lowest cost available one. Numerical tests
performed on a very crude approximation of the French electricity
market using only two fuels (gas and oil) provide an illustration
of the potential interest of this model. This is a joint work
with Luciano Campi, Adrien Nguyen Huu and Nizar Touzi.
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Alvaro Cartea (Universidad Carlos III & Madrid)
Mean reversion, measure changes and stochastic risk premia in
commodity markets
The stylised facts of the behaviour of spot prices of commodities
have been extensively studied. The main two features are: the
presence of a seasonal trend; and that deviations from this trend,
for instance jumps, tend to mean revert in the short-term. But
what are the stylised facts under the risk-neutral measure? In
this paper we set out to answer this question. We analyse how
risk averse investors adjust the statistical measure of price
dynamics when pricing risky securities written on commodities.
Spot (or day-ahead) price dynamics are much easier to model than
dynamics under the risk-neutral measure because the former are
observed whereas the latter can only be inferred from the price
dynamics of instruments written on the spot commodity; for instance
forward contracts. We model the market price of risk so that market
participants bearing spot commodity risk are compensated for:
jump arrival risk; jump size risk; and speed of mean reversion
risk of both diffusion and/or jumps. Our approach can also be
viewed as a special case of stochastic discount factors that not
only affect the mean of the process but also its variance via
the persistence of shocks to the economy.
We consider three models: the Schwartz (1997) pure diffusion
model, the short- and long-term two-factor model of Schwartz and
Smith (2000) and a three-factor arithmetic model that includes
positive and negative jumps in the prices of commodities. We show
that when spot dynamics exhibit mean reversion to a seasonal trend
market participants are averse to deviations from this seasonal
trend. Consequently, when pricing under the risk-adjusted measure
agents will: over-state the time it takes to return to the seasonal
trend; alter the mean of the process; and change the intensity
of the jumps and their average size.
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Michael Coulon (Princeton University)
The Electricity Bid Stack: Linking the dynamics of fuel, power
and carbon prices
Evidence from the 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.
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Alex Eydeland (Morgan Stanley)
Commodity Modeling: View from the trenches
In this presentation we will present various issues and challenges
facing commodity quants and suggest a number of modeling methodologies
designed to address these issues. We will also give a brief introduction
of standard commodity structures as well as new products and recent
developments in commodity markets.
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Gilles Edouard Espinosa (Ecole Polytechnique)
A model of emissions and the price of carbon (joint with R. Carmona
and N. Touzi)
Since the Kyoto Protocol, the carbon tax has aroused such a large
interest from economists, financial analysts as well as mathematicians
because its modelling is not classical. In this work, we propose
a model consisting of n producers that have the ability to reduce
their carbon emissions in return for a certain cost. We assume
that they can also trade on the certificates market. Our aim is
to determine the price of the carbon certificates, which will
lead us to a forward-backward SDE with some irregularity. Using
a method of approximation, we are able to show the existence and
uniqueness of a solution. As expected, we find that with the introduction
of such a tax, the emissions decrease.
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Max Fehr (London School of Economics)
Option Pricing in the European Unions Emission Trading Scheme
We propose a model for risk neutral futures price dynamics in
the European Unions Emissions Trading Scheme (EU ETS). Historical
price dynamics suggests that both allowance prices for different
compliance periods and CER prices for different compliance periods
are significantly related. To obtain a realistic price dynamics
we take into account the specific details of the EU ETS compliance
regulations, such as banking and the link to the Clean Development
Mechanism (CDM), and exploit arbitrage relationships between futures
on EU allowances and Certified Emission Reductions.
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Helyette Geman (Birbeck and ESCP Europ)
Inventory, Commodity Forward Curve and Spot price Volatility:
The case of Crude Oil and Natural Gas
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 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. The goal
of our talk is threefold: i) review some interesting features
of energy commodity forward curves 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 major commodities the relationship
between inventory and price volatility. In agreement with the
theory of storage, we find 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
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Georg Gruell (Duisburg-Essen)
Pricing CO2 permits using approximation approaches
Equilibrium models have been widely used in literature with the
aim of showing theoretical properties of emission trading systems.
First, a new equilibrium model is derived. Second, it is shown
that the theoretical permit price is related to changes in the
expectation of how long regulated companies will need to exhaust
the remaining permits. Third, by application to real data we demonstrate
that emission trading systems are inherently prone to jumps.
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Sebastian Jaimungal (University of Toronto)
A New Approach to Commodity Market Models
Market models and so-called string models for commodities have
been widely used in industry. These approaches exactly match the
current forward price term structure and traditionally model forward
prices as multivariate GBMs. Here, I introduce a new approach
where I model the discrete forward cost-of-carry directly rather
than forward prices themselves. Through a sequence of measure
changes, I demonstrate how to calibrate calendar spread options,
options on forwards and forward prices all at once. Furthermore,
I explain how unspanned stochastic volatility and jumps can be
easily added into the modeling framework.
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Walid Mnif (University of Western Ontario)
Pricing and Hedging Strategies for Contingent Claims in an Incomplete
Hybrid Emissions Market
We propose a stochastic approach for trading and pricing emission
permits under an incomplete hybrid system in which credits are
partially fungible from one period to another. Our approach is
based on the filtering theory proposed by Follmer and Schweizer
(1991). Under our framework exotic options can easily be priced
with the use of straightforward derivatives pricing algorithms
such as Monte Carlo simulation.
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Ehud Ronn (Morgan Stanley & Co.)
The Valuation and Information Content of Options on Crude-Oil
Futures Contracts
Using market prices for crude-oil futures options and the prices
of their underlying futures contracts, we estimate the volatility
skew in two ways. As a benchmark for our theoretical model, on
each date we first estimate a cross-sectional polynomial structure
for each maturity to demonstrate the strength and weaknesses of
a purely-mechanical model. We then apply to the empirical data
a Merton-style jump-diffusion model, with a rich structure of
cross-sectional constraints on the parameters. Both models are
tested with respect to their mark-to-market accuracy over time,
as well as their efficacy in hedging intertemporal option price
changes. The postulated Merton-style model is shown to yield useful
parameters from which market prices can be computed, option prices
can be marked-to-market and (imperfectly) hedged, as well as an
informationally-rich structure covering the time period of the
turbulent past six months.
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Marliese Uhrig-Homburg (Universität Karlsruhe)
Understanding the Price Dynamics of Emission Permits: A Model
for Multiple Trading Periods
Emission Trading Systems, like the EU Emission Trading System
(EU ETS), are currently established in many different countries
all over the world. Those are characterized by different regulatory
rules which have shown to be appropriate in the recent past. The
regulatory framework clearly impacts the characteristics of the
spot price dynamics of such emission permits, a dependency not
yet fully understood. In fact, there exists no theoretical model
accounting for the main stylized facts like the existence of multiple
trading periods, the allowance of banking, and the later delivery
of lacking certificates, although insights into this subject are
worthy for risk management, derivative pricing, and energy-related
investment decisions. Therefore, we develop a dynamic stochastic
equilibrium model for multiple trading periods, which makes it
possible to account for those features. We present a solution
for the equilibrium spot price and analyze the spot price dynamics
and volatilities as well as sensitivities towards market design
parameters. Moreover, we can show that the spot price can be subdivided
into different ratios, each one resulting from one trading period
of a particular setting.
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