New Directions and Products in Energy Markets: Oil, Electricity, Carbon

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New Directions and Products in Energy Markets
Oil, Electricity, Carbon

• Hélyette Geman
• Professor of Finance
• Birkbeck, University of London ESSEC Business
  School
• To be presented at the Europlace Finance
  Conference -
• Paris - June 22, 2006

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Growth of 100 1991-1999
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Growth of 100 2000-2004
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Is Mean-Reversion Dead ?
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Mean-Reversion versus "Random Walk"Representation
in Oil and Natural Gas Prices

• The objective is to check whether, in the
  representation
• the coefficient r is significantly different
  from 1
• The H0 hypothesis is the existence of a unit root
  (i.e., r 1)
• A p-value smaller than 0.05 allows one to reject
  the H0 hypothesis with a confidence level higher
  than 0.95, in which case the process is of the
  mean-reverting type
• Otherwise, a unit root is uncovered and the
  process is of the "random walk" type
• The higher the p-value, the more the random walk
  model is validated

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Mean-Reversion Properties for Oil and Natural
Gas Prices(Geman (2005) J. of Alternative
Investments)

• For crude oil
• a mean-reversion pattern prevails over the period
  1994-2000
• it changes into a random walk (arithmetic
  Brownian motion) as of 2000
• For natural gas
• there is a mean-reversion pattern until 2001
• since 2002, a change into a random walk accurs
• during both periods, seasonality of gas prices
  tends to blur the signals

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US Natural Gas Prices over the periodJanuary
1994 - October 2004

• Spot prices are proxied by the Nymex one month
  Futures contract
• Over the period Jan 94 - Oct 04
• ADF p-value 0.712
• Phillips Perron p-value 0.1402
• Over the period Jan 1999 - Oct 2004
• ADF p-value 0.3567
• Phillips Perron p-value 0.3899

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• Taking instead log-prices, we obtain
• Over the last five years of the period, the
  arithmetic Brownian motion assumption clearly
  prevails

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WTI Spot Prices over the period January 1994 -
October 2004

• Again, spot prices are proxied by Nymex one-month
  Futures prices
• Whole period 1994-2004
• Augmented Dickey Fuller 0.651
• Phillips Perron 0.5048
• Period January 1999 - October 2004
• Augmented Dickey Fuller 0.7196
• Phillips Perron 0.5641
• The mean-reversion assumption is strongly
  rejected over the whole period and even more so
  over the recent one
• Because of absence of seasonality, the property
  is more pronounced than in the case of natural gas

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The Literature on Mean-Reversion in Commodity
Prices

• Bessembinder, Coughenour, Seguin and Smoller
  (JOF, 1995) test the term structure of Futures
  prices over the period January 1982 to December
  1991 and find mean-reversion in the 11 markets
  they examine. They also conclude that the
  magnitude of mean-reversion is large for
  agricultural commodities and crude oil, and
  substantially less for metals.
• Rather than examining evidence of ex post
  reversion using time series of asset prices, they
  use price data from futures contracts with
  various horizons to test whether investors expect
  prices to revert. The authors analyze the
  relation between price levels and the slope of
  the futures term structure an inverse relation
  between prices and this slope constitutes
  evidence that investors expect mean reversion in
  spot prices, as it implies a lower rate of
  expected intertemporal price appreciation when
  prices rise

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• Pindyck (1999) analyses 127 years of data (period
  1870-1996) on crude oil and bituminous coal,
  obtained from the US Department of Commerce
• Using a unit root test, he exhibits that prices
  mean revert to stochastically fluctuating trend
  lines these lines, which represent long-run
  total marginal costs, are themselves unobservable
• Pindyck finds that during the time period of
  analysis, the random walk distribution for
  log-prices, i.e., the geometric Brownian motion
  for spot prices, is a much better approximation
  for coal and gas than oil
• The recent period (2000-2006) has been quite
  different!

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Other Relevant Models forElectricity Price
Processes

• A three-factor mean reverting process with
  stochastic volatility
• where the three Brownian motions may be
  correlated

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Using Pure Jump Lévy Processes for Commodity
Prices

• Prices moves are represented as a succession of
  jumps many small jumps, a few big jumps upward
  or downward
• The Lévy density k(x) translates the arrival
  intensity of jumps of size x in a unit time
  interval
• The CGMY process
• (Carr-Geman-Madan-Yor, Journal of Business 2002)
• It is successfully implemented in many financial
  institutions. Its Lévy density k is defined by

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Modelling the Electricity Price Processthe
Markov property

• One way of generating the downward part of a
  spike is to introduce a non-Markovian process
  the spot price needs to "remember" its past
  values as well as the current value to know that
  if should be going down (with a high probability)
• Nearly all option pricing models in finance
  involve Markovian processes
• Another way of handling the representation of the
  spike is to introduce in the dynamics of the
  price process a component which triggers a
  downward jump (in probability) only on the basis
  of the current spot price and preserve the Markov
  property!

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Checking that a Model forPower Spot Price is
Acceptable

• Realize that going from 4500 to 70 is
  definitely a jump downward.
• Hence, the model should allow for positive and
  negative jumps
• Generate with the model a variety of trajectories
  and check that a least some of them look like
  real trajectories
• Trajectorial Adequacy of the model
• Compute the first 4 moments of the calibrated
  model and of the real trajectory and verify that
  they are similar
• Statistical Adequacy of the model
• For instance, introducing only upward jumps
  generates a very highly positive skewness which
  is not observed in practice

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A Jump-Reversion ModelGeman - Roncoroni, J. of
Business (2006)

• A/ Empirical observations
• Trajectories Descriptive statistics
• Class of models to account for heterogeneity
  across market
• B/ Modeling by marked point processes
• C/ Calibration
• "Structural" elements
• Parameters
• Path Properties
• Statistics

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MAIN
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Logarithmic
Prices
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A Power Plant has the Option to transform Fuel
into Power
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Valuation of Physical Assets in the Energy
Industry

• Scenario A
• Gas 3.5 /MMbtu
• Power 40 /MWh
• Heat Rate 10 MMbtu/MWh
• Profit 40 - 3.5 x 10 5
• Scenario b
• Gas 4.5 /MMbtu
• Do not operate
• The ownership of the physical assets amounts to
  a series of options over the lifetime of the
  plant

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Pricing the Exchange Option

• It is the option to exchange at time T one asset
  S2 for another asset S1
• Pay-off at time T max (0, S1(T) S2(T))
• the right numéraire is S2
• the relevant volatility is the volatility S of
• Margrabe's formula holds under the sole
  hypothesis of a deterministic volatility for
  no assumption on interest rates, stochastic or
  not
• where
• and
• The correlation coefficient plays a key role in
  the option price

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Kyoto-Pricing Power Plant Valuation and Emission
Rights

• Spark Spread Pe (Pt/e) (PCO2 / e / c)
• where
• Pe Price of power in /MWh
• Pf Price of fuel
• E Efficiency
• PCO2 Price of Permits in /ton
• c Specific carbon content (ton CO2 / ton)

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• Kyoto pricing of a coal plant

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The European Carbon Market

• The 2005 emissions of the 21 States representing
  88 of the European allowances were announced on
  May 15, with a surplus over 3.4. The allowances
  allocated to these countries were 62.8 Mt higher
  than the declared emissions
• The allowance price reacted very strongly to the
  anticipated disclosure of compliance data and
  dropped by 65 between April 24 and May 12, and
  the price of the 2008 contract fell by 25
• The disclosure of compliance data did not give
  the market enough information to limit its
  instability price volatility in late May was
  significantly greater than the one observed in
  the 8 previous month
• It seems necessary to improve the information in
  direction of the market, by increasing
  transparency and harmonization between countries
  in their reporting