Matt Davison

1
Natural Gas Markets Spot, Forward, and Real
Options

• Matt Davison

Departments of Applied Mathematics and of
Statistical Actuarial Sciences, The University
of Western Ontario
2
Natural Gas
On NYMEX, Natural Gas futures is based on 10,000
mm Btu (million btus). The price is quoted in
dollars per mm Btu.
3
Natural Gas Price
Time-series of Henry Hub natural gas prices
1995-1999
4
Outline

• Stylized features of natural gas markets
• Some simple spot models
• An example full forward curve model
• Untidy reality
• Real options natural gas storage
• A natural gas trading disaster
• Conclusions electricity preview

5
Mean Reverting Spot Models

• Mean-reversion models are common for modelling
  commodity spot prices
• the one factor Pilipovic Model
• Comparing with GBM

6
Two-factor Pilipovic Model

• Where the two Brownian risk factors are
  correlated

7
The Solutions of Pilipovic Model

• The explicit solution of one-factor model
• The explicit solution of two-factor model
• In the special case ,two-factor
  reduce to one-factor Pilipovic model

8
European Option Pricing Formula

• Using one-factor Pilipovic model

9
Stylized Features of Nat Gas Markets

• Highly seasonal, with oscillating forward curve
• High volatility levels, in 30-100 range

10
Natural Gas Volatility Features

• High volatility levels 30-100
• Volatilities of futures increase as maturity
  approaches (Samuelson effect)
• Considerable volatility skew, esp. for short
  maturities
• Skew is positive for OTM calls and negative for
  OTM puts
• ATM volatilities display seasonal effects

11
Capturing this with spot models

• Pilipovic one and 2 factor models
• Last weeks model (Ribiero Hodges 2004)
• Spot model ? forward model (solution of a PDE)
  BUT
• Very hard to construct such models which have all
  the right features.

12
Modeling

• Suggests modeling the entire term structure of
  the forward curve, like HJM or like Jara (2000)
• Approach I describe here comes from Powojowski
  (2007).
• Can introduce jumps to capture the volatility
  skew (Merton 1976 Cont and Tankov 2003)

13
Powojowski Model
14
More specification

• chosen in order to guarantee that the solution of
  (Miro1) is a martingale.
• Assume specific form for volatility functions

15
Solution of the Model
16
Characteristic Function of F(t,T)

• The characteristic function of F(t, T)
• The factor can be computed
  through a combination of analytical and numerical
  integration.
• The random variable G(t,T) is normally
  distributed
• Hence,
• And

17
Can Price Vanilla Options

• Also price strips of forward starting options
  cliquet or ratchet options
• Can also price swaptions and calendar spreads

18
Covariance Structure of Forward Curve

• Let

19
Untidy Reality

• Like all commodity markets Natural Gas markets
  involve real things.
• But Natural gas is more so.
• Local in Space
• Local in Time
• Demand and Supply are weather dependent

20
Where is Natural Gas produced?

• In Canada Natural gas is produced chiefly in
  Alberta and Saskatchewan (but also to a limited
  extent in SW Ontario) in the US also in the Gulf
  of Mexico, Texas, California and to a limited
  extent in Appalachia
• Worldwide it is produced in the North Sea, in the
  Middle East, and in Russia

21
Where is Natural Gas consumed?

• Everywhere, but in Canada to a great extent in
  the Eastern half (population density higher).
• Natural gas must be transported on a pipeline
  network and refined (in Canada, often in Sarnia)
  before being used.
• Liquidity in North American Markets Henry Hub
  in Louisiana, AECO hub in Alberta

22
Not Really A World Market

• No pipelines between here and the mid-East
• However Liquified Natural Gas (LNG) can be
  transported by ship.
• This is expensive but worth the effort since, for
  a fossil fuel, natural gas is very clean (short
  hydrocarbon chains ? less pollutants and
  greenhouse gas emissions per unit burned).

23
LNG Storage
24
Local in Time

• Natural Gas is difficult to store (about which
  more later)
• Demand for Nat Gas is highly seasonal (in winter
  for heating in summer for electricity
  generation/air conditioning)
• This explains the bumps in the forward curve

25
Weather Dependent

• Seasonality is a function of temperature
  dependence as temperature rises above a
  threshold (18 Celsius) or below a similar
  threshold, gas use increases dramatically.
• Links with Heating/Cooling Degree Day derivatives

26
Weather Dependence (II)

• Production of gas from the Gulf of Mexico (as
  well as refining) is also weather dependent
• Hurricane Katrina devastated oil and gas
  production for several months.
• For oil markets this was a small problem because
  of the local nature of gas markets it was a
  proportionally much bigger problem.

27
Financial vs. Fundamental

• I have heard that 5 years ago the fundamentals of
  gas markets were equally important to the
  financial aspects but that more recently the
  balance is more like 70 financial, 30
  fundamental.
• But 30 is still a lot and if you miss it you can
  get into deep trouble, about which more later.

28
Natural Gas Storage Facilities

• Natural gas can be stored underground in
• salt caverns
• mines
• aquifers
• depleted oil/gas reservoirs
• hard rock mines

29
Resources in Ontario
30
Storage, Injection, and Withdrawal

• An aggregate US-level picture of storage and
  withdrawal is available from the US Energy
  Information Administration.

31
Aggregate Inject/Withdraw
32
Modeling a single facility

• Use Mertons application of Bellmans principle
  to finance
• Incorporate engineering details

33
Physics/Engineering pVnRT

• Base gas capacity
• Required for reservoir pressure
• Never removed
• Working gas capacity
• Amount of gas available to produce and sell
• Deliverability
• Rate at which gas can be released
• Depends on gas level
• Injection capacity
• Rate at which natural gas can be added
• Depends on gas level
• Cycling
• Salt caverns are HDMC
• Reservoir seepage
• Cost to pump gas

34
Variables in General Gas Storage Equations

• P price per unit of natural gas
• I current amount of working natural gas
  inventory
• c control variable gas injected (c gt 0) /
  stored (c lt 0)
• Imax max storage capacity of facility
• Imin -- base gas capacity
• cmax(I) max deliverability rate as function of
  storage
• level
• cmin(I) min injection rate as function of
  storage level
• a(I,c) amount of gas lost given c units of gas
• released/injected

35
Optimization Framework I
The objective function
Subject to
Change in I obeys ODE
Change in P obeys Markov process
36
Optimization Framework II
To simultaneously determine optimal strategy c(P,
I, t) and corresponding optimal value V(p, I, t),
let
Split integral to get
Moving towards Bellmans equation
37
Standard Taylor Series arguments
Employ Itos lemma to obtain Taylor series
Eliminate all higher order terms and simplify
Take expectations and divide by dt
38
The PDE

• The optimal value for c maximizes

Subject to

• The PDE

Initial condition
Boundary conditions
39
The Numerical Difficulties

• Hyperbolic in I
• direction of information flow
• upwind finite differencing
• Total variation diminishing schemes
• Slope limiting method works best
• Method of lines approach (Mukadam)

40
A Sample Problem

• The Stratton Ridge facility
• Working gas capacity of 2000 MMcf
• Base gas requirement 50 MMcf
• Minimum capacity injectivity 80 MMcf/day
• Injection pump requirement 1.7MMcf /day
• No seepage from reservoir
• Ideal gas law and Bernoulli's law apply
• Prices in MMbtus
• Time measured in years
• Discount rate 10

41
The PDE
The function a
The PDE
Then
42
Natural Gas Control Surface
43
Natural Gas Value Surface
44
Put and Call
45
A Natural Gas Trading Disaster

• Perhaps because of the volatility and complexity
  of natural gas markets, large amounts have been
  made or lost trading them.
• BMO lost a bundle in summer 2001 others more
  recently.
• Most famous story, however, is that of Amaranth
  Partners

46
The Tale of Amaranth Partners

• Amaranth LLP was a Connecticut Hedge fund but
  with a strong Canadian connection
• Their star trader, Brian Hunter, was based in
  Calgary.
• In 2005 he had made several billion dollars (!)
  trading gas

47
A Spread Trade Gone Bad

• In early fall 2006 Hunter put on a spread trade
  between September and October natural gas futures
• This was a bet that, as in fall 2005, a
  hurricane would hit the Gulf drilling platforms
  also gas storage levels were inadequate.
• But no hurricane and trade went bad, losing
  US3.5 billion and bringing down the fund.

48
Conclusions

• Real Options are important for Natural Gas
• As well as gas storage facilities natural gas
  electricity generating facilities are important
  natural gas real options.
• These embody the so-called spark spread
• Gas is burned to generate electrical power
• So to value these generation real options a model
  for electricity prices is also needed
• That will be the topic of next weeks lecture.