View
77
Download
0
Category
Tags:
Preview:
DESCRIPTION
Natural Gas Markets – Spot, Forward, and Real Options. Matt Davison. Departments of Applied Mathematics and of Statistical & Actuarial Sciences, The University of Western Ontario. Natural Gas. - PowerPoint PPT Presentation
Citation preview
Natural Gas Markets Natural Gas Markets – Spot, Forward, and – Spot, Forward, and
Real OptionsReal Options
Matt DavisonMatt Davison
Departments of Applied Mathematics Departments of Applied Mathematics and of Statistical & Actuarial Sciences,and of Statistical & Actuarial Sciences,
The University of Western OntarioThe University of Western Ontario
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.
Natural Gas Price
Time-series of Henry Hub natural gas prices 1995-1999
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
Mean Reverting Spot Models
Mean-reversion models are common for modelling commodity spot prices
the one factor Pilipovic Model:
Comparing with GBM
( )t t t t tdS S dt S dW
t t t tdS S dt S dW
Two-factor Pilipovic Model
Where the two Brownian risk factors are correlated:
(1)
(2)
( )t t t t t
t t t t
dS L S dt S dW
d L dt L dW
(1) (2),d W W dt Here
the spot price
the equilibrium price
time of observation
rate of price mean reversion
volatility
drift of long-term equilibrium price
volatility in the long-term equilibrium price
S
L
t
dW
(1)
(1)
random stocahstic variable defining the randomness in the spot price
random stocahstic variable defining the randomness in the equilibrium pricedW
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
(1) (1) ( 2) (1)2 2 2 2( 1/ 2 ) ( 1/ 2 ) ( 1/ 2 ) ( 1/ 2 )0 0 0
t t x xtt W t W x W x W
tS S e L e e e dx
2 2 2
( )2 2 2
0 0
t tt t s
tt W t W W sstS S e Le e e ds
0
European Option Pricing Formula
Using one-factor Pilipovic model
( )rTT TC e E S K
2
2 2 2
0
2
2 2 2
0
( ) ( )
0
1max (0)
2
,0]
(0) ( ) ( ) [( 1) ( )],
( ) :
where
distribution function with characteristic
Ty TrT aT
T
T S yTy T y SaT aS
yr a T rT r a T aTT
T
C e S e e
aLe e e e dS K e dy
e S y e K y Le e zF dz
F dz
2
( 1)
0 0
2
: :
1( ) .
2
function
and
aTi e
yx
e
y T y y y
x e dy
Stylized Features of Nat Gas Markets
Highly seasonal, with ‘oscillating’ forward curve
High volatility levels, in 30%-100% range
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
Capturing this with spot models
Pilipovic one and 2 factor models Last week’s 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.
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)
Powojowski Model
2 *log ( , ) [ ( , ) 0.5 ( , )] ( , ) ( , ) ( ) (Miro1)d F t T t T t T dt t T dW t T dQ t
( , ) : ;
( , ) :
( , ) :
( ) :
( ) :
Where
forward price at time for delivery at
positive function; continuous volumn
positive function; jump volumn
Standard Brownian Motion
Marked Poisson Process wit
F t T t T
t T
t T
W t
Q t
2, ( , ) ( )
h intensity and jumps
of random size distributed independent of Y N W t
More specification
chosen in order to guarantee that the solution of (Miro1) is a martingale.
Assume specific form for volatility functions
2 2
( , ) [exp( ( , ) ) 1]
exp[ ( , ) 0.5 ( , );
B t T E t T Y
t T t T
( , ) ( ) exp[ ( )], 0
( , ) ( ) exp[ ( )], 0
t T T y T t y
t T T T t
Solution of the Model
( , )
0
( )2
0 01
log ( , ) log (0, )
1( , ) ( , ) ( ) ( , )( )
2
( , ) ( , ) ( , )
t T
N tt t
i ii
F t T F T e d t
T d T dW t T
I t T G t T V t T
( , ) 2
0 0
0
( ) :
, 1, ( ) :
:
1( , ) log (0, ) ( , )
2
( , ) ( , ) ( )
( , ) ( , )( )
where
the number of Poisson events
arrival times
independent standard normal variatesi
i
t tT
t
i ii
N t
t i N t
I t T F T e d t T d
G t T T dW
V t T t T
L
( )
1
N t
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
log ( , ) ( , ) ( , )( ) [exp( log ( , ))] exp( ( , )) ( ) ( )F t T G t T V t Tu E iu F t T iuI t T u u
exp( ( , ))iuI t T
22 ( ) 2( )
(0, ( ))2
T t TTe e
%
2 22 ( ) 2
( , )
( )( ) exp ( ) .
4T t T
G t T
u Tu e e
%
( ) ( )0( ) exp ( )
t
V t Au u d t
Can Price Vanilla Options
Also price strips of forward starting options – cliquet or ratchet options
Can also price swaptions and calendar spreads
Covariance Structure of Forward Curve
Let log( ( , )) log( (0, ))i i iX F t T F T
1 2
1 2
1 2 1 2
( )1 2 1 2 0
( )1 2 0
( ) ( )1 2 1 2
( , ) ( ) ( ) var( )
( ) ( ) var( )
( ( ) ( ) ( ) ( ) )
T Tt
T Tt
T T T T
cov X X T T e W W
T T e Q Q
T T e T T e t
% %
% %
% % % %
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
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
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
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).
LNG Storage
Above-Ground Natural Gas Storage FacilitiesSource: Energy Information Administration, “The Global Liquified Natural Gas Market: Status and Outlook”, December 2003
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
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
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.
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.
Natural Gas Storage Facilities
Natural gas can be stored underground in (A) salt caverns (B) mines (C) aquifers(D) depleted oil/gas reservoirs(E) hard rock mines
Resources in Ontario
Storage, Injection, and Withdrawal
An aggregate US-level picture of storage and withdrawal is available from the US Energy Information Administration.
Aggregate Inject/Withdraw
Modeling a single facility
Use Merton’s application of Bellman’s principle to finance
Incorporate engineering details
Physics/Engineering pV=nRT 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
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 > 0) / stored (c < 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;
Optimization Framework I
The objective function
Subject to
Change in I obeys ODE
Change in P obeys Markov process
0( , , )max ( ( , ))
T
c P I tE e c a I c Pd
min max( ) ( )c I c c I
[ ( , )]dI c a I c dt
1 1 11
( , ) ( , ) ( , , )N
k k kk
dP P t dt P t dX P t J dq
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 Bellman’s equation
( )( , , , ; ) max ( ( , ))T t
tcV P W f t c E e c a I c Pd
( )max ( ( , )) ( , , )t dt t dt
tcV E e c a I c Pd e V P dP I dI t dt
Standard Taylor Series arguments
Employ Ito’s lemma to obtain Taylor series
Eliminate all higher order terms and simplify
Take expectations and divide by dt
21 1
1 11
1max ( ( , )) (1 ) (1 )( ( ( , )) )
2
(1 ) ( ) (1 )( )
t PP P Ic
N
k k Pk
V E c a I c Pdt dt V dt V V V c a I c V dt
dt V V dq dt V dX
21 1 1 1
1
1max ( ( ( , )) ( ( , )) ) ( ) ( ) 0
2
N
t PP P I k k Pc
k
E V V V c a I c V c a I c P V dt V V dq V dX
21 1
1
1max ( ( ( , )) ( ( , )) [ ] 0
2
N
t PP P I k kc
k
E V V V c a I c V c a I c P E V V
The PDE
• The optimal value for c maximizes
Subject to
• The PDE
Initial condition:
Boundary conditions:
min max( ) ( )c I c c I
max ( ( , )) ( ( , ))Ic
c a I c V c a I c P
21 1
1
1( ( , )) ( ( , )) [ ] 0
2
N
t PP P I k kk
V V V c a I c V c a I c P E V V
( , , ; ) 0V P I T c
0
0 0
for large
for PP
PP
V P
V P
The Numerical Difficulties
Hyperbolic in I direction of information flow upwind finite differencing
Total variation diminishing schemesSlope limiting method works best
Method of lines approach (Mukadam)
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%
0.25 (2.5 ) 0.2 ( )
0 1 2(6,4)
1 2
with probability
with probability
dP P dt PdX J P dq
dtdq J N
dt
The PDE
The function a
The PDE
Then
0 0( , ) ( )
1.7 365 0
for
for
ca I c a c
c
2
2
( 6) /8
0
1(0.04 ) 0.25(2.5 ) ( ( )) 0.1 1000( ( ))
21
2 ( ( , , ) ( , , ) 08
t PP P I
J
V P V P V c a c V V c a c P
V J I t V P I t e dJ
max
min 1 2
( ) 2040.41
1( )
b
c I I
c I K KI I
Natural Gas Control Surface
Natural Gas Value Surface
Put and Call
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
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
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 US$3.5 billion and bringing down the fund.
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 week’s lecture.
Recommended