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.