Commodities and Energy Markets Supplementary
Notes: Common Structures
Princeton RTG summer school in Financial Mathematics
Presenters: Michael Coulon and Glen Swindle
17 April 2013
c Glen Swindle: All rights reserved
1 / 67
Outline
Orders of magnitude
Global Landscape
Motivations and Common Structures
Themes
2 / 67
Orders of magnitude
Units
We will be focusing on the three most traded groups of energy
commodities.
- Crude oil and refined products
- Natural gas
- Power (electricity)
Energy commodities are typically discussed/traded in the
following units:
- Oil and Refined Products
- Barrels, gallons, metric tons
- Natural Gas
- MMBtu’s, decatherms, gigaJoules, cubic feet
- Power
- Megawatt-hours (MWh)
3 / 67
Orders of Magnitude:
Units
1 barrel of crude oil:
- 42 gallons
- ≈ 5.4 MMBtu
- .1364 metric tons
1 MMBtu of natural gas
- ≈1000 cubic feet (cf) of natural gas
- .025 metric tons of crude oil
1 MWh
- ≈ 7 MMBtu
- This is actual generation including thermal inefficiency; not heat
content
- 1 MWh powers about 1000 U.S. homes for 1 hour
- One U.S. household consumes electricity at the rate of roughly 1kW.
- To store 1kW-day as potential energy this would lift a 1000kg over
8.8km.
4 / 67
Global Landscape
World Energy Consumption:
Daily U.S. crude oil consumption is approximately 20m barrels
= 120m MMBtu
Total world oil consumption is ≈ 85m barrels per day.
Annual U.S. natural gas consumption is ≈ 25 Tcf = 25b
MMBtu
World energy consumption in 2009 was roughly 11 billion
ton-oil-equivalent (toe) annually.
-
This is ≈ 80 billion barrels of crude oil equivalent.
At $100 / barrel, this is 8 trillion USD.
World GDP is ≈ 60 trillion USD.
This is a very rough calculation. Market price / MMBtu heat content
varies dramatically between fuels.
5 / 67
Global Landscape
World Energy Consumption:
Historically dramatic increase in world energy consumption by
source
- Source: BP Statistical Review of World Energy June 2010
Energy Consumption By Type
14000
12000
Crude Oil
Natural Gas
Coal
Nuclear
Hydro
Mtoe
10000
8000
6000
4000
2000
0
1980
1985
1990
1995
2000
2005
2010
6 / 67
Global Landscape
Regional Imbalances
Viewed by Geographic Region
- Source: BP Statistical Review of World Energy June 2010
Note the increase in Asia Pacific, coincident with the increase
in rate of coal consumption in the previous slide.
Energy Consumption By Region
14000
12000
North America
Central & South America
Europe
Middle East
Africa
Asia Pacific
Mtoe
10000
8000
6000
4000
2000
0
1980
1985
1990
1995
2000
2005
2010
2015
7 / 67
Global Landscape
Crude Oil Consumption:
Historical Crude Oil Consumption By Region
- Source: BP Statistical Review of World Energy June 2010
Crude Oil Consumption
90
80
Million Barrels / Day
70
North America
Central & South America
Europe
Middle East
Africa
Asia Pacific
60
50
40
30
20
10
0
1980
1985
1990
1995
2000
2005
2010
8 / 67
Global Landscape
Crude Oil Production:
Historical Crude Oil Production By Region
- Source: BP Statistical Review of World Energy June 2010
Crude Oil Production
90
80
Million Barrels / Day
70
North America
Central & South America
Europe
Middle East
Africa
Asia Pacific
60
50
40
30
20
10
0
1980
1985
1990
1995
2000
2005
2010
9 / 67
Global Landscape
Crude Oil Imbalances:
Net production for three regions
- Source: BP Statistical Review of World Energy June 2010
Crude Oil Net Production
20
15
Million Barrels / Day
10
5
0
−5
−10
−15
−20
−25
−30
1980
Middle East
North America
Asia Pacific
1985
1990
1995
2000
2005
2010
10 / 67
Global Landscape
Crude Oil Consumption: Monthly:
Monthly world oil demand (source EIA).
The regression shows growth rates as estimated through 2007.
- Note the impact of the credit crisis.
- Note also the seasonality as implied by the residuals in the lower plot.
- The residuals are driven by higher winter heating demand in the northern
hemisphere.
World Crude Oil Consumption By Month
Million Barrels/Day
90
85
Annual Increase: 1.33 mmBpd
80
75
70
65
1994
1996
1998
2000
2002
2004
2006
2008
2010
World Crude Oil Consumption Residuals By Month
Million Barrels/Day
2
1
0
−1
−2
−3
1
2
3
4
5
6
7
8
9
10
11
12
11 / 67
Global Landscape
Crude Oil: Historical Pricing:
Price of crude oil in 2009 USD.
Concatenation of historical U.S., Arabian and Brent prices.
- Source: BP Statistical Review of World Energy June 2010
Historical Spot Prices (2011 USD/Barrel)
120
100
$/Barrel
80
60
40
20
0
1860
1880
1900
1920
1940
1960
1980
2000
12 / 67
Motivations and Common Structures
Balancing Supply and Demand
As seen above:
-
Energy consumption globally continues to grow.
There are significant locational imbalances.
Consumption is not uniform in time.
What is extracted as a resource is ultimately consumed as other products.
- Crude oil is converted into refined products.
- Natural gas, coal and uranium are converted into electricity.
Significant investment is required to:
- Fund the continuing extraction of new resources.
- Building the physical optionality to convert commodities from one form
(type,location,time) to another.
These activities can be categorized as:
-
Exploration and production
Transportation
Conversion
Storage
Consumption
13 / 67
Motivations and Common Structures
Exploration and Production
Producers of oil and related products or natural gas are
natural longs—term sellers.
- These include large integrated energy producers as well as small
exploration and development companies.
Motivations:
- Expectations of investors regarding exposure of expected future
production to energy prices
- A view of future evolution of energy prices
- Sale of reserves (future production) to raise capital for more exploration
- Volumetric Production Payments (VPPs)
- The lender (purchaser) hedges the forward value sexpected future
production to protect the value of the commodity purchase.
14 / 67
Motivations and Common Structures
Exploration and Production
Standard transaction is a vanilla swap or forward sale for some
tenor/volume of expected production.
Often options structures are often embedded.
- Some would argue that swaps/forwards are enough.
Options Structures: Collars (fences)
Producer is long a low strike put and short a high strike call.
- Protecting more extreme downward moves in price.
- Funded by the sale of the high strike call.
- The swap is a degenerate case of the put/call at the same strike
15 / 67
Motivations and Common Structures
Exploration and Production
Producer hedge from the dealer’s perspective.
Swap and Collar Hedges
2
Swap
Collar
1.5
Payoff ($/MMBtu)
1
0.5
0
−0.5
−1
−1.5
−2
8
8.5
9
9.5
10
10.5
11
11.5
12
Forward Price ($/MMBtu)
16 / 67
Motivations and Common Structures
Exploration and Production
Options Structures: ”Price Enchancement”
- The producer sells the commodity at an off-market (higher) price in
return for a short position in a low strike option.
- A common structure has been a European ”knock-out.”
- The risk to the hedger is that the hedge vanishes when it is needed
most, as experienced by some producers in 2008.
KO Swap
4
Swap
KO Swap
3
Payoff ($/MMBtu)
2
Market Price: $6.33
1
0
−1
Enhanced Price: $7.33
−2
−3
−4
Knockout Strike: $5.33
3
4
5
6
7
8
9
10
Forward Price ($/MMBtu)
17 / 67
Motivations and Common Structures
Transportation
Building the infrastructure to move a commodity from one
location to another is creating physical optionality.
- A tanker (crude or LNG) and associated infrastructure to move oil or
natural gas.
- A pipeline to transport crude oil, refined products or natural gas.
- A transmission line to move power.
In its purest form the option payoff is:
Q max [Fsink (T + τ, T + τ ) − Fsource (T , T ) − K , 0]
-
Q is the notional or rate
T is the time of ”loading”
τ is the delay in transportation
K is cost of transport per unit notional
18 / 67
Motivations and Common Structures
Transportation
All physical transportation involves a constraints and
complexities which make the above an idealization.
- Costs can be a function of price Fsource and of rate Q.
- Q can vary with environment (transmission capacity as a function of
temperature).
- Transmission operators can adjust Q based on operational limitations.
- Some pipelines can carry a variety of refined products (making this a
much more complex option).
Transportation is designed to exploit (and thereby reduce)
locational specialness.
This is our first example of a spread option.
- Valuation involves vols and correlations between the respective forward
prices.
19 / 67
Motivations and Common Structures
Conversion
Building infrastructure to convert one commodity to another
is creating physical optionality.
The two most common occurences are:
Refining
Power Generation
The cost of infrastructure is considerable.
- Purchase of existing assets or construction of new assets involves
significant financing.
- Lenders typically require that the value of the asset in question be hedged
for some duration (often 5-7 years).
- The provider of the hedge:
- Receives an option of a form designed to be a reasonable facsimile
of the asset.
- Typically pays a monthly ”capacity payment” as an option premium
for the structure.
20 / 67
Motivations and Common Structures
Conversion: Refining
Conversion of crude oil to a vast array of petroleum products.
Inputs: Specifications of crude oil which vary by:
- Sulfur content: sour (high sulfur crude) is harder to refine to than sweet
(low sulfur).
- Density/Gravity: light (low density) crudes have a higher percentage of
light (volatile) hydrocarbons which yields premium products such as
gasoline; heavy crudes yield more residuals which are heavy, cheaper and
more polluting.
Outputs:
- Vary by grade of input crude
- Yields can be altered adjustment of refining process to accommodate
seasonal demand.
- High density residuals resulting from simple distillation are subsequently
processed by catalysis/crackers which increase yield of low density
products.
21 / 67
Motivations and Common Structures
Conversion: Refining
There are many different crude oils of varying specification.
The figure shows crude oil ”streams” versus API density and
sulfur content.
- The API density is inversely related to actual density and is, therefore, an
increasing indication of value.
- Note the negative correlation between sulphur content and API density.
0.035
Sour
MAYA
COLD LAKE
0.03
SULPHUR
0.025
TIA JUANA HEAVY
MORICHAL
PILON
MEREY
BCF−17
MENEMOTA
0.02
Russian Export Blend (REBCO)
Iranian Light
LEONA
Saudi Light
LAGOTRECO
MESA 28 LAGOMEDIO
ANS Tia Juana Light
Furrial
MESA
0.015
0.01
Sweet
BASRAH LIGHT
BCF−24
0.005
CABINDA
0
10
15
20
Heavy
25
30
API DENSITY
35
BRENT
WTI
Bonny Light
40
Anaco Wax
TAPIS
45
50
Light
22 / 67
Motivations and Common Structures
Conversion: Refining
Historical spot prices for a few specific crudes.
Observations:
- All prices clearly follow global macro crude oil trends.
- The low level of Maya which is high sulfur (sour) and heavy, versus say
WTI or Brent which are low sulfur (sweet) and light.
Historical Spot Prices
160
140
WTI
Brent
Dubai
Maya
$/Barrel
120
100
80
60
Maya
40
20
2005
2006
2007
2008
2009
2010
2011
2012
2013
23 / 67
Motivations and Common Structures
Conversion: Refining
Basis of the crude prices to WTI as a reference.
Drivers:
- Relative value of sulfur and density.
- Shipping rates.
$/Barrel
Historical Brent−WTI Spot Basis
20
0
−20
2005
2006
2007
2008
2009
2010
2011
2012
2013
$/Barrel
Historical Dubai−WTI Spot Basis
20
0
−20
Q1−05
Q2−05
Q3−05
Q4−05
Q1−06
Q2−06
Q3−06
Q4−06
Q1−07
Q2−07
Q3−07
Q4−07
Q1−08
Q2−08
Q3−08
Q4−08
Q1−09
Q2−09
Q3−09
Q4−09
Q1−10
Q2−10
Q3−10
Q4−10
Q1−11
Q2−11
Q3−11
Q4−11
Q1−12
Q2−12
Q3−12
Q4−12
Q1−13
$/Barrel
Historical Maya−WTI Spot Basis
20
0
−20
Q1−05
Q2−05
Q3−05
Q4−05
Q1−06
Q2−06
Q3−06
Q4−06
Q1−07
Q2−07
Q3−07
Q4−07
Q1−08
Q2−08
Q3−08
Q4−08
Q1−09
Q2−09
Q3−09
Q4−09
Q1−10
Q2−10
Q3−10
Q4−10
Q1−11
Q2−11
Q3−11
Q4−11
Q1−12
Q2−12
Q3−12
Q4−12
Q1−13
24 / 67
Motivations and Common Structures
Conversion: Refining
The following figure shows typical yield by stream
- (Source: EIA)
25 / 67
Motivations and Common Structures
Conversion: Refining
The following figure shows average U.S. refining output.
- (Source: EIA)
Refineries are designed to:
- Produce more of the the low density (valuable) products.
- Change the relative composition of products to meet seasonal demand.
26 / 67
Motivations and Common Structures
Conversion: Refining
A refiner is trying to make money converting crude oil into a
palette of refined products.
Loosely speaking the products can be categorized (mapped)
to one of three kinds:
- Gasoline
- Heating oil
- Residual fuel oil
These three products products have well indexed spot prices
and liquid forward/futures markets.
The spreads between the prices of these products and the
input crude oil are called crack spreads.
27 / 67
Motivations and Common Structures
Conversion: Refining
The following figure shows spot prices for these three products
at NY Harbor (a common delivery point) versus WTI.
Again, note the global price trends.
Spot Prices ($/barrel)
180
Gasoline
Heating Oil
Resid
160
140
$/Barrel
120
100
80
60
40
20
2006
2007
2008
2009
2010
2011
2012
2013
28 / 67
Motivations and Common Structures
Conversion: Refining
This figure shows spot crack spreads.
This is what a refiner is making from three products.
Note that residual loses money consistently.
- For any given crude oil, there are engineering limitations as to how little
resid is produced.
Crack Spreads ($/barrel)
50
Gasoline
Heating Oil
Resid
40
30
$/Barrel
20
10
0
−10
−20
−30
−40
−50
2006
2007
2008
2009
2010
2011
2012
2013
29 / 67
Motivations and Common Structures
Conversion: Refining
Observe that monthly demand of heating oil and gasoline
exhibit seasonality.
- Heating oil demand peaks in winter.
- Gasoline demand peaks in summer (”driving season”).
Distillate Monthly Demand
Million Barrels / Day
5
4.5
4
3.5
3
2000
2002
2004
2006
2008
2010
Gasoline Monthly Demand
Million Barrels / Day
10
9.5
9
8.5
8
7.5
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
30 / 67
Motivations and Common Structures
Conversion: Refining
This is demand averaged by calendar month.
Million Barrels / Day
Distillate Monthly Demand Average Consumption By Month
4.4
4.2
4
3.8
3.6
3.4
1
2
3
4
5
6
7
8
9
10
11
12
Million Barrels / Day
Gasoline Monthly Demand Average Consumption By Month
10
9.5
9
8.5
8
1
2
3
4
5
6
7
8
9
10
11
12
31 / 67
Motivations and Common Structures
Conversion: Refining
Refiners tune their product palette to produce (and store) the
required product at the right time.
Distillate Average Inventory By Month
Million Barrels
150
140
130
120
110
1
2
3
4
5
6
7
8
9
10
11
12
Gasoline Average Inventory By Month
Million Barrels
240
230
220
210
200
190
1
2
3
4
5
6
7
8
9
10
11
12
32 / 67
Motivations and Common Structures
Conversion: Refining
In theory a refinery is a spread option between multiple legs:
+
sup w̄ t (T )F̄ (T , T ) − G (T , T ) − V
w̄ (·)∈A
where:
-
F̄ denotes the set of forward prices for the variety of refined products.
G denotes the price of the input crude oil.
V is the cost of refining one unit of crude oil.
w̄ denotes defines the relative quantity of the variety of refined products
to be produced.
- A denotes constraints on both the product mix at time T as well as
operational constraints on how w can change with T .
The relevant crack spreads are Fj − G .
In practice the visibility of the forwards F̄ at long tenors is
limited and the volatilities of many of the products is
altogether unobservable.
33 / 67
Motivations and Common Structures
Conversion: Generation
Types of Generation
Natural: Energy source is geophysical in some way
- Hydro: reservoirs (controllable) or run-of-river (uncontrollable).
- Wind: turbines driven by wind and undispatchable.
- Solar: very small part of the supply stack.
Fuel Driven:
- Baseload: cheap but inflexible, typically running 7x24.
- Nuclear: uranium.
- Coal: variety of coal grades with varying specifications and logistics.
34 / 67
Motivations and Common Structures
Conversion: Generation
Types of Generation
Fuel Driven: (cont)
- Mid-merit: efficient with increased flexibility
- Combined cycle (CC) natural gas power generation.
- Sometimes has fuel switching capabilities (residual fuel oil).
- Has comprised the vast majority on increased generation capacity
since the mid-90s.
- The recent glut of shale gas has resulted in CC generation
comprising an increasing slice of electricity output.
- Peaking: expensive but highly flexible.
- Basically jet engines.
- Typically natural gas or kerosene/jet fuel driven.
35 / 67
Motivations and Common Structures
Conversion: Generation
Types of Generation
The following figure shows U.S. Generation by Type for 2009
- (Source: EIA)
U.S. Generation By Type (2009)
Renewables 4%
Oil 1%
NG 24%
Other 1%
Hydro 6%
Nuclear 20%
Coal 45%
36 / 67
Motivations and Common Structures
Conversion: Generation
U.S. Markets
The following map shows the U.S. markets by control region.
The ISO (independent systems operators) administer market
clearing of power and ancillary (reliability) products.
The most actively traded markets are:
-
PJM
Other northeast markets: NY, New England
ERCOT (Texas)
California/West
37 / 67
Motivations and Common Structures
Conversion: Generation
38 / 67
Motivations and Common Structures
Conversion: Generation
Demand
Power demand (load) is driven by many factors.
Time of day.
Temperature and other weather variables.
Macroeconomic effects
The following shows two views of hourly PJM classic load.
PJM Classic Hourly Demand
4
7
x 10
6
MWh
5
4
3
2
1
1998
2000
4
6
x 10
2002
2004
2006
2008
2010
PJM Classic Hourly Demand (Jul10)
MWh
5
4
3
2
07/01 07/02 07/03 07/04 07/05 07/06 07/07 07/08 07/09 07/10 07/11 07/12 07/13 07/14 07/15 07/16
39 / 67
Motivations and Common Structures
Conversion: Generation
Demand
The collection of generation in a region provides the physical
optionality to meet power demand.
The ”stack” is a graphical display of the marginal cost of
incremental generation versus generation capacity.
- Given arrays of capacity [C1 , . . . , CN ] sorted so that the associated costs
of generation in $/MWh are increasing [p1 , . . . , pN ], the stack is a plot of
P
pn vs 1≤k≤n Ck .
40 / 67
Motivations and Common Structures
Conversion: Generation
The following is a picture of the PJM stack circa 2010.
PJM System Stack (2010)
350
Hydro
Nuclear
Coal
Natural Gas
Petroleum
Other
300
$/MWh
250
200
150
100
50
0
0
5
10
MW
15
4
x 10
41 / 67
Motivations and Common Structures
Conversion: Generation
Price Clearing
Power markets effectively clear the load at each moment with
the supply (the stack).
Note:
- In reality most power markets generate hundreds of locational prices
factoring in transmission constraints.
- The stack prices p̄ are functions of fuel prices at the moment and bidding
behavior of generators.
The results are price series, such as the hourly PJM Western
Hub price below.
42 / 67
Motivations and Common Structures
Conversion: Generation
Price Clearing
Note the dramatic variation in price level as well as the
occasional negative price.
Hourly PJM Spot Prices
1200
1000
$/MWh
800
600
400
200
0
−200
2000
2002
2004
2006
2008
2010
43 / 67
Motivations and Common Structures
Conversion: Generation
Market Mechanics
Trading hourly forwards would be prohibitively cumbersome at
long tenors.
- Hourly forwards trade day-ahead and day-of.
Term swaps reference buckets of hours.
- Given the dramatic differences in demand (and price) over the course of a
day market convention has evolved to using swaps that reference ratable
delivery over the following buckets:
- Peak (5x16): Business days (usually M-F) 7AM - 11PM.
- Offpeak: The complement of peak.
- Offpeak is occasional split into 7x8 and 2x16
- Note: Western power markets define Peak as 6x16 (M-Sat).
44 / 67
Motivations and Common Structures
Conversion: Generation
Tolling Deals
Motivations are as hedges for earnings or borrowing for
acquisition or construction of generation.
Tradeoff between complexity of the derivative and the match
with the asset.
The basic structure has payoff:
max [FB (T , T ) − H∗ G (T , T ) − V , 0]
- T denotes delivery day (this is discrete time).
- B denotes the delivery bucket (e.g. 5x16.)
- F and G denote the prices of power and natural gas respectively, typically
at liquid pricing hubs.
- H∗ is the generator or deal heatrate (conversion rate between gas and
power.)
- V is VOM (variable operation and maintenance).
45 / 67
Motivations and Common Structures
Conversion: Generation
Tolling Deals
Spark Spread for a generator or toll with heatrate H∗ is:
FB (T , T ) − H∗ G (T , T )
The generator should generate when the spark spread exeeds
the cost of generation.
Assuming V = 0 this amounts to:
H(T , T ) ≡
FB (T , T )
> H∗
G (T , T )
- The ratio of the market price of power to natural gas H(T , T ) is called
the ”market heatrate.”
46 / 67
Motivations and Common Structures
Conversion: Generation
Tolling Deals
Mismatch: Actual generation abounds with complexities
- Typically units receive spot prices unique to their locations, referred to as
-
locational marginal price (LMP) which in general is at best highly
correlated with a liquid hub.
Greater flexibility than block dispatch: can change generation level on
short time scales.
- Lesser flexibility: minimum down time.
- Heatrate H varies with generation level and ambient temperature.
- Multiple fuel choices:
max [FB (t, t) − min [H1 G1 (t, t), H2 G2 (t, t)] − V , 0]
Structuring a hedge involves balancing matching the asset
with simplicity of the derivative.
47 / 67
Motivations and Common Structures
Conversion: Generation
Tolling Deals
Valuation Issues:
- The basic structure is a spread option, requiring returns
correlation as inputs as well as forwards and vols.
- What are returns for a non-storable such as power?
- Can you dynamically hedge into the cash month (daily or even
hourly exercise)?
- These structures are typically not auto-exercise (as one would
think from the equations above)
-
Usually exercise is day-ahead
For example by 9:45 the desk must exercise by phone.
This is hours before the actual index price is posted by the ISO.
The information available at exercise next-day power swaps and
partial information on the gas price fixing.
- How do you handle hourly flexibility/constraints when the
smallest commoditized risks are daily?
48 / 67
Motivations and Common Structures
Storage
We have discussed seasonal demand in crude oil and refined
products.
Facilities exist to store surplus quantities of each to
accomodate anticipated seasonal demand fluctuations.
The same is true for natural gas, though weather driven
demand fluctuations are even more extreme.
U.S. Basic Facts:
- Annual gas consumption is roughly 25 Tcf with roughly 4 Tcf of imports.
- Gas consumption is highly seasonal due to winter heating requirements.
- Approximately 4 Tcf of natural gas storage facilitates accommodation of
winter demand.
49 / 67
Motivations and Common Structures
Storage: Consumption/Production Mismatch
North American demand is highly seasonal.
Observations:
- High winter peak demand and the relatively mild summer peaks (air
conditioning power demand met with CC generation)
- Non-seasonal production profile.
- Recent increase in domestic production—shale gas glut.
U.S. Natural Gas Production and Consumption
110
Consumption
Production
100
Bcf/day
90
80
70
60
50
40
2001
2003
2005
2007
2009
2011
50 / 67
Motivations and Common Structures
Storage: Inventory History
Roughly 4.2bcf of storage capacity resolves the
production/consumption mismatch.
Compare historical inventory levels versus ”normal”
- ”Normal” is a Fourier fit with the number of modes used determined by
an out-of-sample selection method with estimates of working capacity.
U.S. Working Storage
4000
Actual
Normal
3500
3000
Bcf
2500
2000
1500
1000
500
2000
2002
2004
2006
2008
2010
2012
51 / 67
Motivations and Common Structures
Storage: Inventory History
This is the departure of inventory levels from ”normal”:
R(t) ≡ S(t) − S̄(t) with
S̄(t) = α + βt +
K
X
[γk sin(2πkt) + δk cos(2πkt)]
(1)
k=1
U.S. Working Storage Residual
1000
800
600
400
Bcf
200
0
−200
−400
−600
−800
−1000
2000
2002
2004
2006
2008
2010
2012
52 / 67
Motivations and Common Structures
Storage: Inventory and Price Dynamics
The storage residual R(t) is an important variable in price
dynamics.
Forward curves on January 2001 (deficit) and January 2002
(surplus).
- Note the higher prices, backwardation and greater seasonality in 2001.
NG Forward Curves
10
05Jan01
07Jan02
9
8
$/MMBtu
7
6
5
4
3
2
0
0.5
1
1.5
2
2.5
3
Tenor (Y)
53 / 67
Motivations and Common Structures
Storage: Inventory and Price Dynamics
Calendar strip forward yields versus storage residual.
Calendar strips are used to “strip out” seasonal effects.
NG 1st/2nd Cal Strip Carry Versus Inventory Residuals
40
Annualized Carry (%)
30
20
Backwardation (Carry<0)
10
0
−10
Contango (Carry>0)
−20
−30
−800
−600
−400
−200
0
200
400
600
800
1000
NG Storage Residuals (Bcf)
54 / 67
Motivations and Common Structures
Storage: Inventory Drivers
Weather is a key driver.
- This is the weekly change in inventory scattered against average
temperature at LaGuardia Airport in New York City.
Weekly Injection Versus Temperature (KLGA)
150
100
50
0
Heating Demand
Bcf
−50
−100
Air Conditioning Demand
−150
−200
−250
−300
10
20
30
40
50
60
70
80
90
o
Average Daily Temperature ( F)
55 / 67
Motivations and Common Structures
Storage: Facilities
Two common types:
- Salt cavern storage is flexible (“high-turn”)
- Aquifer storage is less flexible (“low turn”) and exhibits hysteresis effects
which complicate modeling and valuation.
Derivative renditions can provide financial exposure to storage
optionality and can be used to facilitate financing for
construction of new storage or improvement of existing
storage facilities.
Storage facilities are sold/leased and can embed very
challenging option structures.
56 / 67
Motivations and Common Structures
Storage: Park and Loans
A physical forward contract in which physical delivery of
natural gas (fully laden with title, tax and insurance issues) is
taken and subsequently stored to be resold at a specific future
date and prescribed price.
The implied interest rate is:
1
F (t, T )
log
T −S
F (t, S)
which is compared to the cost of funding of the storing
institution.
57 / 67
Motivations and Common Structures
Storage: CSOs
Calendar Spread Options (CSOs) are the simplest ”storage
options”.
- Also called time-spread options.
A CSO straddle between two months has pay-off:
[F (T1 , T2 ) − F (T1 , T1 )]+
- This can be interpreted (slightly modified) as the simplest of storage
transactions.
h
i+
F (τ, T + U) − e rU F (τ, T ), 0
- This straddle embodies the option to withdraw/inject a unit quantity of
the commodity at time T and inject/withdraw at time T + U.
- Note that valuation of CSOs involves vols of both legs and correlations.
- We will see later that estimation and hedging of correlations in this case
are highly nontrivial.
58 / 67
Motivations and Common Structures
Storage: Dynamic Optimization
Actual storage affords the owner the option of injecting and
withdrawing on a daily basis.
- Each decision effects the future optionality due to effects in existing
inventory.
The universal valuation constraint is that inventory S(t) must
satisfy: 0 ≤ S(t) ≤ Smax .
Other commonly encountered constraints:
-
Injection/withdrawal rate constraints: s∗ ≤ S 0 (t) ≤ s ∗ .
s∗ and s∗ are functions of inventory level S(t).
Hysteresis: s∗ and s∗ are functions of [S(u) : u ≤ t].
Ratchets: S(τ̄ ) >= S̄level and similar upper bounds.
59 / 67
Motivations and Common Structures
Storage: Dynamic Optimization
The basic inventory problem is to find the optimal rule
s(·) ∈ A for the payoff:
Z
T
d(t, u) [−s(u)F (u, u) + κ (s(u), S(u), F (u, u))] du
t
where:
-
S(t) is the current inventory level: 0 ≤ S(·) ≤ Smax .
S 0 (t) = s(t).
κ denotes costs associated with injection and withdrawal.
A denotes allowed controls, accommodating the constraint above as well
-
as constraints on the magnitude of s(·) and bounds on inventory levels at
particular times.
d(t, u) is the discount factor.
This is a very hard problem.
60 / 67
Motivations and Common Structures
Consumers
Consumers of these products are natural shorts—buyers.
These include:
- Large industrials
- Transportation, especially airlines
- Some utilities, specifically their collection of customers.
Motivations for hedging include:
- Risk management of exposure to future energy prices.
- Hedging programs driven by investor expectations or regulatory
requirements.
As with producers, vanillas swaps constitute the most
common hedging instrument.
Differences from producers:
- The tenor of hedging is often shorter and driven by regulatory impetus.
- Many natural shorts (especially in power) can simply pass through costs
and are, therefore, less inclined to engage in hedging programs.
61 / 67
Motivations and Common Structures
Consumers
A Key Distinction: Many natural shorts have daily or hourly
exposure that is outside of their direct control.
- Recall the load time series earlier.
- The figure below shows the series versus KPHL (Philadelphia) average
temperature.
4
6.5
PJM Load at 4PM vs KPHL
x 10
6
5.5
5
MW
4.5
4
3.5
3
2.5
2
10
20
30
40
50
DegF
60
70
80
90
62 / 67
Motivations and Common Structures
Consumers
Utilities are effective short:
L(h) [p(h) − pfixed ]
- L(h) is the hourly load.
- p(h) is the hourly spot price for power.
- pfixed is the fixed contract price with the end user(s).
This payoff is heavily temperature-dependent and has a
nontrivial correlation (quanto) between load and price, as
evidenced by the follow plot of clearing prices versus load.
63 / 67
Motivations and Common Structures
Consumers
PJM Western Hub spot prices versus load.
PJM Hourly Spot Price versus Load
800
700
600
500
400
300
200
100
0
−100
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
4
x 10
64 / 67
Themes
Supply and Demand Drivers
Unlike securities markets, infrastructure is required to get a
desired commodity from where it is in space and time, to
where it is needed.
This infrastructure can either be inadequate or can fail,
resulting in large price variations.
This is the origin of specialness.
Dimensionality Mismatch
There are many grades, locations and delivery times for
commodities.
There are relatively few liquid benchmarks to trade/hedge
risks.
65 / 67
Themes
Time Scale Mismatch
Producer/Consumer hedges can involve risks on monthly or
annual (swaption) time scales.
Generation and load on time scale of months, days or hours.
Few swaps markets and no options markets exist in energy on
time-scales shorter than daily.
Blunt Instruments
Swaps available for hedging often trade in monthly or
seasonal/annual strips.
Extremely limited market in correlation products—either
intra-commodity or inter-commodity.
66 / 67
Themes
Engineering Underpinnings
Common hedging structures resemble (appropriately)
engineering features
Typical response is to evolve (devolve) into ever more intricate
complete-markets analysis even in the absence of required
liquid options markets.
Nonstationarity
Changes in supply/demand balance coupled with engineering
constraints can result in dramatic changes in price dynamics.
Macroeconomic effects and regulatory changes (past and
present) make inference challenging.
67 / 67