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IMPA Commodities Course:Introduction
Sebastian [email protected]
Department of Statistics andMathematical Finance Program,
University of Toronto, Toronto, Canadahttp://www.utstat.utoronto.ca/sjaimung
February 19, 2008
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
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Table of contents
1 Basics Issues
2 DataCrude OilHeating OilData Observations
3 ProductsSwapSingle Name OptionsTwo-Name OptionsExotic Options
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
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Basic Issues
Commodities are NOT like equities, interest rates, orcurrencies!
Commodities are real assets: produced, consumed,transported and stored
Owing a commodity at point A at time T is not the same asowing it at point B at time S!
Customers of commodity derivatives are typically
Industrial producers / consumersGovernments
Customers tend to be very risk averse due to
Required consumptionLegal risks
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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Basic Issues
Spot markets per se do not exist
Instead there are major futures marketElectricity
day-ahead / day-of / hour-aheadbalance-of-week /monthyears ahead
Crude Oil, Heating Oil, Nat Gas: months / years
Contracts
May be for physical delivery or financial settlementMay require delivery over a period of timeMay be interruptibleMay vary the amount delivered
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Crude Oil Forward Price Data
Forward price curves from Jan-2002 to Dec-2006
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Crude Oil Forward Price Data
Weekly deviations in Forward price curves Jan-2002 and 2003
0 1 2 3 4 5 6 718
18.5
19
19.5
20
20.5
21
21.5
22Jan−2002
Term (years)
For
war
d P
rice
0 1 2 3 4 5 6 722
24
26
28
30
32
34
36
38Jan−2003
Term (years)
For
war
d P
rice
week 1week 2week 3week 4week 5
week 1week 2week 3week 4week 5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Crude Oil Forward Price Data
Weekly deviations in Forward price curves Jan-2004 and 2005
0 1 2 3 4 5 6 726
28
30
32
34
36
38Jan−2004
Term (years)
For
war
d P
rice
0 1 2 3 4 5 6 738
40
42
44
46
48
50Jan−2005
Term (years)
For
war
d P
rice
week 1week 2week 3week 4week 5
week 1week 2week 3week 4week 5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Crude Oil Forward Price Data
Weekly deviations in Forward price curves Jan-2006 and Dec-2006
0 1 2 3 4 5 6 758
60
62
64
66
68
70Jan−2006
Term (years)
For
war
d P
rice
0 1 2 3 4 5 6 755
60
65
70Nov−2006
Term (years)
For
war
d P
rice
week 1week 2week 3week 4week 5
week 1week 2week 3week 4week 5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Crude Oil Forward Price Data
Long end and short end of the Forward price curves from Jan-2002to Dec-2006
3.72 3.74 3.76 3.78 3.8 3.82 3.84 3.86 3.88 3.9 3.92
x 104
10
20
30
40
50
60
70
80
Time
Fo
rwar
d P
rice
Short EndMiddleLong End
Notice periods of Backwardation and Contango.Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Crude Oil Forward Price Data
Term Structure of volatilities
0 1 2 3 4 5 6 70.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
Term
Vo
lati
lity
Exponentially decaying volatilitiesSebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Crude Oil Forward Price Data
Correlation of constant maturity forward price returns
02
46
8
0
2
4
6
8
0.65
0.7
0.75
0.8
0.85
0.9
0.95
TermTerm
Co
rrel
atio
n
As expected, terms further apart less correlated
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Heating Oil Forward Price Data
Forward price curves from Jan-2002 to Dec-2006
3.743.76
3.783.8
3.823.84
3.863.88
3.9
x 104
0
0.5
1
1.50.5
1
1.5
2
2.5
Trading DateTerm (years)
Fo
rwar
d P
rice
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Heating Oil Forward Price Data
Weekly deviations in Forward price curves Jan-2002 and 2003
0 0.5 1 1.50.5
0.52
0.54
0.56
0.58
0.6
0.62Jan−2002
Term (years)
For
war
d P
rice
0 0.5 1 1.5
0.7
0.8
0.9
1
1.1Jan−2003
Term (years)
For
war
d P
rice
week 1week 2week 3week 4week 5
week 1week 2week 3week 4week 5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Heating Oil Forward Price Data
Weekly deviations in Forward price curves Jan-2004 and 2005
0 0.2 0.4 0.6 0.8 1 1.2 1.40.7
0.75
0.8
0.85
0.9
0.95
1
1.05Jan−2003
Term (years)
For
war
d P
rice
0 0.5 1 1.51.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45Jan−2005
Term (years)
For
war
d P
rice
week 1week 2week 3week 4week 5
week 1week 2week 3week 4week 5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Heating Oil Forward Price Data
Weekly deviations in Forward price curves Jan-2006 and Dec-2006
0 0.5 1 1.51.7
1.75
1.8
1.85
1.9
1.95
2
2.05
2.1Jan−2006
Term (years)
For
war
d P
rice
0 0.5 1 1.51.5
1.6
1.7
1.8
1.9
2
2.1
2.2Nov−2006
Term (years)
For
war
d P
rice
week 1week 2week 3week 4week 5
week 1week 2week 3week 4week 5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Heating Oil Forward Price Data
Long end and short end of the Forward price curves from Jan-2002to Dec-2006
3.72 3.74 3.76 3.78 3.8 3.82 3.84 3.86 3.88 3.9 3.92
x 104
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Time
Fo
rwar
d P
rice
Short EndMiddleLong End
Notice periods of Backwardation and Contango.Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Heating Oil Forward Price Data
Term Structure of volatilities
0 0.2 0.4 0.6 0.8 1 1.2 1.40.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
Term
Vo
lati
lity
Exponentially decaying volatilities
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
Products
Crude OilHeating OilData Observations
Heating Oil Forward Price Data
Correlation of constant maturity forward price returns
0
0.5
1
1.5
0
0.5
1
1.50.8
0.85
0.9
0.95
1
TermTerm
Co
rrel
atio
n
As expected, terms further apart less correlated
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
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Crude OilHeating OilData Observations
Observations: Arbitrage Opportunity?
There are no simple “calendar spread” arbitrage opportunitiese.g. if curve is upward sloping: purchase short-term contract; sell
long-term contract → what’s wrong?
The contract with the shortest maturity is called the firstnearby
The contract with the next shortest maturity is called thesecond nearby, etc..
When the first nearby matures, it is said to roll off
Many exotics have nearbys as underliers – as opposed to aspecific contract
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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Crude OilHeating OilData Observations
Observations: Shape
Most commodities have volatile short-ends, andquasi-stable long-ends
Long-end is “determined” by marginal cost of production
Short-end governed by short termed supply/demand
When there are excesses of commodity :curve is upward sloping (contango)When there are shortages of commodity :curve is downward sloping (backwardation)
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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Crude OilHeating OilData Observations
Observations: Seasonality
Seasonality occurs in many commodities (crude oil is themain exception)
If storage capacity exceeds the wavelength, then no humps
Natural gas has large hump in winter, small hump in summer
Gasoline has large hump in summer
Electricity has humps in winter and summer, negative humpon weekends, and intra-day structure
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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Crude OilHeating OilData Observations
Observations: Bias
Trading volume is clumped in the long-end of the curve
Short-end used to cover unexpected demand
Short-end is positively biased to avoid large negative impact ifdemand is not met – risk averse investors
Hedge funds becoming larger players in this market
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Swap
Swaps entail exchanging a fixed payment stream for a floatingpayment stream– Floating typically linked to a commodity spot price or price index
Payer
Fixed Price
ReceiverPayer
Commodity Price
Receiver
The floating-leg can be viewed as series of forward contracts
The fixed-leg can be viewed as a series of “coupon” payments
fixed-leg payments determined such that both legs have equalvalue
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Call Option
Call options give the holder the right (but not obligation) topurchase a commodity (or a forward on commodity) at a specifiedfuture date for a specified price
ϕ = max(FT (T1)− K )+
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
30
35
40
45
50
Underlying
Pay
off
Stochastic models must be built to obtain the evolution andvaluation
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Call Option
Simplest model is to assume forward price at maturity islog-normal :
FT (T1) = FT (0) exp{−1
2σ2 T +σ
√T Z} and Z ∼ N (0, 1)
20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5
4x 10
4
Forward Price
Fre
qu
ency
σ = 10%σ = 20%σ = 30%
(a) Forward Price
0 5 10 15 20 25 30 35 400
5000
10000
15000
Payoff
Fre
qu
ency
σ = 10%σ = 20%σ = 30%
(b) Call Payoff
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Asian Call Option
Asian call options are call options on the average of thecommodity (forward) price on several daysA(t1, t2; T ) , 1
t2−t1
∫ t2t1
FT (u)du
Two main classes:
Average Price: ϕ = (A(t1, t2,T )− K )+
Average Strike: ϕ = (FT (t1)− A(t1, t2,T ))+
Tend to be cheaper than a regular option – averaged out thevolatility
Moment match approximations are sometimes used to obtain“closed-form” results: ∼ log-normal or ∼ inverse-gaussian
Geometric averaging used to analytically compute price andcorrected via control variate
PDE / transform methods
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Asian Call Option
0 50 100 150 200 250 300 350 4000
500
1000
1500
2000
2500
3000
3500
4000
Payoff
Fre
qu
ency
(c) Average years 0− 5
0 50 100 150 200 250 300 350 4000
500
1000
1500
2000
2500
3000
3500
4000
PayoffF
req
uen
cy
(d) Year 5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Floating Strike Call
Floating Strike Call Options are call options with the strike levelset at a future date based on some price index
ϕ = (FT (T1)− IT )+
There are two sources of uncertainty here – the commodityprice and the index price
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Calendar Spread Options
A calendar spread option is an option to exchange aT2-maturity forward contract for a T1-maturity forwardcontract at a cost of K at time T :
ϕ = (FT (T1)− FT (T2)− K )+ .
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Calendar Spread Options
Payoff profile ( F0(T1) = 10, F0(T2) = 9, K = 1, T = 1yr.,σ1 = σ2 = 50%)
0 5 10 150
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Payoff
Fre
qu
ency
ρ = 0ρ = −0,5ρ = 0.5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
Basics IssuesData
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Calendar Spread Options
Payoff profile ( F0(T1) = 10, F0(T2) = 11, K = 1, T = 1yr.,σ1 = σ2 = 50%)
0 5 10 150
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Payoff
Fre
qu
ency
ρ = 0ρ = −0,5ρ = 0.5
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Intercommodity Spread
Intercommodity Spread Options are options based on thedifference in two commodities
ϕ =(F
(1)T (T1)− αF
(2)T (T1)
)+
Crack Spread is the option on the spread between crude oiland refined products (such as heating oil)
Spark Spread is the option on the spread between electricityand fuel (such as coal/gas)
– α is then referred to as the heat rate (BTUs needed to create 1
kWh of electricity)
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Barrier Option
Barrier options are like other options except they are turned on(knock-in) or turned off (knock-out) when the commodity priceenters a given price D
Typically, the region corresponds to the asset dropping aboveor below a critical price level
The barrier may be single sided or double sided
Hitting one barrier may turn on additional barriers – onionoptions
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction
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SwapSingle Name OptionsTwo-Name OptionsExotic Options
Swing Option
Swing options are options on contract volume. They give theholder the right to change the volume of commodity delivered tothem.
Delivery occurs over several days
The holder is allowed to vary volume within a specified range
The holder has K < N “swing” opportunities (N is number ofdays during which delivery occurs)
These are complex options with embedded American likefeatures
Tree implementations require solving a forest – not a singletree
Sebastian Jaimungal [email protected] IMPA Commodities Course: Introduction