IMPA Commodities Course: Introduction · Basics Issues Data Products Table of contents 1 Basics...

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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

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Table of contents

1 Basics Issues

2 DataCrude OilHeating OilData Observations

3 ProductsSwapSingle Name OptionsTwo-Name OptionsExotic Options


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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


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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


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Crude OilHeating OilData Observations

Crude Oil Forward Price Data

Forward price curves from Jan-2002 to Dec-2006


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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



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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




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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




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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

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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

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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


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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


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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




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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




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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




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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

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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


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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


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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


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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)


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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


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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


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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


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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


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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


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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


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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


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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


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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 )+ .


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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


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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


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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)


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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


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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


IMPA Commodities Course: Introduction · Basics Issues Data Products Table of contents 1 Basics...

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