IMPA Commodities Course: Introduction Sebastian Jaimungal sebastian.jaimungal@utoronto.ca Department of Statistics and Mathematical Finance Program, University of Toronto, Toronto, Canada http://www.utstat.utoronto.ca/sjaimung February 19, 2008
Table of contents Basics Issues 1 Basics Issues 2 Observations 3 Swap Single Name Options Two-Name Options Exotic Options
Basic Issues Commodities are NOT like equities, interest rates, or currencies! Commodities are real assets: produced, consumed, transported and stored Owing a commodity at point A at time T is not the same as owing it at point B at time S! Customers of commodity derivatives are typically Industrial producers / consumers Governments Customers tend to be very risk averse due to Required consumption Legal risks
Basic Issues Spot markets per se do not exist Instead there are major futures market Electricity day-ahead / day-of / hour-ahead balance-of-week /month years ahead,, Nat Gas: months / years Contracts May be for physical delivery or financial settlement May require delivery over a period of time May be interruptible May vary the amount delivered
Forward Price Observations Forward price curves from Jan-2002 to Dec-2006
Forward Price Observations Weekly deviations in Forward price curves Jan-2002 and 2003 Forward Price 22 21.5 21 20.5 20 19.5 19 18.5 Jan 2002 week 1 week 2 week 3 week 4 week 5 Forward Price 18 0 1 2 3 4 5 6 7 38 36 34 32 30 28 26 24 Term (years) Jan 2003 22 0 1 2 3 4 5 6 7 Term (years) week 1 week 2 week 3 week 4 week 5
Forward Price Observations Weekly deviations in Forward price curves Jan-2004 and 2005 Forward Price 38 36 34 32 30 28 Jan 2004 week 1 week 2 week 3 week 4 week 5 Forward Price 26 0 1 2 3 4 5 6 7 50 48 46 44 42 40 Term (years) Jan 2005 week 1 week 2 week 3 week 4 week 5 38 0 1 2 3 4 5 6 7 Term (years)
Forward Price Observations Weekly deviations in Forward price curves Jan-2006 and Dec-2006 Forward Price 70 68 66 64 62 60 Jan 2006 week 1 week 2 week 3 week 4 week 5 Forward Price 58 0 1 2 3 4 5 6 7 70 65 60 Term (years) Nov 2006 week 1 week 2 week 3 week 4 week 5 55 0 1 2 3 4 5 6 7 Term (years)
Forward Price Observations Long end and short end of the Forward price curves from Jan-2002 to Dec-2006 80 70 Short End Middle Long End Forward Price 60 50 40 30 20 10 3.72 3.74 3.76 3.78 3.8 3.82 3.84 3.86 3.88 3.9 3.92 Time x 10 4 Notice periods of Backwardation and Contango.
Forward Price Term Structure of volatilities Observations 0.34 0.32 0.3 Volatility 0.28 0.26 0.24 0.22 0.2 0.18 0 1 2 3 4 5 6 7 Term Exponentially decaying volatilities
Forward Price Observations Correlation of constant maturity forward price returns Correlation 0.95 0.9 0.85 0.8 0.75 0.7 0.65 8 6 4 Term 2 0 0 2 4 Term 6 8 As expected, terms further apart less correlated
Observations Forward Price Forward price curves from Jan-2002 to Dec-2006 2.5 Forward Price 2 1.5 1 0.5 1.5 Term (years) 1 0.5 0 3.74 3.76 3.78 3.86 3.84 3.82 3.8 Trading Date 3.88 x 10 4 3.9
Forward Price Observations Weekly deviations in Forward price curves Jan-2002 and 2003 Forward Price 0.62 0.6 0.58 0.56 0.54 0.52 Jan 2002 week 1 week 2 week 3 week 4 week 5 Forward Price 0.5 0 0.5 1 1.5 1.1 1 0.9 0.8 0.7 Term (years) Jan 2003 week 1 week 2 week 3 week 4 week 5 0 0.5 1 1.5 Term (years)
Forward Price Observations Weekly deviations in Forward price curves Jan-2004 and 2005 Forward Price 1.05 1 0.95 0.9 0.85 0.8 0.75 Jan 2003 week 1 week 2 week 3 week 4 week 5 Forward Price 0.7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.45 1.4 1.35 1.3 1.25 1.2 1.15 Term (years) Jan 2005 week 1 week 2 week 3 week 4 week 5 1.1 0 0.5 1 1.5 Term (years)
Forward Price Observations Weekly deviations in Forward price curves Jan-2006 and Dec-2006 Forward Price 2.1 2.05 2 1.95 1.9 1.85 1.8 1.75 Jan 2006 week 1 week 2 week 3 week 4 week 5 Forward Price 1.7 0 0.5 1 1.5 2.2 2.1 2 1.9 1.8 1.7 1.6 Term (years) Nov 2006 week 1 week 2 week 3 week 4 week 5 1.5 0 0.5 1 1.5 Term (years)
Forward Price Observations Long end and short end of the Forward price curves from Jan-2002 to Dec-2006 2.4 2.2 2 Short End Middle Long End Forward Price 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 3.72 3.74 3.76 3.78 3.8 3.82 3.84 3.86 3.88 3.9 3.92 Time x 10 4 Notice periods of Backwardation and Contango.
Forward Price Term Structure of volatilities Observations 0.38 0.36 0.34 Volatility 0.32 0.3 0.28 0.26 0.24 0.22 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Term Exponentially decaying volatilities
Forward Price Observations Correlation of constant maturity forward price returns 1 Correlation 0.95 0.9 0.85 0.8 1.5 1 Term 0.5 0 0 0.5 Term 1 1.5 As expected, terms further apart less correlated
Observations Observations: Arbitrage Opportunity? There are no simple calendar spread arbitrage opportunities e.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 first nearby The contract with the next shortest maturity is called the second nearby, etc.. When the first nearby matures, it is said to roll off Many exotics have nearbys as underliers as opposed to a specific contract
Observations: Shape Basics Issues Observations Most commodities have volatile short-ends, and quasi-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)
Observations: Seasonality Observations Seasonality occurs in many commodities (crude oil is the main 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 hump on weekends, and intra-day structure
Observations: Bias Basics Issues Observations 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 if demand is not met risk averse investors Hedge funds becoming larger players in this market
Swap Basics Issues Swap Single Name Options Two-Name Options Exotic Options Swaps entail exchanging a fixed payment stream for a floating payment stream Floating typically linked to a commodity spot price or price index Fixed Price Payer Receiver Commodity Price 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 equal value
Call Option Basics Issues Swap Single Name Options Two-Name Options Exotic Options Call options give the holder the right (but not obligation) to purchase a commodity (or a forward on commodity) at a specified future date for a specified price ϕ = max(f T (T 1 ) K) + 50 45 40 35 30 Payoff 25 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 Underlying Stochastic models must be built to obtain the evolution and valuation
Call Option Basics Issues Swap Single Name Options Two-Name Options Exotic Options Simplest model is to assume forward price at maturity is log-normal : F T (T 1 ) = F T (0) exp{ 1 2 σ2 T + σ T Z} and Z N (0, 1) 3.5 4 x 104 σ = 10% σ = 20% σ = 30% 15000 σ = 10% σ = 20% σ = 30% 3 Frequency 2.5 2 1.5 Frequency 10000 5000 1 0.5 0 20 30 40 50 60 70 80 90 100 Forward Price (a) Forward Price 0 0 5 10 15 20 25 30 35 40 Payoff (b) Call Payoff
Asian Call Option Basics Issues Swap Single Name Options Two-Name Options Exotic Options Asian call options are call options on the average of the commodity (forward) price on several days A(t 1, t 2 ; T ) 1 t2 t 2 t 1 t 1 F T (u)du Two main classes: Average Price: ϕ = (A(t 1, t 2, T ) K) + Average Strike: ϕ = (F T (t 1 ) A(t 1, t 2, T )) + Tend to be cheaper than a regular option averaged out the volatility Moment match approximations are sometimes used to obtain closed-form results: log-normal or inverse-gaussian Geometric averaging used to analytically compute price and corrected via control variate PDE / transform methods
Asian Call Option Basics Issues Swap Single Name Options Two-Name Options Exotic Options 4000 4000 3500 3500 3000 3000 Frequency 2500 2000 1500 Frequency 2500 2000 1500 1000 1000 500 500 0 0 50 100 150 200 250 300 350 400 Payoff 0 0 50 100 150 200 250 300 350 400 Payoff (c) Average years 0 5 (d) Year 5
Floating Strike Call Basics Issues Swap Single Name Options Two-Name Options Exotic Options Floating Strike Call Options are call options with the strike level set at a future date based on some price index ϕ = (F T (T 1 ) I T ) + There are two sources of uncertainty here the commodity price and the index price
Calendar Spread Options Swap Single Name Options Two-Name Options Exotic Options A calendar spread option is an option to exchange a T 2 -maturity forward contract for a T 1 -maturity forward contract at a cost of K at time T : ϕ = (F T (T 1 ) F T (T 2 ) K) +.
Calendar Spread Options Swap Single Name Options Two-Name Options Exotic Options Payoff profile ( F 0 (T 1 ) = 10, F 0 (T 2 ) = 9, K = 1, T = 1yr., σ 1 = σ 2 = 50%) 5000 4500 ρ = 0 ρ = 0,5 ρ = 0.5 4000 3500 Frequency 3000 2500 2000 1500 1000 500 0 0 5 10 15 Payoff
Calendar Spread Options Swap Single Name Options Two-Name Options Exotic Options Payoff profile ( F 0 (T 1 ) = 10, F 0 (T 2 ) = 11, K = 1, T = 1yr., σ 1 = σ 2 = 50%) 5000 4500 ρ = 0 ρ = 0,5 ρ = 0.5 4000 3500 Frequency 3000 2500 2000 1500 1000 500 0 0 5 10 15 Payoff
Intercommodity Spread Basics Issues Swap Single Name Options Two-Name Options Exotic Options Intercommodity Spread Options are options based on the difference in two commodities ( ) ϕ = F (1) T (T 1) α F (2) T (T 1) Crack Spread is the option on the spread between crude oil and refined products (such as heating oil) Spark Spread is the option on the spread between electricity and fuel (such as coal/gas) α is then referred to as the heat rate (BTUs needed to create 1 kwh of electricity) +
Barrier Option Basics Issues Swap Single Name Options Two-Name Options Exotic Options Barrier options are like other options except they are turned on (knock-in) or turned off (knock-out) when the commodity price enters a given price D Typically, the region corresponds to the asset dropping above or below a critical price level The barrier may be single sided or double sided Hitting one barrier may turn on additional barriers onion options
Swing Option Basics Issues Swap Single Name Options Two-Name Options Exotic Options Swing options are options on contract volume. They give the holder the right to change the volume of commodity delivered to them. 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 of days during which delivery occurs) These are complex options with embedded American like features Tree implementations require solving a forest not a single tree