Theory of Storage and the Dynamics of Metals Forward Curves
Helyette GemanDirector, Commodity Finance Centre
University of London and ESCP EuropeScientific Advisor to the European Commission
To be presented at the Vale Conference on CommoditiesGetulio Vargas Foundation - Rio de Janeiro
August 16 & 17, 2012
Metals, Energy, Agriculturals : A Multi or Unique Asset Class?
→ Commodities have displayed over the last 30 years
. a period of low prices in the 1980s and 1990s, strictly declining if adjusted for inflation, and low volatility
. low correlations between metals, energy, agriculturals
→ Then much higher prices prevailed as of 2002 for crude oil, as of 2004 for copper, 2005 for agriculturals. In the case of iron ore, long-term contracts imposed by steelmakers broke down after the concerted action of Vale, Rio Tinto and BHP Billiton
High correlations appeared, created. by the massive arrival of financial actors buying at the same time several
commodities such as copper, gold, crude oil in the form of a commodity index
. by the effects of substitution between commodities and competition for the same rare resources, called electricity , water, land
CRB Commodity Index – 1988 to 2011
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Data from Yahoo Finance via Matlab Datafeed toolbox
Brazil Equity versus UK Equity
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
S&P data from Yahoo (ticker ^GSPC) via Matlab Data Source ToolkitAgriculture Index from World Bank http://data.worldbank.org/data-catalog/commodity-price-data
US Equity versus the Metals and Minerals Index:‘Risk on/ Risk off ‘ behaviour recently !
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
S&P data from Yahoo (ticker ^GSPC) via Matlab Data Source ToolkitMetals & Minerals Index from World Bank http://data.worldbank.org/data-catalog/commodity-price-
data
US Equity versus Agriculture Index
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
S&P data from Yahoo (ticker ^GSPC) via Matlab Data Source ToolkitEnergy Index from World Bank http://data.worldbank.org/data-catalog/commodity-price-data
US Equity versus Energy Index
Metal Reserves
New mining techniques, deeper drilling and mining in untapped places such as Greenland, Mongolia and the Arctic should lead to years in mineral reserves, at current production rates, estimated at 590 for iron ore, 136 for copper, 610 for potash; versus 18.9 for gold, 46.2 for crude oil and 82 for metallurgical coal
Hotelling in his (1931) paper on exhaustible commodities had established that the shadow price of the resource, which is an economic measure of its scarcity, should grow at least at the rate of interest
→ Young (1992) applies Hotelling model to Canadian copper mining firms and finds it poorly depicts the database he analyzes; but the period of analysis ended in 1990 and was the period of price mean- reversion ( G. 2005 : Is Mean Reversion in Commodity Prices Dead? )
→ It is useful to recognize that the possible decline in the quality of the
BDI, Copper and the world economy growth
Dislocation between BDI and Copper Prices
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Commodiity Monthly Prices from World Bank http://data.worldbank.org/data-catalog/commodity-price-data
Copper versus Crude Oil
Copper Volatility Smile - April 6, 2012
Merrill Lynch, “Modelling the Implied Volatility Surface”, http://finmath.stanford.edu/seminars/docs/ml2004win.pdf
Implied Volatility “Surface”(bottom axes are price and time to maturity)
Implied vol
Strike Price
Time to Maturity
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15
Impl
ied
Vola
tility
K/S
Gold 1st Contract (22 Jun 2012)
Theory of StorageKeynes (1936), Kaldor (1939), Working (1949)
Three fundamentals results:
→ The holder of the physical commodity receives an implicit dividend called convenience yield
→ The volatility of the commodity spot price is high when inventory is low
→ Traditionally, forward curves used to be mostly declining with the maturity ( ‘normal backwardation’) and sometimes in contango. Today, we even get humps
→ The dynamics of the global forward curve matters, in hedging activities in particular, since one never hedges with the prompt- month
The Forward Curve
→ The set {FT (t) , T > t} is the forward curve prevailing at date t for a givencommodity in a given location
→ It is the fundamental tool when trading commodities, as spot prices may beunabservable and options not always liquid
→ It allows to identify the prices forecasted by the market at future dates since realtrades did take place at these prices
→ The shape of the forward curve is a crucial piece of financial information to becompared to all the other sources!
Crude Oil in Backwardation in September 2007
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70
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M1 M5 M9 M13 M17 M21 M25 M29 M33 M37 M41 M45 M49 M53 M57 M61
WTI Oil Prices Jan 2002 - Oct 2007
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2002
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Spot-Forward Relationship for a Storable Commodity
Under no arbitrage
( ) ( ) ( ) ( ) ( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−−+−+=43421321321
dividendimplicitstorageoftcosfinancingoftcos
tTytTctTr1tStf 1T
If we define a convenience yield net of cost of storage
( ) ( ) ( )( )[ ]tTyr1tStf T −−+=
Or in continuous time, at a fixed date t (today), for a given maturity T
( ) ( ) ( )( )tTyrT etStf −−=
Copper Forward Curve - 27 May 2008 : Backwardation
345
350
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380
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M24
Copper COMEX
Crude Oil Future curve (17/11/2008)
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M1 M4 M7 M10 M13 M16 M19 M22 M25 M28 M31 M34 M37 M40 M43 M46 M49 M52 M55 M58 M61
Forward Curve
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Maturity
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ard
Pric
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WTI Forward Bid WTI Forward Offer
A hump in the Oil Forward Curve (bid/ ask) - March 2006
Copper Forward Curve, Oct 2009
3.1
3.15
3.2
3.25
3.3
3.35
3.4
3.45
1-Au
g-12
1-Oc
t-12
1-De
c-12
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b-13
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1-Au
g-13
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1-Fe
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1-Ap
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g-14
1-Oc
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1-Fe
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1-Ap
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1-Au
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1-Oc
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1-Fe
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1-Au
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1-Oc
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1-Fe
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1-Ap
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n-17
Copper 14 Aug 2012
Copper
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Coal 14 Aug 2012
Coal
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c-20
NG 14 Aug 2012
NG
From CME Group, http://www.cmegroup.com/trading/metals/ferrous/iron-ore-62pct-fe-cfr-china-tsi-swap-futures_quotes_settlements_futures.html
Iron Ore Futures Curve
China import Iron Ore Fines 62% FE spot (CFR Tianjin port), US Dollars per Metric Tonhttp://www.indexmundi.com/commodities/?commodity=iron-ore&months=60
Iron Ore – Evolution of Spot Price
Inventory, Volatility and Shape of the Forward Curve
→Working (1949) proposed to use the spread of the forward curve (long term forward – short term forward) as a proxy for inventory : when the spread is negative, inventory is low
→ Fama and French (1988) use LME Future prices over the period 1972 to1983 to test five base metals (copper, aluminium, copper, lead, tin and zinc) and find that the variance of spot prices declines with high inventories. In the case of gold, forward curve spreads provided little forecast for price volatility.
→Ng and Pirrong (1994) analyze four base metals over the period 1986 to 1992 and find persistence of the property that both spot and forward variance declines with inventory in the case of metals
→G - Nguyen ( 2005) reconstruct a world inventory of soybeans over several years and directly exhibit a quasi- perfect inverse relationship between inventory and spot price volatility
→G- Ohana (2009) . Examine at US crude oil and natural gas markets. Show that indeed the spread of the forward curve is a good proxy for
inventory. Exhibit that the correlation between the spread of the forward curve and low
inventory is particular significant during periods of scarcity
G – Smith (2012) . Reconstruct inventory for copper, lead, iron, tin from the addition of the LME and SHFE data. Validate the use of the spread of the forward curveas a measure of inventory. Display directly an affine relationship between inverse inventory and spot price volatility
Copper Inventory – 1985 to 2011
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Why Should Different Commodities Be Correlated?
Correlation (A,B) > 0
Economic Effects
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Why Should Different Commodities Be Correlated?
Correlation (A,B) > 0
Water for Agriculture or Mining or Oil & Gas “Fracking”
Shipping for Metal ore or Agriculturals
Land for Residential Land or Agriculture or Mining
Offshore Experts for Windfarms or Offshore Oil
Resource Competition
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Commodiity Montly Prices from World Bank http://data.worldbank.org/data-catalog/commodity-price-data
Agriculture Index versus Base Metals Index
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Why Should Different Commodities Be Correlated?
Aluminium Ore(Bauxite)
Aluminium
A is needed to produce B
Correlation (A,B) > 0
Lots of Electricity
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Why Should Different Commodities Be Correlated?
Substitution : A or B
Oil
Natural Gas
Electricity
Correlation (A,B) > 0
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Corn and Wheat Prices - 2000 to 2011
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
World Fertilizer Index – 2000 to 2011
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Why Should Different Commodities Be Correlated?
Correlation (B,C) > 0due to competition
Alternative Output : A can produce B or C
orSugar Cane
Sugar
Ethanol as Biofuel
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Commodiity Monthly Prices from World Bank http://data.worldbank.org/data-catalog/commodity-price-data
Sugar versus Crude Oil
?
Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith
Why Should Different Commodities Be Correlated?
Correlation (A,B) > 0 since supplies of both occur at the same time.
Co-Production : A and B produced together
togetherwith
Zinc
Lead
COMEX Gold Prices - 2002 to 2010
Gold Prices since 1975
COMEX Gold 28/2/2007
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s-07
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COMEX Gold
Gold Forward Curve - 27 May 2008
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M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18
Gold
Gold Forward Curve – Feb 2010
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Gold 4/ 9/ 2012
Gold
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Gold 14 Aug 2012
Gold
0.15
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0.23
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0.8 0.85 0.9 0.95 1 1.05 1.1 1.15
Impl
ied
Vola
tility
K/S
Gold 1st Contract (22 Jun 2012)
References
H. Geman and W. Smith (2012) “Inventories and Base Metals Forward Curves”, Resources PolicyH.Geman and S. Sarfo (2012) “Seasonality in Cocoa Spot and Forward Markets: Empirical Evidence”, Journal of Agricultural Expansion and Rural Development H.Geman ( 2011) “ Volatility in Commodity Spot Markets: Speculation or Scarcity?”, Swiss Derivatives ReviewH.Geman (2010) “Commodities and Numéraire”, Encyclopedia of Quantitative FinanceH. Geman and Yfong Shi (2009) “ The CEV model for Commodity Prices”, Journal of Alternative InvestmentsH. Geman and S. Kourouvakalis (2008) "A Lattice-Based Method for Pricing Electricity Derivatives under the Geman-Roncoroni Model", Applied Mathematical FinanceH. Geman and C. Kharoubi( 2008) “Diversification with Crude Oil Futures : the Time-to- Maturity Effect, Journal of Bankingand FinanceS. Borovkova and H. Geman (2006) "Seasonal and Stochastic Effects in Commodity Forward Curves", Review of Derivatives ResearchH. Geman and A. Roncoroni (2006) "Understanding the Fine Structure of Electricity Prices", Journal of BusinessH. Geman (2005) "Energy Commodity Prices: Is Mean Reversion Dead?", Journal of Alternative InvestmentsH. Geman and S. Ohana (2009) "Inventory, Reserves and Price volatility in Oil and Natural Gas Markets“,Energy EconomicsH. Geman (2005) "Commodities and Commodity Prices: Pricing and Modeling for Agriculturals, Metals and Energy", Wiley FinanceH. Geman and V. Nguyen (2005) "Soybean inventory and forward curves dynamics", Management ScienceH.Geman (2004) “Water as the Next Commodity”, Journal of Alternative InvestmentsH. Geman and M. Yor (1993) "An Exact Valuation for Asian Option", Mathematical FinanceA. Eydeland and H. Geman (1999) "Fundamentals of Electricity options" in Energy Price Modellng, Risk BooksH. Geman and O. Vasicek (2001) "Forwards and Futures on Non Storable Commodities", RISKH. Geman (2003) "DCF versus Real Option for Pricing Energy Physical Assets" Conference of the International Energy Agency - Paris