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Matt Davison

Natural Gas Markets – Spot, Forward, and Real Options. Matt Davison. Departments of Applied Mathematics and of Statistical & Actuarial Sciences, The University of Western Ontario. Natural Gas.

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Matt Davison

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  1. Natural Gas Markets – Spot, Forward, and Real Options Matt Davison Departments of Applied Mathematics and of Statistical & Actuarial Sciences, The University of Western Ontario

  2. Natural Gas On NYMEX, Natural Gas futures is based on 10,000 mm Btu (million btus). The price is quoted in dollars per mm Btu.

  3. Natural Gas Price Time-series of Henry Hub natural gas prices 1995-1999

  4. Outline • Stylized features of natural gas markets • Some simple spot models • An example full forward curve model • Untidy reality • Real options: natural gas storage • A natural gas trading disaster • Conclusions; electricity preview

  5. Mean Reverting Spot Models • Mean-reversion models are common for modelling commodity spot prices • the one factor Pilipovic Model: • Comparing with GBM

  6. Two-factor Pilipovic Model Where the two Brownian risk factors are correlated:

  7. The Solutions of Pilipovic Model • The explicit solution of one-factor model • The explicit solution of two-factor model • In the special case: ,two-factor reduce to one-factor Pilipovic model

  8. European Option Pricing Formula Using one-factor Pilipovic model

  9. Stylized Features of Nat Gas Markets • Highly seasonal, with ‘oscillating’ forward curve • High volatility levels, in 30%-100% range

  10. Natural Gas Volatility Features • High volatility levels – 30%-100% • Volatilities of futures increase as maturity approaches (Samuelson effect) • Considerable volatility skew, esp. for short maturities • Skew is positive for OTM calls and negative for OTM puts • ATM volatilities display seasonal effects

  11. Capturing this with spot models • Pilipovic one and 2 factor models • Last week’s model (Ribiero & Hodges 2004) • Spot model  forward model (solution of a PDE) BUT • Very hard to construct such models which have all the right features.

  12. Modeling • Suggests modeling the entire term structure of the forward curve, like HJM or like Jara (2000) • Approach I describe here comes from Powojowski (2007). • Can introduce jumps to capture the volatility skew (Merton 1976; Cont and Tankov 2003)

  13. Powojowski Model

  14. More specification chosen in order to guarantee that the solution of (Miro1) is a martingale. Assume specific form for volatility functions

  15. Solution of the Model

  16. Characteristic Function of F(t,T) • The characteristic function of F(t, T): • The factor can be computed through a combination of analytical and numerical integration. • The random variable G(t,T) is normally distributed: • Hence, • And

  17. Can Price Vanilla Options • Also price strips of forward starting options – cliquet or ratchet options • Can also price swaptions and calendar spreads

  18. Covariance Structure of Forward Curve • Let

  19. Untidy Reality • Like all commodity markets Natural Gas markets involve “real” things. • But Natural gas is “more so”. • Local in Space • Local in Time • Demand and Supply are weather dependent

  20. Where is Natural Gas produced? • In Canada Natural gas is produced chiefly in Alberta and Saskatchewan (but also to a limited extent in SW Ontario); in the US also in the Gulf of Mexico, Texas, California and to a limited extent in Appalachia • Worldwide it is produced in the North Sea, in the Middle East, and in Russia

  21. Where is Natural Gas consumed? • Everywhere, but in Canada to a great extent in the Eastern half (population density higher). • Natural gas must be transported on a pipeline network and refined (in Canada, often in Sarnia) before being used. • Liquidity in North American Markets: Henry Hub in Louisiana, AECO hub in Alberta

  22. Not Really A World Market • No pipelines between here and the mid-East • However Liquified Natural Gas (LNG) can be transported by ship. • This is expensive but worth the effort since, for a fossil fuel, natural gas is very clean (short hydrocarbon chains  less pollutants and greenhouse gas emissions per unit burned).

  23. LNG Storage

  24. Local in Time • Natural Gas is difficult to store (about which more later) • Demand for Nat Gas is highly seasonal (in winter for heating; in summer for electricity generation/air conditioning) • This explains the bumps in the forward curve

  25. Weather Dependent • Seasonality is a function of temperature dependence: as temperature rises above a threshold (18 Celsius) or below a similar threshold, gas use increases dramatically. • Links with Heating/Cooling Degree Day derivatives

  26. Weather Dependence (II) • Production of gas from the Gulf of Mexico (as well as refining) is also weather dependent • Hurricane Katrina devastated oil and gas production for several months. • For oil markets this was a small problem; because of the local nature of gas markets it was a proportionally much bigger problem.

  27. Financial vs. Fundamental • I have heard that 5 years ago the fundamentals of gas markets were equally important to the financial aspects but that more recently the balance is more like 70 financial, 30 fundamental. • But 30% is still a lot and if you miss it you can get into deep trouble, about which more later.

  28. Natural Gas Storage Facilities • Natural gas can be stored underground in • salt caverns • mines • aquifers • depleted oil/gas reservoirs • hard rock mines

  29. Resources in Ontario

  30. Storage, Injection, and Withdrawal • An aggregate US-level picture of storage and withdrawal is available from the US Energy Information Administration.

  31. Aggregate Inject/Withdraw

  32. Modeling a single facility • Use Merton’s application of Bellman’s principle to finance • Incorporate engineering details

  33. Physics/Engineering pV=nRT • Base gas capacity • Required for reservoir pressure • Never removed • Working gas capacity • Amount of gas available to produce and sell • Deliverability • Rate at which gas can be released • Depends on gas level • Injection capacity • Rate at which natural gas can be added • Depends on gas level • Cycling • Salt caverns are HDMC • Reservoir seepage • Cost to pump gas

  34. Variables in General Gas Storage Equations P – price per unit of natural gas; I – current amount of working natural gas inventory; c – control variable gas injected (c > 0) / stored (c < 0); Imax – max storage capacity of facility; Imin -- base gas capacity; cmax(I) – max deliverability rate as function of storage level; cmin(I) – min injection rate as function of storage level; a(I,c) – amount of gas lost given c units of gas released/injected;

  35. Optimization Framework I The objective function Subject to Change in I obeys ODE Change in P obeys Markov process

  36. Optimization Framework II To simultaneously determine optimal strategy c(P, I, t) and corresponding optimal value V(p, I, t), let Split integral to get Moving towards Bellman’s equation

  37. Standard Taylor Series arguments Employ Ito’s lemma to obtain Taylor series Eliminate all higher order terms and simplify Take expectations and divide by dt

  38. The PDE • The optimal value for c maximizes Subject to • The PDE Initial condition: Boundary conditions:

  39. The Numerical Difficulties • Hyperbolic in I • direction of information flow • upwind finite differencing • Total variation diminishing schemes • Slope limiting method works best • Method of lines approach (Mukadam)

  40. A Sample Problem The Stratton Ridge facility • Working gas capacity of 2000 MMcf • Base gas requirement 50 MMcf • Minimum capacity injectivity 80 MMcf/day • Injection pump requirement 1.7MMcf /day • No seepage from reservoir • Ideal gas law and Bernoulli's law apply • Prices in MMbtus • Time measured in years • Discount rate 10%

  41. The PDE The function a The PDE Then

  42. Natural Gas Control Surface

  43. Natural Gas Value Surface

  44. Put and Call

  45. A Natural Gas Trading Disaster • Perhaps because of the volatility and complexity of natural gas markets, large amounts have been made or lost trading them. • BMO lost a bundle in summer 2001; others more recently. • Most famous story, however, is that of Amaranth Partners

  46. The Tale of Amaranth Partners • Amaranth LLP was a Connecticut Hedge fund but with a strong Canadian connection • Their star trader, Brian Hunter, was based in Calgary. • In 2005 he had made several billion dollars (!) trading gas

  47. A Spread Trade Gone Bad • In early fall 2006 Hunter put on a spread trade between September and October natural gas futures • This was a bet that, as in fall 2005, a hurricane would hit the Gulf drilling platforms; also gas storage levels were inadequate. • But no hurricane and trade went bad, losing US$3.5 billion and bringing down the fund.

  48. Conclusions • Real Options are important for Natural Gas • As well as gas storage facilities natural gas electricity generating facilities are important natural gas real options. • These embody the so-called ‘spark spread’ • Gas is burned to generate electrical power • So to value these generation real options a model for electricity prices is also needed • That will be the topic of next week’s lecture.

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