Real Options in Energy Finance May 27 2016

1. Real Options in Energy FinanceMay 27 2016 Canada Research Chair in Quantitative Finance, Western University Canada Talk to Universita degli Studi Verona Matt Davison

2. Energy Markets Collaborators • Former PhD students • Matt Thompson, Faculty of Business, Queen's University • Natasha Kirby, Energy Desk, Royal Bank of Canada. • Current PhD Student Christian Maxwell • My senior colleague and valued mentorHenning Rasmussen, 1939-2009

3. Energy Finance • Energy is a fundamental human need • Scientists and Engineers play essential roles • Energy comes from the wrong places, is available at the wrong time, and in fuel of the wrong form • So diplomats and policy makers are also critical! • Energy Finance helps with all these solutions

4. Energy Markets The wrong time • Weekly loads • Daily loads

5. The wrong place • We are currently having lots of political battles in Canada about pipelines between Alberta and Saskatchewan and a) the US b) British Columbia and c) Eastern Canada • For years Western political leaders have bewailed that oil comes from Iran, Saudi Arabia, and Venezuela

6. In the wrong form • There is lots of energy in coal and uranium • We know how to turn this energy into electricity (physics and engineering both!) • But a nuclear powered aircraft carrier is one thing, a nuclear powered car another. • Plus they are both, in their own ways, “dirty”

7. Real Options in Energy Finance • Energy Finance real options allow energy to be moved from place to place (pipeline, oil tanker; not discussed today) • From time to time (natural gas storage, discussed today) • From form to form (corn ethanol plants, discussed today)

8. Resulting problems require • Knowledge of physics, chemistry, and engineering behind the processes and plant design • Incorporation of lots of messy reality • Powerful mathematical derivations leading to complicated mathematical problems that must be solved numerically

9. Benefits of these solutions include • Better ways to operate power plants/storage facilities/factories (optimal ways, in fact, if the underlying models can be believed!) • Ways to think about energy policy

10. Agenda • Natural Gas Storage • Ethanol plants • Some speculations

11. 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.

12. 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

13. 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

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

15. Resources in Ontario

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

17. Aggregate Inject/Withdraw

18. Modeling a single facility • Use Merton’s application of Bellman’s principle to finance • Incorporate engineering details • Details in Thompson, Davison, & Rasmussen, Naval Research Logistics, 2009 • Related papers are: • Chen & Forsyth, SIAM J. Sci. Comp 2007 • Carmona & Ludkowski, Quant Finance 2010

19. 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 • Max release rate of gas; Depends on gas level • Injection capacity • Max injection rate of gas; Depends on gas level • Cycling • Salt caverns are HDMC • Reservoir seepage • Cost to pump gas

20. 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;

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

22. 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

23. 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

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

25. 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)

26. 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%

27. The PDE The function a The PDE Then

28. Natural Gas Control Surface

29. Natural Gas Value Surface

30. Put and Call

31. Gas Storage Conclusions • Energy storage allowed gas at one time to be converted into gas at another time • How to do this requires physics and engineering knowledge as well as finance • The fact that seasonality persists in the forward curve proves storage is still only part of the puzzle.

32. A European Take • With Stephan Schlueter, then an economics PhD student from University of Erlangen who visited my team for a time in 2008, we repeated all this for a Dutch nat gas hub and European gas price models. • Fewer jumps, more “garch”. • Played with various lease final conditions. • Results are in his thesis.

33. Ethanol • In recent decades, ethanol biofuel production in North America has become popular • Energy independence from Middle East (?) • Reduced environmental impact (?) • Powerful agricultural lobby • Have the resulting subsidies led to an increased correlation between food and fuel prices? What impact might this have?

34. An Ethanol Plant

35. What is the real option?: • Run plant when ethanol is worth more than cost of inputs; idle plant when it is not • There are switching costs between “on” and “off” – this will lead to hysteresis

36. Assembling the model • A model for policy impact on ethanol production requires: • A model for the plant including flexibility of control, capitalized construction costs, profit as a function of ethanol/corn prices, costs to pause/resume production • Stochastic model for corn & ethanol prices • Optimal operating rule which maximizes future profits

37. Management Flexibility • Management has two options: • Enter into investment with capitalized cost of $B per gallon of production capacity • Start/Stop production to avoid running at a loss over sustained periods or to recapture profits if price spread becomes favourable again • Two main states: on (1), off (0). • Turn production “on” from “off”: penalty $D01 per gallon • Turn production “off” from “on”: penalty $D10 per gallon

38. Profit process • In regime I running profit is fi(…) • Corn  ethanol + by-products • Yield (gallons ethanol/bushel corn): κ • Ethanol $Xt/gallon; Corn $Yt (/bushel) • Running costs – subsidy = $K1/gallon • So f1 = κ(Xt-K1)-Yt • Also f0 = -κ(K0)

39. Price Processes

40. Modelling This • Kirby & Davison (2010) found weak evidence for mean reversion or seasonality in monthly corn/gasoline prices, so we use: • (Ethanol) dXt = μXt dt + σXt dW1t • (Corn) dYt = aYt dt + bYt dW2t • E[dW1t dW2t] = ρdt • Parameters from basic regression analysis

41. Expected Discounted Earnings Ji(t,α,x,y) = E{ ʃtT exp[-r(s-t)]fis(s,Xs,Ys)ds - Σk=1n exp[-r(τk – t)]Dik-1,ik | Ft} • Here α = (τk , ik) is a switching control between on and off states • t < τk <T an increasing sequence of switching times, ik corrresponding states • Assume facility has no salvage value

42. The Value Function • Value function in state I at time t assuming subsequent optimal control is • Vi(t,x,y) = supα Ji(t,α,x,y) Vi(t,x,y) = sup τ E{ ʃtτ exp-r(s-t)fis(s,Xs,Ys)ds + max Σ1n e-r(τ – t)[Vj(τ,Xτ,Yτ)- Dij]I(τ < T) | Ft} • Where the max is over taking the value in the same state as time t or in the opposite state and paying the appropriate change penalty

43. Dynamic Programming  PDE • Using the same kind of dynamic programming argument as shown in the nat gas storage problem above, yields the following system of free boundary PDE: • Except here we choose between staying and switching thus:

44. The Moving Boundary PDE • Either it is optimal to stay in the current state i, in which case this PDE is satisfied: δVi/δt + L[Vi] + fi(t,x,y) – rVi = 0, Vi ≥ Vj – Dij • Or it is optimal to switch in which case: Vi(t,x,y) = Vj(t,x,y) – Dij • Here L is the generator associated with the Diffusion (the usual 2D Black Scholes parts)

45. Variational inequality • Can also write this as a variational inequality: Max{δVi/δt + L[Vi] + fi(t,x,y) – rVi, Vj(t,x,y) – Dij} Switching boundaries areδSij = {t,x,y: Vi = Vj- Dij} (there will be two: a switch on boundary and a switch off boundary). Smooth pasting condition holds along δSij

46. Finite Difference • Discretize in time and space (Time step h). • L = difference operator corresponding to differential operator (usual centred difference approximation for first and second ‘spatial’ derivatives). • Leads to Matrix System: (Vk+1 – Vk)/h + LVk + fk ≤ 0

47. linear complementarity problem • This leads to the coupled linear complementarity problem: • MV – b ≤ 0, V ≥ g where at least one equality holds, so: • (MV-b)T(V-g) = 0. • g =V*-D* (switch bdy), b = -Vk+1 – fk, M = hL-I • Equiv to: minx (xTMx – qx) st. x ≥ 0, q = V - g

48. The data structure

49. Vij = function (itself, neighbours)

50. Outline of policy investigation • We investigate the effects of: • Increased corn-ethanol correlation • Increased corn prices • Reduced ethanol subsidy • On: • Plant value • Operating Characteristics • Decision to enter into investment