Petroleum Concessions with Extendible Options Using Mean Reversion with Jumps to Model Oil Prices

1. Petroleum Concessions with Extendible Options Using Mean Reversion with Jumps to Model Oil Prices By: Marco A. G. Dias (Petrobras) & Katia M. C. Rocha (IPEA) . 3rd Annual International Conference on Real Options - Theory Meets Practice Wassenaar/Leiden, The Netherlands June 1999

2. Presentation Highlights Paper has two new contributions: Extendible maturity framework for real options Use of jump-reversion process for oil prices Presentation of the model: Petroleum investment model Concepts for options with extendible maturities Thresholds for immediate investment and for extension Jump + mean-reversion process for oil prices Topics: systematic jump, discount rate, convenience yield C++ software interactive interface Base case and sensibility analysis Alternative timing policies for Brazilian National Agency Concluding remarks

3. E&P Is a Sequential Options Process Drill the pioneer? Wait? Extend? Revelation, option-game: waiting incentives

4. Economic Quality of a Developed Reserve Concept by Dias (1998): q = ?V/?P q = economic quality of the developed reserve V = value of the developed reserve ($/bbl) P = current petroleum price ($/bbl) For the proportional model, V = q P, the economic quality of the reserve is constant. We adopt this model. The option charts F x V and F x P at the expiration (t = T)

5. The Extendible Maturity Feature

6. Options with Extendible Maturity Options with extendible maturities was studied by Longstaff (1990) for financial applications We apply the extendible option framework for petroleum concessions. The extendible feature occurs in Brazil and Europe Base case of 5 years plus 3 years by paying a fee K (taxes and/or additional exploratory work). Included into model: benefit recovered from the fee K Part of the extension fee can be used as benefit (reducing the development investment for the second period, D2) At the first expiration, there is a compound option (call on a call) plus a vanilla call. So, in this case extendible option is more general than compound one

7. Extendible Option Payoff at the First Expiration At the first expiration (T1), the firm can develop the field, or extend the option, or give-up/back to govern For geometric Brownian motion, the payoff at T1 is:

8. Poisson-Gaussian Stochastic Process We adapt the Merton (1976) jump-diffusion idea but for the oil prices case: Normal news cause only marginal adjustment in oil prices, modeled with a continuous-time process Abnormal rare news (war, OPEC surprises,...) cause abnormal adjustment (jumps) in petroleum prices, modeled with a discrete time Poisson process Differences between our model and Merton model: Continuous time process: mean-reversion instead the geometric Brownian motion (more logic for oil prices) Uncertainty on the jumps size: two truncated normal distributions instead the lognormal distribution Extendible American option instead European vanilla Jumps can be systematic instead non-systematic

9. Stochastic Process Model for Oil Prices Model has more economic logic (supply x demand) Normal information causes smoothing changes in oil prices (marginal variations) and means both: Marginal interaction between production and demand (inventory levels is an indicator); and Depletion versus new reserves discoveries (the ratio of reserves/production is an indicator) Abnormal information means very important news: In a short time interval, this kind of news causes a large variation (jumps) in the prices, due to large variation (or expected large variation) in either supply or demand Mean-reversion has been considered a better model than GBM for commodities and perhaps for interest rates and for exchange rates. Why? Economic logic; term structure of futures prices; volatility of futures prices; spot prices econometric tests

10. Nominal Prices for Brent and Similar Oils (1970-1999)

11. Equation for Mean-Reversion + Jumps The stochastic equation for the petroleum prices (P) Geometric Mean-Reversion with Random Jumps is:

12. Mean-Reversion and Jumps for Oil Prices The long-run mean or equilibrium level which the prices tends to revert P is hard to estimate Perhaps a game theoretic model, setting a leader-follower duopoly for price-takers x OPEC and allies A future upgrade for the model is to consider P as stochastic and positively correlated with the prices level P Slowness of a reversion: the half-life (H) concept Time for the price deviations from the equilibrium-level are expected to decay by half of their magnitude. H = ln(2)/(h P ) The Poisson arrival parameter l (jump frequency), the expected jump sizes, and the sizes uncertainties. We adopt jumps as rare events (low frequency) but with high expected size. So, we looking to rare large jumps (even with uncertain size). Used 1 jump for each 6.67 years, expecting doubling P (in case of jump-up) or halving P (in case of jump-down). Let the jump risk be systematic, so is not possible to build a riskless portfolio as in Merton (1976). We use dynamic programming

13. Dynamic Programming and Options

14. A Motivation for Using Dynamic Programming First, see the contingent claims PDE version of this model:

15. Boundary Conditions

16. The C++ Software Interface: Main Window

17. The C++ Software Interface: Progress Calculus Window

18. Main Results Window

19. Parameters Values for the Base Case The more complex stochastic process for oil prices (jump-reversion) demands several parameters estimation The criteria for the base case parameters values were: Looking values used in literature (others related papers) Half-life for oil prices ranging from less than a year to 5 years For drift related parameters, is better a long time series than a large number of samples (Campbell, Lo & MacKinlay, 1997 ) Looking data from an average oilfield in offshore Brazil Oilfield currently with NPV = 0; Reserves of 100 millions barrels Preliminary estimative of the parameters using dynamic regression (adaptative model), with the variances of the transition expressions calculated with Bayesian approach using MCMC (Markov Chain Monte Carlo) Large number of samples is better for volatility estimation Several sensibility analysis were performed, filling the gaps

20. Jump-Reversion Base Case Parameters

21. The First Option and the Payoff Note the smooth pasting of option curve on the payoff line The blue curve (option) is typical for mean reversion cases

22. The Two Payoffs for Jump-Reversion

23. The Options and Payoffs for Both Periods

24. Options Values at T1 and Just After T1

25. The Thresholds Charts for Jump-Reversion

26. Alternatives Timing Policies for Petroleum Sector The table presents the sensibility analysis for different timing policies for the petroleum sector Option values are proxy for bonus in the bidding Higher thresholds means more investment delay Longer timing means more bonus but more delay (tradeoff) Results indicate a higher % gain for option value (bonus) than a % increase in thresholds (delay) So, is reasonable to consider something between 8-10 years

27. Alternatives Timing Policies for Petroleum Sector The first draft of the Brazilian concession timing policy, pointed 3 + 2 = 5 years The timing policy was object of a public debate in Brazil, with oil companies wanting a higher timing In April/99, the notable economist and ex-Finance Minister Delfim Netto defended a longer timing policy for petroleum sector using our paper: In his column from a top Brazilian newspaper (Folha de S�o Paulo), he commented and cited (favorably) our paper conclusions about timing policies to support his view! The recent version of the concession contract (valid for the 1st bidding) points up to 9 years of total timing, divided into two or three periods So, we planning an upgrade of our program to include the cases with three exploration periods

28. Comparing Dynamic Programming with Contingent Claims Results show very small differences in adopting non-arbitrage contingent claims or dynamic programming However, for geometric Brownian motion the difference is very large

29. Sensibility Analysis: Jump Frequency Higher jump frequency means higher hysteresis: higher investment threshold P* and lower extension threshold PE

30. Sensibility Analysis: Volatility

31. Comparing Jump-Reversion with GBM Is the use of jump-reversion instead GBM much better for bonus (option) bidding evaluation? Is the use of jump-reversion significant for investment and extension decisions (thresholds)? Two important parameters for these processes are the volatility and the convenience yield d. In order to compare option value and thresholds from these processes in the same basis, we use the same d In GBM, d is an input, constant, and let d = 5%p.a. For jump-reversion, d is endogenous, changes with P, so we need to compare option value for a P that implies d = 5%:

32. Comparing Jump-Reversion with GBM

33. Concluding Remarks The paper main contributions are: Use of the options with extendible maturities framework for real assets, allowing partial recovering of the extension fee K We use a more rigourous and more logic but more complex stochastic process for oil prices (jump-reversion) The main upgrades planned for the model: Inclusion of a third period (another extendible expiration), for several cases of the new Brazilian concession contract Improvement on the stochastic process, by allowing the long-run mean P to be stochastic and positively correlated First time a real options paper cited in Brazilian important newspaper Comparing with GBM, jump-reversion presents: Higher options value (higher bonus); higher thresholds for short lived options (concessions) and lower for long lived one

34. Additional Materials for Support

35. Demonstration of the Jump-Reversion PDE Consider the Bellman for the extendible option (up T1):

36. Finite Difference Method Numerical method to solve numerically the partial differential equation (PDE) The PDE is converted in a set of differences equations and they are solved iteratively There are explicit and implicit forms Explicit problem: convergence problem if the �probabilities� are negative Use of logaritm of P has no advantage for mean-reverting Implicit: simultaneous equations (three-diagonal matrix). Computation time (?) Finite difference methods can be used for jump-diffusions processes. Example: Bates (1991)

37. Explicit Finite Difference Form Grid: Domain space DP x Dt Discretization F(P,t) � F( iDP, jDt ) � Fi, j With 0 � i � m and 0 � j � n where m = Pmax/DP and n = T / Dt

38. Finite Differences Discretization The derivatives approximation by differences are the central difference for P, and foward-difference for t: FPP � [ F i+1,j - 2Fi,j + Fi-1,j ] / (DP)2 FP � [ F i+1,j - Fi-1,j ] / 2DP Ft � [ F i,j+1 - Fi,j ] / Dt Substitutes the aproximations into the PDE

39. Economic Quality of a Developed Reserve

40. Sensibility analysis show that the options values increase in case of: Increasing the reversion speed h (or decreasing the half-life H); Decreasing the risk-adjusted discount rate r, because it decreases also d, due the relation r = h(P - P) + d , increasing the waiting effect; Increasing the volatility s do processo de revers�o; Increasing the frequency of jumps l; Increasing the expected value of the jump-up size; Reducing the cost of the extension of the option K; Increasing the long-run mean price P; Increasing the economic quality of the developed reserve q; and Increasing the time to expiration (T1 and T2) Others Sensibility Analysis

41. Sensibility Analysis: Reversion Speed

42. Sensibility Analysis: Discount Rate r

43. Estimating the Discount Rate with Market Data A practical �market� way to estimate the discount rate r in order to be not so arbitrary, is by looking d with the futures market contracts with the longest maturity (but with liquidity) Take both time series, for d (calculated from futures) and for the spot price P. With the pair (P, d) estimate a time series for r using the equation: r(t) = d (t) + h[P - P (t)]. This time series (for r) is much more stable than the series for d. Why? Because d and P has a high positive correlation (between +0.809 to 0.915, in the Schwartz paper of 1997) . An average value for r from this time series is a good choice for this parameter OBS: This method is different of the contingent claims, even using the market data for r

44. Sensibility Analysis: Lon-Run Mean

45. Sensibility Analysis: Time to Expiration

46. Sensibility Analysis: Economic Quality of Reserve

47. Geometric Brownian Base Case

48. Drawbacks from the Model The speed of the calculation is very sensitive to the precision. In a Pentium 133 MHz: Using DP = 0.5 $/bbl takes few minutes; but using more reasonable DP = 0.1, takes two hours! The point is the required Dt to converge (0.0001 or less) Comparative statics takes lot of time, and so any graph Several additional parameters to estimate (when comparing with more simple models) that is not directly observable. More source of errors in the model But is necessary to develop more realistic models!

49. The Grid Precision and the Results

50. Software Interface: Data Input Window