By: Marco A. G. Dias (Petrobras) & Katia M. C. Rocha (IPEA) .

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 Oil/Gas Success Probability = p • Drill the pioneer? Wait? Extend? • Revelation, option-game: waiting incentives Expected Volume of Reserves = B Revised Volume = B’ • Appraisal phase: delineation of reserves • Technical uncertainty: sequential options Primary focus of our model: undeveloped reserves • Develop? “Wait and See” for better conditions? Extend the option? • Developed reserves. Model: reserves value proportional to the oil prices, V = qP • q = economic quality of the developed reserve • Other (operational) options: not included

4. Economic Quality of a Developed Reserve V = q . P F(t=T) = max (q P - D, 0) F F tg q = q = economic quality tg 45o = 1 45o q D V D/q P • 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) F(t=T) = max (NPV, 0) NPV = V - D

5. The Extendible Maturity Feature t = 0 toT1:First Period T1: First Expiration T1 to T2:Second Period T2: Second Expiration Period Available Options [Develop Now] or [Wait and See] [Develop Now] or [Extend (pay K)] or [Give-up (Return to Govern)] T I M E [Develop Now] or [Wait and See] [Develop Now] or [Give-up (Return to Govern)]

6. Options with Extendible Maturity • Options withextendible 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) Jumps-down Jumps-up • We see oil prices jumps in both directions, depending of the kind of abnormal news: jumps-up in 1973/4, 1978/9, 1990, 1999 (?); and jumps-down in 1986, 1991, 1998(?)

11. Equation for Mean-Reversion + Jumps • The stochastic equation for the petroleum prices (P) Geometric Mean-Reversion with Random Jumps is: ; So, • The jump size/direction are random: f ~ 2N • In case of jump-up, prices are expected to double • In case of jum-down, prices are expected to halve

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 t = 0 toT1:First Period T1: First Expiration T1 to T2:Second Period T2: Second Expiration • The optimization under uncertainty given the stochastic process and given the available options, is performed by using the Bellman-dynamic programming equations: Period Bellman Equations

14. A Motivation for Using Dynamic Programming • First, see the contingent claims PDE version of this model: r estimation is necessary even for contingent claims • Compare with the dynamic programming version: • Even discounting with risk-free rate, for contingent claims, appears the parameter risk-adjusted discount rate r • This is due the convenience yield (d) equation for the mean-reversion process: d = r - h(P - P)[remember r = growth rate + dividend rate] • Conclusion: Anyway we need r for mean-reversion process, because d is a function of r ; d is not constant as in the GBM • So, we let r be an exogenous risk-adjusted discount rate that considers the incomplete markets/systematic jump feature, with dynamic programming a la Dixit & Pindyck (1994) • A market estimation ofr : use the d time-series from futures market

15. Boundary Conditions • Absorbing barrier at P = 0 • First expiration optimally (include extension feature) • Value matching at P* (for both periods) • Second expiration optimally (D2 can be different of D1) • Smooth pasting condition (for both periods) • In the boundary conditions are addressed: • The NPV (payoff for an immediate development = V - D), which is function of q, that is, V = q P NPV = q P - D • The extension feature at T1, paying K and winning another call option • To solve the PDE, we use finite differences in explicit form • A C++ software was developed with an interactive interface

16. The C++ Software Interface: Main Window • Software solves extendible options for three different stochastic processes (two jump-reversion and the GBM)

17. The C++ Software Interface: Progress Calculus Window • The interface was designed using the C-Builder (Borland) • The progress window shows visual and percentage progress and tells about the size of the matrix DP x Dt (grid density)

18. Main Results Window • This window shows only the main results • The complete file with all results is also generate

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 • In our model we allow to recover a part of the extension fee K, by reducing the investment D2 in the second period • The second payoff (green line) has a smaller development investment D2 = 4.85 $/bbl than in the first period (D1 = 5 $/bbl) because we assume to recover 50% of K (e.g.: exploratory well used as injector)

23. The Options and Payoffs for Both Periods t = 0 toT1:First Period T1: First Expiration T1 to T2:Second Period T2: Second Expiration Options Charts Period T I M E

24. Options Values at T1 and Just After T1 • At T1 (black line), the part which is optimal to extend (between ~6 to ~22 $/bbl), is parallel to the option curve just after the first expiration, and the distance is equal the fee K • Boundary condition explains parallel distance of K in that interval • Chart uses K = 0.5 $/bbl (instead base case K = 0.3) in order to highlight the effect

25. The Thresholds Charts for Jump-Reversion • At or above the thresholds lines (blue and red, for the first and the second periods, respectively) is optimal the immediate development. • Extension (by paying K) is optimal at T1 for 4.7 < P < 22.2 $/bbl • So, the extension thresholdPE = 4.7 $/bbl (under 4.7, give-up is optimal)

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 São 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 • OBS: for contingent claims, we adopt r = 10% and r = 5% to compare

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

30. Sensibility Analysis: Volatility • Higher volatility also means higher hysteresis: higher investment threshold P* and lower extension threshold PE • Several other sensibilities analysis were performed • Material available at http://www.puc-rio.br/marco.ind/main.html

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%: • Sensibility analysis points in general higher option values (so higher bonus-bidding) for jump-reversion (see Table 3)

32. Comparing Jump-Reversion with GBM • Jump-reversion points lower thresholds for longer maturity • The threshold discontinuity near T2 is due the behavior of d, that can be negative for lower values of P: d = r - h(P - P) • A necessary condition for early exercise of American option is d > 0

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): • We can rewrite the Bellman equation in a general form: • Where W(P, t)is the payofffunction that can be the extendible payoff (feature considered only at T1) or the NPV from the immediate development. Optimally features are left to the boundary conditions. • We rewrite the equation for the continuation region in return form: (*) • The value E[dF] is calculated with the Itô´s Lemma for Poisson + Itô mix process (see Dixit & Pindyck, eq.42, p.86), using our process for dP: • Substituting E[dF] into (*), we get the PDE presented in the paper

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 Domain Space P t • Grid: Domain space DP x Dt • Discretization F(P,t) º F( iDP, jDt ) º Fi, j • With 0 £ i £ m and 0 £ j £ n • where m = Pmax/DP and n = T / Dt (distribution) “Probabilities” p need to be positives in order to get the convergence (see Hull)

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 • Economic quality of a developed reserve depends of the nature (permo-porosity and fluids quality), taxes, operational cost, and of the capital in-place (by D). • Concept doesn’t depend of a linear model, but it eases the calculus • Schwartz (1997) shows a chart NPV x spot price and gives linear for two and three factors models • For the two factors model, but with time varying production Q(t), the economic quality of a developed reserve q is: • Where A(t) is a non-stochastic function of parameters and time. A(t) doesn’t depend on spot price P • In this example there are 10 years of production • h is the reversion speed of the stochastic convenience yield

40. Others Sensibility Analysis • Sensibility analysis show that the options values increase in case of: • Increasing the reversion speedh (or decreasing the half-life H); • Decreasing the risk-adjusted discount rater, because it decreases also d, due the relation r = h(P - P) + d , increasing the waiting effect; • Increasing the volatilitys do processo de reversão; • Increasing the frequency of jumpsl; • 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)

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 • The precision can be negligible or significant (values from an older base case)

50. Software Interface: Data Input Window