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). 3 rd Annual International Conference on Real Options - Theory Meets Practice Wassenaar/Leiden, The Netherlands June 1999

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

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

4. Economic Quality of a Developed Reserve u Concept by Dias (1998): q =  V/  P l q = economic quality of the developed reserve l V = value of the developed reserve ($/bbl) l P = current petroleum price ($/bbl) u For the proportional model, V = q P, the economic quality of the reserve is constant. We adopt this model. l The option charts F x V and F x P at the expiration (t = T) F VD 45 o tg 45 o = 1 F PD/q  tg  = q = economic quality V = q. P F(t=T) = max (q P  D, 0) F(t=T) = max (NPV, 0) NPV = V  D

5. The Extendible Maturity Feature T 2 : Second Expiration t = 0 to T 1 : First Period T 1 : First Expiration T 1 to T 2 : Second Period [Develop Now] or [Wait and See] [Develop Now] or [Extend (pay K)] or [Give-up (Return to Govern)] T I M E PeriodAvailable Options [Develop Now] or [Wait and See] [Develop Now] or [Give-up (Return to Govern)]

6. Options with Extendible Maturity u Options with extendible maturities was studied by Longstaff (1990) for financial applications u We apply the extendible option framework for petroleum concessions. l The extendible feature occurs in Brazil and Europe l Base case of 5 years plus 3 years by paying a fee K (taxes and/or additional exploratory work). l Included into model: benefit recovered from the fee K è Part of the extension fee can be used as benefit (reducing the development investment for the second period, D 2 ) l 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 u At the first expiration (T 1 ), the firm can develop the field, or extend the option, or give-up/back to govern u For geometric Brownian motion, the payoff at T 1 is:

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

9. Stochastic Process Model for Oil Prices u Model has more economic logic (supply x demand) l Normal information causes smoothing changes in oil prices (marginal variations) and means both: è Marginal interaction between production and demand (inventory levels is an indicator); and è Depletion versus new reserves discoveries (the ratio of reserves/production is an indicator) l Abnormal information means very important news: è 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 u Mean-reversion has been considered a better model than GBM for commodities and perhaps for interest rates and for exchange rates. Why? l 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) u 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(?) Jumps-up Jumps-down

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

12. Mean-Reversion and Jumps for Oil Prices u The long-run mean or equilibrium level which the prices tends to revert P is hard to estimate l Perhaps a game theoretic model, setting a leader-follower duopoly for price-takers x OPEC and allies l A future upgrade for the model is to consider P as stochastic and positively correlated with the prices level P u 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)/(  P )  The Poisson arrival parameter (jump frequency), the expected jump sizes, and the sizes uncertainties. l We adopt jumps as rare events (low frequency) but with high expected size. So, we looking to rare large jumps (even with uncertain size). è Used 1 jump for each 6.67 years, expecting doubling P (in case of jump- up) or halving P (in case of jump-down). l 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 2 : Second Expiration t = 0 to T 1 : First Period T 1 : First Expiration T 1 to T 2 : Second Period Period Bellman Equations u The optimization under uncertainty given the stochastic process and given the available options, is performed by using the Bellman-dynamic programming equations:

14.  estimation is necessary even for contingent claims  Even discounting with risk-free rate, for contingent claims, appears the parameter risk-adjusted discount rate  This is due the convenience yield (  equation for the mean-reversion process:  (P  P) [remember  = growth rate + dividend rate]  Conclusion: Anyway we need  for mean-reversion process, because  is a function of  ;  is not constant as in the GBM  So, we let  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 of  : use the  time-series from futures market A Motivation for Using Dynamic Programming u First, see the contingent claims PDE version of this model: u Compare with the dynamic programming version:

15. Boundary Conditions u 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 l The extension feature at T 1, paying K and winning another call option è Absorbing barrier at P = 0 è First expiration optimally (include extension feature) è Smooth pasting condition (for both periods) è Value matching at P* (for both periods) è Second expiration optimally (D 2 can be different of D 1 ) u To solve the PDE, we use finite differences in explicit form u A C++ software was developed with an interactive interface

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

17. The C++ Software Interface: Progress Calculus Window u 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  P x  t (grid density)

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

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

20. Jump-Reversion Base Case Parameters

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

22. The Two Payoffs for Jump-Reversion u In our model we allow to recover a part of the extension fee K, by reducing the investment D 2 in the second period l The second payoff (green line) has a smaller development investment D 2 = 4.85 $/bbl than in the first period (D 1 = 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 I M E Options Charts T 2 : Second Expiration t = 0 to T 1 : First Period T 1 : First Expiration T 1 to T 2 : Second Period Period

24. Options Values at T 1 and Just After T 1 u At T 1 (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 l Boundary condition explains parallel distance of K in that interval l 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 u At or above the thresholds lines (blue and red, for the first and the second periods, respectively) is optimal the immediate development. l Extension (by paying K) is optimal at T 1 for 4.7 < P < 22.2 $/bbl l So, the extension threshold P E = 4.7 $/bbl (under 4.7, give-up is optimal)

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

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

28. Comparing Dynamic Programming with Contingent Claims u Results show very small differences in adopting non- arbitrage contingent claims or dynamic programming l However, for geometric Brownian motion the difference is very large  OBS: for contingent claims, we adopt  = 10% and r = 5% to compare

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

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

31. Comparing Jump-Reversion with GBM u Is the use of jump-reversion instead GBM much better for bonus (option) bidding evaluation? u 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 . In order to compare option value and thresholds from these processes in the same basis, we use the same   In GBM,  is an input, constant, and let  = 5%p.a.  For jump-reversion,  is endogenous, changes with P, so we need to compare option value for a P that implies  = 5%: u Sensibility analysis points in general higher option values (so higher bonus-bidding) for jump-reversion (see Table 3)

32. Comparing Jump-Reversion with GBM u Jump-reversion points lower thresholds for longer maturity  The threshold discontinuity near T 2 is due the behavior of , that can be negative for lower values of P:  ( P  P) A necessary condition for early exercise of American option is  > 0

33. Concluding Remarks u The paper main contributions are: l Use of the options with extendible maturities framework for real assets, allowing partial recovering of the extension fee K l We use a more rigourous and more logic but more complex stochastic process for oil prices (jump-reversion) u The main upgrades planned for the model: l Inclusion of a third period (another extendible expiration), for several cases of the new Brazilian concession contract l Improvement on the stochastic process, by allowing the long-run mean P to be stochastic and positively correlated u First time a real options paper cited in Brazilian important newspaper u Comparing with GBM, jump-reversion presents: l 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 u Consider the Bellman for the extendible option (up T 1 ): u We can rewrite the Bellman equation in a general form:  Where  (P, t)  is the payoff  function that can be the extendible payoff (feature considered only at T 1 ) or the NPV from the immediate development. Optimally features are left to the boundary conditions. u We rewrite the equation for the continuation region in return form: u 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: u Substituting E[dF] into (*), we get the PDE presented in the paper (*)

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

37. Explicit Finite Difference Form  Grid: Domain space  P x  t Discretization F(P,t)  F( i  P, j  t  F i, j With 0  i  m and 0  j  n  where m = P max /  P and n = T /  t “Probabilities” p need to be positives in order to get the convergence (see Hull) P t Domain Space (distribution)

38. Finite Differences Discretization u The derivatives approximation by differences are the central difference for P, and foward-difference for t: F PP  [ F i+1,j  2F i,j + F i-1,j ] / (  P) 2 F P  [ F i+1,j  F i-1,j ] / 2  P F t  [ F i,j+1  F i,j ] /  t u Substitutes the aproximations into the PDE

39. Economic Quality of a Developed Reserve u Schwartz (1997) shows a chart NPV x spot price and gives linear for two and three factors models l 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 l In this example there are 10 years of production  is the reversion speed of the stochastic convenience yield u 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). è Concept doesn’t depend of a linear model, but it eases the calculus

40. u Sensibility analysis show that the options values increase in case of: Increasing the reversion speed  (or decreasing the half-life H); Decreasing the risk-adjusted discount rate , because it decreases also  due the relation  (P  P) + , increasing the waiting effect; Increasing the volatility  do processo de reversão; Increasing the frequency of jumps ; l Increasing the expected value of the jump-up size; l Reducing the cost of the extension of the option K; l Increasing the long-run mean price P; l Increasing the economic quality of the developed reserve q; and l Increasing the time to expiration (T 1 and T 2 ) Others Sensibility Analysis

41. Sensibility Analysis: Reversion Speed

42. Sensibility Analysis: Discount Rate 

43. Estimating the Discount Rate with Market Data  A practical “market” way to estimate the discount rate  in order to be not so arbitrary, is by looking  with the futures market contracts with the longest maturity (but with liquidity) Take both time series, for  (calculated from futures) and for the spot price P. With the pair (P,  ) estimate a time series for  using the equation:  (t)  (t)  [P  P (t)]. This time series (for  ) is much more stable than the series for . Why? Because  and P has a high positive correlation (between +0.809 to 0.915, in the Schwartz paper of 1997). An average value for  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 

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

49. The Grid Precision and the Results u The precision can be negligible or significant (values from an older base case)

50. Software Interface: Data Input Window