1. Petroleum Concessions with Extendible Options Using Mean Reversion with Jumps to Model Oil Prices Workshop on Real Options in Petroleum and Energy September 19-20, 2002, Mexico City By: Marco A. G. Dias (Petrobras) & Katia M. C. Rocha (IPEA)

2. Presentation Highlights u Paper has two new contributions: l Extendible maturity framework for real options è Kemna (1993) model is too simplified (European option, etc.) l Use of jump-reversion process for oil prices è First presented in May/98 (Workshop on RO, Stavanger) u Presentation topics: l Concepts for options with extendible maturities è Thresholds for immediate development and for extension l Main stochastic processes for oil prices è Jump + mean-reversion: + real application on Marlim field l Dynamic programming x contingent claims l Model discussion, charts, C++ software interface l Public debate on timing policy for oil sector in Brazil l Concluding remarks

3. E&P Is a Sequential Options Process u Drill the pioneer? Wait? Extend? u Revelation and technical uncertainty modeling Oil/Gas Success Probability = p Expected Volume of Reserves = B Revised Volume = B’ u Appraisal phase: delineation of reserves u Invest in additional information? u Develop? “Wait and See” for better conditions? Extend the option? u Developed Reserves. u Expand the production? Stop Temporally? Abandon? Primary focus of our model: undeveloped reserves

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

5. Brazilian Timing Policy for the Oil Sector u With the Brazilian petroleum sector opening in 1997, the new regulation for exploratory areas is: l Fiscal regime of concessions, first-price sealed bid (like USA) l Adopted the concept of extendible options (two or three periods). è The time extension is conditional to additional exploratory commitment (1-3 wells), established before the bid. è Let K be the cost (or exercise price) to extend the exploratory concession term. The benefit is another option term to explore and/or to develop. l The extendible feature occurred also in USA (5 + 3 years, for some areas of GoM) and in Europe (see paper of Kemna, 1993) l American options with extendible maturities was studied by Longstaff (1990) for financial applications l The timing for exploratory phase (time to expiration for the development rights) was object of a public debate è The National Petroleum Agency posted the first project for debate in its website in February/1998, with 3 + 2 years, time we considered too short

6. 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 National Agency u For the geometric Brownian motion, the payoff at T 1 is:

7. Main Stochastic Processes for Oil Prices u There are many models of stochastic processes for oil prices in real options literature. I classify them into three classes. u The nice properties of Geometric Brownian Motion (few parameters, homogeneity) is a great incentive to use it in real options applications. u We (Dias & Rocha) used the mean-reversion process with jumps of random size and also the geometric Brownian motion for comparison

8. Mean-Reversion + Jump: the Sample Paths  100 sample paths for mean-reversion + jumps ( =  jump each 5  years)

9. Nominal Prices for Brent and Similar Oils (1970-2001) u With an adequate long-term scale, we can see that oil prices jump 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, 1997, 2001 Jumps-up Jumps-down

10. 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

11. 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 level 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 few months, this kind of news causes jumps in the prices, due the 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 Microeconomic logic; term structure and volatility of futures prices; econometric tests with long time-span l However, reversion in oil prices is slow (Pindyck, 1999)

12. 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, price is expected to double l In case of jum-down, price is expected to drop by half ; So,

13. Real Case with Mean-Reversion + Jumps u A similar process of mean-reversion with jumps was used by Dias for the equity design (US$ 200 million) of the Project Finance of Marlim Field (oil prices-linked spread) l Equity investors reward: è Basic interest-rate + spread (linked to oil business risk) l Oil prices-linked: transparent deal (no agency cost) and win-win: è Higher oil prices  higher spread, and vice versa (good for both) u Deal was in December 1998 when oil price was 10 $/bbl l We convince investors that the expected oil prices curve was a fast reversion towards US$ 20/bbl (equilibrium level) l Looking the jumps-up & down, we limit the spread by putting both cap (maximum spread, protecting Petrobras) and floor (to prevent negative spread, protecting the investor) l This jumps insight proved be very important: è Few months later the oil prices jumped-up (price doubled by Aug/99) –The cap protected Petrobras from paying a very high spread

14. Parameters Values for the Base Case u The more complex stochastic process for oil prices (jump- reversion) demands several parameters estimation l Jumps frequency: counting process with a jump criteria l The jumps data were excluded in order to estimate mean- reversion (jumps and reversion processes are independent) u The criteria for the base case parameters values were: l Looking values used in literature for mean-reversion è For drift related parameters, is better a long time series than a large number of samples (Campbell, Lo & MacKinlay, 1997 ) è Large number of samples is better for volatility estimation l Econometric 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) l Used other econometric (classical) approaches u Several sensibility analysis were performed, filling the gaps

15. Jump-Reversion Base Case Parameters

16. Mean-Reversion and Jumps Parameters u The long-run mean or equilibrium level which the prices tends to revert can be estimated by econometric way l Another idea is 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 (GBM) and positively correlated with the prices level P u Slowness of a reversion: the half-life (H) concept l Time for the price deviations from the equilibrium-level are expected to decay by half of their magnitude. Range: 1-5 years  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. Poisson is a counting process and we consider only large-jumps to set this frequency. l We allow also the jump risk be systematic, so is not possible to build a riskless portfolio as in Merton (1976). We use dynamic programming

17. 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, was first performed by using the Bellman-dynamic programming equations:

18.  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 yield]  Conclusion: Anyway we need  for mean-reversion process, because  is a function of  ;  is not constant as in the GBM u As in Dixit & Pindyck (1994), we use dynamic programming Let  be an exogenous risk-adjusted discount rate that considers the incomplete markets/systematic jump feature l We compare the results dynamic programming x contingent claims A Motivation for Using Dynamic Programming u First, see the contingent claims PDE version of this model: u Compare with the dynamic programming version:  estimation is necessary even for contingent claims

19. Boundary Conditions u In the boundary conditions are addressed: Payoff for an immediate development is NPV/bbl = V  D. l Developed reserve value is proportional to P: V = q P 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 u A C++ software was developed with an interactive interface

20. C++ Software Interface: The Main Window u Software solves extendible options for 3 different stochastic processes and two methods (dynamic programming and contingent claims)

21. 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

22. 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)

23. Debate on Exploratory Timing Policy u The oil companies considered very short the time of 3 + 2 years that appeared in the first draft by National Agency l It was below the international practice mainly for deepwaters areas (e.g., USA/GoM: some areas 5 + 3 years; others 10 years) l During 1998 and part of 1999, the Director of the National Petroleum Agency (ANP) insisted in this short timing policy l The numerical simulations of our paper (Dias & Rocha, 1998) concludes that the optimal timing policy should be 8 to 10 years l In January 1999 we sent our paper to the notable economist, politic and ex-Minister Delfim Netto, highlighting this conclusion l In April/99 (3 months before the first bid), Delfim Netto wrote an article at Folha de São Paulo (a top Brazilian newspaper) defending a longer timing policy for petroleum sector l Delfim used our paper conclusions to support his view! l Few days after, the ANP Director finally changed his position! è Since the 1 st bid most areas have 9 years. At least it’s a coincidence!

24. Alternatives Timing Policies in Dias & Rocha u The table below presents the sensibility analysis for different timing policies for the petroleum sector l Option values (F) are proxy for bonus in the bid l Higher thresholds (P*) means more delay for investments è Longer timing means more bonus but more delay (tradeoff) u Table indicates a higher % gain for option value (bonus) than a % increase in thresholds (delay) l So, is reasonable to consider something between 8-10 years

25. 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 was large  OBS: for contingent claims, we adopt  = 10% and r = 5% to compare

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

27. 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/

28. 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 low oil prices P:  ( P  P) A necessary condition for American call early exercise is  > 0

29. Concluding Remarks u The paper main contributions were: l Use of the American call options with extendible maturities framework for real assets l We use a more rigourous and 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 be stochastic and positively correlated with P u Comparing with GBM, jump-reversion presented: l Higher options value (higher bonus); higher thresholds for short lived options (concessions) and lower for long lived one u First time a real options paper contributed in a Brazilian public debate being cited by a top newspaper

30. Additional Materials for Support

31. 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 (*)

32. 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)

33. 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)

34. 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

35. 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)

36. 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

37. 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

38. 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)

39. 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

40. 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)

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

42. Software Interface: Data Input Window

43. u Sensibility analysis show that the options values increase in case of: Increasing the reversion speed  (or decreasing the half-life H). But note that P 0 < P in the base case; 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

44. Sensibility Analysis: Reversion Speed

45. Sensibility Analysis: Discount Rate 

46. 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 

47. Sensibility Analysis: Lon-Run Mean

48. Sensibility Analysis: Time to Expiration

49. Sensibility Analysis: Economic Quality of Reserve

50. Geometric Brownian Base Case

51. 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!

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