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

• 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

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

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EP Is a Sequential Options Process
Oil/Gas Success Probability p

• Drill the pioneer? Wait? Extend?
• Revelation and technical uncertainty modeling

Expected Volume of Reserves B
Revised Volume B

• Appraisal phase delineation of reserves
• Invest in additional information?

Primary focus of our model undeveloped reserves

• Develop? Wait and See for better conditions?
  Extend the option?

• Developed Reserves.
• Expand the production? Stop Temporally? Abandon?

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The Extendible Maturity Feature (2 Periods)
Develop Now or Wait and See
Develop Now or Extend (commit K) or Give-up
(Return to Government)
Develop Now or Wait and See
Develop Now or Give-up (Return to Government)
5
Brazilian Timing Policy for the Oil Sector

• With the Brazilian petroleum sector opening in
  1997, the new regulation for exploratory areas
  is
• Fiscal regime of concessions, first-price sealed
  bid (like USA)
• Adopted the concept of extendible options (two or
  three periods).
• The time extension is conditional to additional
  exploratory commitment (1-3 wells), established
  before the bid.
• 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.
  
• The extendible feature occurred also in USA (5
  3 years, for some areas of GoM) and in Europe
  (see paper of Kemna, 1993)
• American options with extendible maturities was
  studied by Longstaff (1990) for financial
  applications
• The timing for exploratory phase (time to
  expiration for the development rights) was object
  of a public debate
• 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

• At the first expiration (T1), the firm can
  develop the field, or extend the option, or
  give-up/back to National Agency
• For the geometric Brownian motion, the payoff at
  T1 is

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Main Stochastic Processes for Oil Prices

• There are many models of stochastic processes for
  oil prices in real options literature. I classify
  them into three classes.

• The nice properties of Geometric Brownian Motion
  (few parameters, homogeneity) is a great
  incentive to use it in real options applications.
  
• We (Dias Rocha) used the mean-reversion process
  with jumps of random size and also the geometric
  Brownian motion for comparison

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Mean-Reversion Jump the Sample Paths

• 100 sample paths for mean-reversion jumps (l
  1 jump each 5 years)

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Nominal Prices for Brent and Similar Oils
(1970-2001)

• 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-down
Jumps-up
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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

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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 level is an indicator) and
• Depletion versus new reserves discoveries (the
  ratio of reserves/production is an indicator)
• Abnormal information means very important news
• In few months, this kind of news causes jumps in
  the prices, due the 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?
• Microeconomic logic term structure and
  volatility of futures prices econometric tests
  with long time-span
• However, reversion in oil prices is slow
  (Pindyck, 1999)

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Equation for Mean-Reversion Jumps

• The stochastic equation for the petroleum prices
  (P) Geometric Mean-Reversion with Random Jumps is

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Real Case with Mean-Reversion Jumps

• 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)
• Equity investors reward
• Basic interest-rate spread (linked to oil
  business risk)
• Oil prices-linked transparent deal (no agency
  cost) and win-win
• Higher oil prices ? higher spread, and vice
  versa (good for both)
• Deal was in December 1998 when oil price was 10
  /bbl
• We convince investors that the expected oil
  prices curve was a fast reversion towards US
  20/bbl (equilibrium level)
• 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)
• This jumps insight proved be very important
• Few months later the oil prices jumped-up (price
  doubled by Aug/99)
• The cap protected Petrobras from paying a very
  high spread

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Parameters Values for the Base Case

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

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Jump-Reversion Base Case Parameters
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Mean-Reversion and Jumps Parameters

• The long-run mean or equilibrium level which the
  prices tends to revert can be estimated by
  econometric way
• Another idea is 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 (GBM) 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. Range 1-5 years
• 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. Poisson is a
  counting process and we consider only large-jumps
  to set this frequency.
• 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

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Dynamic Programming and Options

• The optimization under uncertainty given the
  stochastic process and given the available
  options, was first performed by using the
  Bellman-dynamic programming equations

Period
Bellman Equations
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A Motivation for Using Dynamic Programming

• First, see the contingent claims PDE version of
  this model

• 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 yield
• Conclusion Anyway we need r for mean-reversion
  process, because d is a function of r d is not
  constant as in the GBM
• As in Dixit Pindyck (1994), we use dynamic
  programming
• Let r be an exogenous risk-adjusted discount
  rate that considers the incomplete
  markets/systematic jump feature
• We compare the results dynamic programming x
  contingent claims

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Boundary Conditions

• In the boundary conditions are addressed
• Payoff for an immediate development is NPV/bbl
  V - D.
• Developed reserve value is proportional to P V
  q P
• The extension feature at T1, paying K and winning
  another call option

• To solve the PDE, we use finite differences
• A C software was developed with an interactive
  interface

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C Software Interface The Main Window

• Software solves extendible options for 3
  different stochastic processes and two methods
  (dynamic programming and contingent claims)

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The Options and Payoffs for Both Periods
Options Charts
Period
T I M E
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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
  lt P lt 22.2 /bbl
• So, the extension threshold PE 4.7 /bbl
  (under 4.7, give-up is optimal)

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Debate on Exploratory Timing Policy

• The oil companies considered very short the time
  of 3 2 years that appeared in the first draft
  by National Agency
• It was below the international practice mainly
  for deepwaters areas (e.g., USA/GoM some areas 5
  3 years others 10 years)
• During 1998 and part of 1999, the Director of the
  National Petroleum Agency (ANP) insisted in this
  short timing policy
• The numerical simulations of our paper (Dias
  Rocha, 1998) concludes that the optimal timing
  policy should be 8 to 10 years
• In January 1999 we sent our paper to the notable
  economist, politic and ex-Minister Delfim Netto,
  highlighting this conclusion
• 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
• Delfim used our paper conclusions to support his
  view!
• Few days after, the ANP Director finally changed
  his position!
• Since the 1st bid most areas have 9 years. At
  least its a coincidence!

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Alternatives Timing Policies in Dias Rocha

• The table below presents the sensibility analysis
  for different timing policies for the petroleum
  sector

• Option values (F) are proxy for bonus in the bid
• Higher thresholds (P) means more delay for
  investments
• Longer timing means more bonus but more delay
  (tradeoff)
• Table indicates a higher gain for option value
  (bonus) than a increase in thresholds (delay)
• So, is reasonable to consider something between
  8-10 years

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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 was large

• OBS for contingent claims, we adopt r 10 and
  r 5 to compare

26
Sensibility Analysis Jump Frequency

• Higher jump frequency means higher hysteresis
  higher investment threshold P and lower
  extension threshold PE

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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/

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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 low oil
  prices P d r - h( P - P)
• A necessary condition for American call early
  exercise is d gt 0

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Concluding Remarks

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

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Additional Materials for Support
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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 payoff function 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 EdF is calculated with the Itôs
  Lemma for Poisson Itô mix process (see Dixit
  Pindyck, eq.42, p.86), using our process for dP

• Substituting EdF into (), we get the PDE
  presented in the paper

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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)

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Explicit Finite Difference Form

• 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)
34
Finite Differences Discretization

• The derivatives approximation by differences are
  the central difference for P, and
  foward-difference for t
• FPP F i1,j - 2Fi,j Fi-1,j / (DP)2
  FP
  F i1,j - Fi-1,j / 2DP
• Ft F i,j1 - Fi,j / Dt
• Substitutes the aproximations into the PDE

35
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
  5p.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)

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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)

F(tT) max (NPV, 0) NPV V - D
37
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

38
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)

39
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

40
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)

41
Main Results Window

• This window shows only the main results
• The complete file with all results is also
  generate

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Software Interface Data Input Window
43
Others Sensibility Analysis

• Sensibility analysis show that the options values
  increase in case of
• Increasing the reversion speed h (or decreasing
  the half-life H). But note that P0 lt P in the
  base case
• 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
  reversão
• 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)

44
Sensibility Analysis Reversion Speed
45
Sensibility Analysis Discount Rate r
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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) hP - 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

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

• 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!

52
The Grid Precision and the Results

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