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
EP 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

• 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
5
The Extendible Maturity Feature
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 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)

• 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

• 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 of r use the d time-series
  from futures market

15
Boundary Conditions

• 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
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
  lt P lt 22.2 /bbl
• So, the extension threshold PE 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
  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)

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 gt 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 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

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) 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 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

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 doesnt 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) doesnt 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 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
  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)

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) 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

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