Transcript Investment under Uncertainty in E&P M.Sc. Thesis Special

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
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
 Jump
for immediate investment and for extension
+ mean-reversion process for oil prices
Topics:
systematic jump, discount rate, convenience yield
 C++
software interactive interface
 Base case and sensibility analysis
Alternative
 Concluding
timing policies for Brazilian National Agency
remarks
E&P Is a Sequential Options Process
Oil/Gas Success
Probability = p
Expected Volume
of Reserves = B
Revised
Volume = B’
 Drill
the pioneer? Wait? Extend?
 Revelation, option-game: waiting incentives
 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
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.

F
The option charts F x V and F x P at the expiration (t = T)
F(t=T) = max (NPV, 0)
NPV = V - D
tg
45o
F
V=q.P
F(t=T) = max (q P - D, 0)
tg q = q = economic quality
=1
q
45o
D
V
D/q
P
The Extendible Maturity Feature
Period
Available Options
t = 0 to T1:
[Develop Now] or [Wait and See]
T I M E
First Period
T1: First
Expiration
T1 to T2:
Second Period
T2: Second
Expiration
[Develop Now] or [Extend (pay K)]
or [Give-up (Return to Govern)]
[Develop Now] or [Wait and See]
[Develop Now] or
[Give-up (Return to Govern)]
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
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:
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
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
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(?)
Jumps-up
Jumps-down
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
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 )
Poisson arrival parameter l (jump frequency), the
expected jump sizes, and the sizes uncertainties.
 The


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 jumpup) 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
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
t = 0 to T1:
First Period
T1: First
Expiration
T1 to T2:
Second Period
T2: Second
Expiration
Bellman Equations
A Motivation for Using Dynamic Programming
 First,
see the contingent claims PDE version of this model:
r
 Compare

estimation is necessary even for contingent claims
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

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

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)
To solve the PDE, we use finite differences in explicit form
 A C++ software was developed with an interactive interface
The C++ Software Interface: Main Window

Software solves extendible options for three different
stochastic processes (two jump-reversion and the GBM)
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)
Main Results Window

This window shows only the main results
 The complete file with all results is also generate
Parameters Values for the Base Case
 The
more complex stochastic process for oil prices (jumpreversion) 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

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

currently with NPV = 0; Reserves of 100 millions barrels
number of samples is better for volatility estimation
Several sensibility analysis were performed, filling the gaps
Jump-Reversion Base Case Parameters
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
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)
The Options and Payoffs for Both Periods
Period
t = 0 to T1:
T I M E
First Period
T1: First
Expiration
T1 to T2:
Second Period
T2: Second
Expiration
Options Charts
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
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 threshold PE = 4.7 $/bbl (under 4.7, give-up is optimal)
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
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
Comparing Dynamic Programming
with Contingent Claims
 Results
show very small differences in adopting nonarbitrage 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
Sensibility Analysis: Jump Frequency
 Higher jump
frequency means higher hysteresis: higher
investment threshold P* and lower extension threshold PE
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
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
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%:
In
 Sensibility
analysis points in general higher option values
(so higher bonus-bidding) for jump-reversion (see Table 3)
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
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
Additional Materials
for Support
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 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
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 jumpdiffusions processes. Example: Bates (1991)
Explicit Finite Difference Form
 Grid:
Domain space DP x Dt
 Discretization
 With

F(P,t)  F( iDP, jDt )  Fi, j
0  i  m and 0  j  n
where
m = Pmax/DP and
n = T / Dt
Fi , j = p  Fi  1 , j - 1  p 0 Fi , j - 1  p - Fi - 1 , j - 1  p jump Fi ( 1 - f ), j
-f)i,j
Domain Space
(distribution)
i+1, j+1
p jump
P
p+
t
p0
i,j
i, j+1
pi-1, j+1
“Probabilities” p need to be
positives in order to get
the convergence (see Hull)
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
F i , j = p  F i  1 , j - 1  p 0 F i , j - 1  p - F i - 1 , j - 1  p jump F i ( 1 - f ), j
Dt   2 i 2 i.( h. P ) i 2 . hDP ilk  Dt   2 i 2 i.( h. P ) i 2 . hDP ilk 
p =

;p =





Dt . r  1  2
2
2
2 
Dt . r  1  2
2
2
2 

p0 =
Dt  1
Dt

2 2

i
l
;
p
=
l
jump


Dt . r  1  D t
Dt . r  1

;


~
k = E f -1
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
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  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)
Sensibility Analysis: Reversion Speed
Sensibility Analysis: Discount Rate r
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
Sensibility Analysis: Lon-Run Mean
Sensibility Analysis: Time to Expiration
Sensibility Analysis: Economic Quality of Reserve
Geometric Brownian Base Case
Drawbacks from the Model
 The
speed of the calculation is very sensitive to
the precision. In a Pentium 133 MHz:
DP = 0.5 $/bbl takes few minutes; but using
more reasonable DP = 0.1, takes two hours!
 Using
point is the required Dt to converge (0.0001 or less)
Comparative statics takes lot of time, and so any graph
The
 Several additional
parameters to estimate (when
comparing with more simple models) that is not
directly observable.
 More
 But
source of errors in the model
is necessary to develop more realistic models!
The Grid Precision and the Results
 The
precision can be negligible or significant
(values from an older base case)
Case 1: higher precision Case 2: lower precision
DP = 0.2 & Dt = 0.0005
Option F @ t = 0
2.060721
(US$/bbl)
Threshold @ t = 0
DP = 0.5 & Dt = 0.001
2.060241
( - 0.023 %)
24
24
19.4
19.5
5.2
6.0 (+15.4%)
P* (US$/bbl)
Threshold to develop @
t= T1 ; P* (US$/bbl)
Threshold to extend (@
t = T1); PE (US$/bbl)
Software Interface: Data Input Window