Transcript Petroleum Concessions with Extendible Options Using Mean

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)
Presentation Highlights
 Paper has
two new contributions:
 Extendible
Kemna
 Use
maturity framework for real options
(1993) model is too simplified (European option, etc.)
of jump-reversion process for oil prices
First
presented in May/98 (Workshop on RO, Stavanger)
 Presentation
 Concepts
for options with extendible maturities
Thresholds
 Main
topics:
for immediate development and for extension
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
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 and technical uncertainty modeling

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?
The Extendible Maturity Feature (2 Periods)
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 (commit K)]
or [Give-up (Return to Government)]
[Develop Now] or [Wait and See]
[Develop Now] or
[Give-up (Return to Government)]
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
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:
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

Mean-Reversion + Jump: the Sample Paths

100 sample paths for mean-reversion + jumps (l = 1 jump each 5 years)
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-up
Jumps-down
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 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)
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, price is
expected to double
 In case of jum-down, price is
expected to drop by half
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
 Deal
prices  higher spread, and vice versa (good for both)
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
Parameters Values for the Base Case
 The
more complex stochastic process for oil prices (jumpreversion) demands several parameters estimation


Jumps frequency: counting process with a jump criteria
The jumps data were excluded in order to estimate meanreversion (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
Jump-Reversion Base Case Parameters
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 leaderfollower 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
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. 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
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
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 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

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

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
 A C++ software was developed with an interactive interface
C++ Software Interface: The Main Window

Software solves extendible options for 3 different stochastic processes
and two methods (dynamic programming and contingent claims)
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
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)
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 it’s a coincidence!
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
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 was 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/
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 > 0
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
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
1-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
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)
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 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)
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 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
Software Interface: Data Input Window
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 < 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  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)