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 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

E&P Is a Sequential Options Process

Drill the pioneer? Wait? Extend? Revelation, option-game: waiting incentives

Oil/Gas SuccessProbability = p

Expected Volumeof Reserves = B

RevisedVolume = B’ Appraisal phase: delineation of reserves

Technical uncertainty: sequential options

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

Primary focus of our model: undeveloped reserves

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

VD

45o

tg 45o = 1

F

PD/q

tg = q = economic quality

V = q . PF(t=T) = max (q P D, 0)

F(t=T) = max (NPV, 0)NPV = V D

The Extendible Maturity Feature

T2: Second Expiration

t = 0 to T1:

First Period

T1: First Expiration

T1 to T2:

Second Period

[Develop Now] or [Wait and See]

[Develop Now] or [Extend (pay K)] or [Give-up (Return to Govern)]

T I

M E

Period Available Options

[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:

The jump size/direction are random: ~ 2N

In case of jump-up, prices are expected to double

In case of jum-down, prices are expected to halve

;

So,

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)/( P )

The Poisson arrival parameter (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

Dynamic Programming and Options

T2: Second Expiration

t = 0 to T1:

First Period

T1: First Expiration

T1 to T2:

Second Period

Period Bellman Equations

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

estimation is necessary even for contingent claims

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 rate]

Conclusion: Anyway we need for mean-reversion process, because is a function of ; is not constant as in the GBM So, we let 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 : use the time-series from futures market

A Motivation for Using Dynamic Programming First, see the contingent claims PDE version of this model:

Compare with the dynamic programming version:

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)

Smooth pasting condition (for both periods)

Value matching at P* (for both periods)

Second expiration optimally (D2 can be different of D1)

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 P x t (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 (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 yearsFor 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

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 PeriodsT

I M

E

Options Charts

T2: Second Expiration

t = 0 to T1:

First Period

T1: First Expiration

T1 to T2:

Second Period

Period

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 non-arbitrage contingent claims or dynamic programming However, for geometric Brownian motion the difference is very large

OBS: for contingent claims, we adopt = 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 P E 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 .

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%:

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 , that can be negative for lower values of P:

( P P) A necessary condition for early exercise of American option is > 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 (P, t)is the payofffunction 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 negativeUse 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)

Explicit Finite Difference Form Grid: Domain space P x t

Discretization F(P,t) F( iP, jtFi, j

With 0 i m and 0 j n where m = Pmax/P and n = T / t

i , j

p jump p +

p 0

p -

i + 1 , j + 1

i , j + 1

i - 1 , j + 1

i , j

j),1(ijump1j,1i1j,i0

1j,1ij,i FpFpFpFpF

“Probabilities” p need to bepositives in order to get the convergence (see Hull)

P

t

Domain Space(distribution)

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 ] / (P)2 FP [ F i+1,j Fi-1,j ] / 2P

Ft [ F i,j+1 Fi,j ] / t

Substitutes the aproximations into the PDE

j),1(ijump1j,1i1j,i0

1j,1ij,i FpFpFpFpF

pt

t

i i P i P i k t

t

i i P i P i k

pt

t ti

t

tjump

.

.( . ) .

.

.( . ) .

. .~

1 2 2 2 2 1 2 2 2 2

1

1

1

2 2 2 2 2 2

0 2 2

; p

; p ; k = E -1

Economic Quality of a Developed Reserve

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 is the reversion speed of the stochastic convenience yield

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

Sensibility analysis show that the options values increase in case of: Increasing the reversion speed (or decreasing the half-life H); 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 ; 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)

Others Sensibility Analysis

Sensibility Analysis: Reversion Speed

Sensibility Analysis: Discount Rate

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

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: 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)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!

The Grid Precision and the Results The precision can be negligible or significant

(values from an older base case)

Software Interface: Data Input Window