Cutting edge: Exploration & production

34 I August 2003

In this article, we aim to describe an efficient option method forthe valuation of an exploration and production project. Considerthe example of a project manager who has the right to start drillingon a given oil reserve in any one of five designated years, called

‘decision’ years. Two years after the drilling starts, crude oil and naturalgas will be produced for 12 consecutive ‘return’ years, all of which, it isassumed, the company will sell on the spot market.

This right resembles an American-style option (Bermudan), and if weare prepared to make the assumptions required for the application ofderivative techniques, we may value it accordingly. However, using tradi-tional approaches and somewhat realistic oil and gas price models, suchan option can be cumbersome to calculate. For example, the construc-tion of the binomial tree and the calculation along it can be tedious andtime-consuming. We aim to show that a recently proposed least-squaresMonte Carlo approach (Longstaff and Schwartz (2001)) can solve thisproblem very efficiently. It is easier to implement and allows moregeneral stochastic processes for state variables (oil and gas prices) thanwith traditional approaches.

Valuing investment projects of this sort represents a classic exampleof real option valuation. The main purpose of this article is to introducea new calculation technique for a familiar industry problem. Hence wewill not consider in detail some common – but important – issues asso-ciated with real option valuation.

That said, we still want to bring attention to these issues and to thelimitations of our approach. One problem area concerns the discountingof future cashflows. For investment commodities such as gold, we canevaluate futures output at current spot prices without discounting. Forconsumption commodities such as oil and gas, future output is not quitethe same as current output. Hence, we need to consider the convenienceyield and cost-of-carry – respectively the benefit and cost of carrying thecommodity – to determine the proper discount rate.

In actual calculation, instead of changing the discount rate, we continueto use the risk-free interest rate and instead adjust down the drift term ofthe spot price process by the convenience yield. If the commodity has afutures market, we can use its futures prices to determine the averageconvenience yield and, therefore, the proper drift (Brennan and Schwartz(1985)). However, when the commodity does not have a futures market orone that extends long enough – more than 18 years in our example – thenwe have to determine the drift differently. In this case, the price processneeds to have a ‘risk-adjusted drift’ – that is, the original drift minus a riskpremium determined from an equilibrium model. Similarly, if output hasan unhedgeable risk – for example, volume uncertainty – we must deductanother risk premium from the drift (Cox, Ingersoll and Ross (1985);Schwartz and Trigeorgis, chapter 1, (2001)).

The deduction of risk premiums is an acknowledgement that underthese conditions, the full value of starting an investment project on a flex-ible date instead of a pre-determined date, is not hedgeable. Therefore thecertainty-equivalent value of it is lower.

In the rest of the article we explain the basics of the least-squaresapproach and report results of its application for our example.

The least-squares approach to calculating an American-style (Bermudan) optionIn each decision year, the project manager would compare the expectedvalue from oil and gas production if drilling were to start immediately(the exercise value) with the expected value if drilling were to start later(the continuation value). He starts drilling if the exercise value is larger.For a given time and price path, the manager calculates the exercise valuefrom the net present value of cashflows, and the project value is thediscounted exercise value averaged over many simulated price paths.

The innovative part of the least-squares approach is in the calculationof continuation value. We may estimate the continuation value by project-ing cashflows realised from continuation on to the space spanned bycurrent prices – we can do this through a simple regression. Detailed stepsfor this process are explained in a simple numerical example in Longstaffand Schwartz (2001, pages 115–120). We provide a simple schematicdescription in the equation below. But to fully understand the process,one should work through their example, which can be easily reproducedusing a spreadsheet or by hand.

As shown above, we assume there are n price-paths. On path 1, thecurrent period price is s1,t, and the continuation value (expected projectvalue) next period is v1,t +1. Suppose next period, t +1, is the final periodto exercise, so one has to exercise. The continuation value, v1,t+1, is simplythe exercise value (discounted cashflow), given s1,t+1. Regressing{v1,t+1,...,vn,t+1} on {s1,t,...,sn,t} yields f (s)= E(v|s), a conditional expecta-tion function approximating the continuation value given price s.1 Usingthis function, we can compare the continuation value v1,t= f (s1,t) with theexercise value at s1,t to decide whether to exercise on path 1 at time t.

For the next step, knowing {v1,t ,...,vn,t} allows us to repeat the aboveprocedure at t –1 and so on, back to time zero. So at the end we willknow the exercise decision and value at each point in time along all pricepaths. By discounting the values at the risk-free interest rate and aver-aging them over all paths, we will obtain the expected project value attime zero.

Hence, the key point here is that to apply this approach to evaluatean exploration & production project, one only needs to define first theprice-generating process and second the exercise value along a pricepath. We go on to define these two components and present the resultsfrom a case study.

1, 1, 1

2, 2, 1

, , 1

Current period Next period

price project value

t t

t t

n t n t

s v

s v

s v

+

+

+

� �

Lukens Energy Group’s Hugh Li sets out an option method for valuing explorationand production projects, using a practical example

Valuing exploration andproduction projects

1 The function f(s) is assumed to be a linear combination of some basis functions such as s2,exp(–s/2) and so on. Our program used simple polynomial functions of s as bases. Calibrating overfrom known American option values such as vanilla calls and puts, these basis functions perform aswell or better than other more elaborate choices proposed in Longstaff and Schwartz (2000)

Cutting edge: Exploration & production

36 I August 2003

The price generating process and exercise valueIn a given return year t, some volumes of crude oil and natural gas areproduced. Denote respectively the volumes as v

c(t) and v

g(t), and the spot

market prices as sc(t) and s

g(t). We assume the two prices to be mean-

reverting and correlated, as in

,(1)

.

Note that here both the means and volatilities may be time-varying. Withthese prices, we can generate revenue s

c(t)v

c(t) + s

g(t)v

g(t), which is what

we actually need. Note that this simple price process implies a certainterm structure of volatility. If we derive implied volatilities based on anoption formula assuming a different price process – notably Black-Scholes – the volatilities will be different. Nevertheless, a big advantageof the least-squares approach is that is does not require a specific priceprocess, so we are free to choose others.

Suppose the drilling starts in year t *. After taking into account theroyalty rate (denoted by a); field operation cost for crude, b; tax rate, d;and discounting, ρ, the project net present value (NPV) of a given samplepath in price space is

.

The exercise value is the expected NPV. Ignoring volumetric uncertainty,the exercise value is

.

(2)

Given (1), from Schwartz (1997),

,

where

,

and

.*

*

2 2

2 ( )11 12( ( )) (1 )2

c t t

t c

c

Var x t e− α −σ + σ= −

α

* *

*

2 2

( ) ( )11 12*( ( )) ln ( ) ( )(1 )

2c ct t t t

t c c c

c

E x t s t e e− − − −+= + − −α ασ σµ

α

* *

*

( ( )) ( ( )) / 2( ( )) t c t cE x t Var x t

t cE s t e+=

*

* * *

*

13

*

2

( 1)

( ) (( ( ( )) ( ) ( ( )) ( )) (1 )

( ) )(1 )

t

t t c c t g g

t t

t

c

E v t E s t v t E s t v t a

v t b d e

+

= +

− −

= + ⋅ − −

⋅ −

∑ρ

*

*

13

( 1)

*

2

( ) (( ( ) ( ) ( ) ( )) (1 ) ( ) )(1 )

t

t

c c g g c

t t

v t s t v t s t v t a v t b d e

+−ρ −

= +

= + ⋅ − − ⋅ −∑

2

2

1

( ) / ( ) ( ( ) ln ( )) ( ) ( )g g g g g j j

j

ds t s t t s t dt t dW t=

= α µ − + σ∑

2

1

1

( ) / ( ) ( ( ) ln ( )) ( ) ( )c c c c c j j

j

ds t s t t s t dt t dW t=

= α µ − + σ∑

The expression for Et *(sg(t)) is symmetric. Therefore equation 2, gives us

Et *v(t*). Given the price-generating process (equation 1) and the exercise

value (equation 2), we can apply the least-squares approach to calculatean expected value.

We can generalise the price process. So far, we have assumed µc(t) = µ

c,

µg(t) = µ

g, and σ

ij(t) = σ

ijdo not vary with time. When they do vary with

time, we need to extend the results in Schwartz (1997). The derivation issimilar, but the specifics more complex. The results are listed below.

An extension: exercise value with varying price and volatilityIn equation 1, let us suppose µ(t) and σ

ij(t) are step functions µ(t) =uk,

σij(t) =σ

ijk, t∈[t

k–1, tk),k =1,...,T ; i =1,2; t

0=0, t

T=T . Et *

(x(t)) and Vart *(x(t))

in equation 2 have the following different expressions. Because theexpressions for crude oil and gas are symmetrical, for exposition purposeswe suppress the subscripts c and g.

Suppose t*∈[tI1 -1, t

I1) for a certain I1, then for t∈[t

*, t

I1)

,

.

For t∈[tI2-1, t

I2) with I2– 1≥ I1,

1 , 1 ,1 2

* * 1 *1

*

1

1 , 2

1 ** 2 *

2 22 2

12 ( )1 1 2 ( ) 2 ( )

1

22

2 ( )1 2 ( ) 2 ( )

( ( )) ( ( 1) ( ( ))2 2

( )) .2

j I j i

I i i

j I

I

It tj j t t t t

t

i I

t tj t t t t

Var x t e e e

e e e

−

−

−α −= = α − α −

= +

α −= α − − α −

σ σ= − + −

α α

σ+ −

α

∑ ∑∑

∑

1 , 1

** 1

* 1

1 ,2

* 1 *

1

1 , 2

1 ** 2 *

2

22

( ))1( )

*

22

11 ( ) ( )

1

22

( )1 ( ) ( )

( ( )) ln( ( )) (( )( 1)2

( )( )2

( )( )) ,2

j I

I

j i

i i

j I

I

t tjt t

t I

Ij t t t t

i

i I

t tj t t t t

I

E x t s t e e

e e

e e e

−

−

α −=−α −

−= α − α −

= +

α −= α − −α −

σ= + µ − −

α

σ+ µ − − +

α

σµ − −

α

∑

∑∑

∑

1 , 1

*

*

22

1 2 ( )( ( )) (1 )

2

j I

j t t

tVar x t e= − α −

σ= −

α

∑

1 , 1

* *

* 1

22

1( ) ( )

*( ( )) ln( ( )) ( )(1 )2

j I

jt t t t

t IE x t s t e e=−α − −α −

σ= + µ − −

α

∑

Input parameter ValueAnnual risk-free interest rate 5%Annualised volatility of crude 60%Annualised volatility of gas 80%Correlation coefficient between crude and gas 30%Royalty rate 12%Expense $6.15/barrelTax rate 35%Mean-reversion rate for crude and gas 0.036Mean price level for crude $20.00Mean price level for gas $3.00Initial crude price $20.4/barrel Initial gas price $3.2/mcf

Source: author

TTaabbllee 11:: Input parameters

Year Crude oil (barrels) Natural gas (mcf)1 15,288,100 29,163,7132 16,285,150 30,742,3753 15,121,925 25,923,3004 13,377,088 22,682,8885 11,466,075 19,442,4756 6,813,175 12,961,6507 5,733,038 8,101,0318 4,777,531 4,860,6199 2,866,519 3,240,41310 1,911,013 1,744,83811 997,050 1,578,66312 913,962 1,578,663

Source: author

TTaabbllee 22:: Annual volumes

www.eprm.com I 37

In both expressions, when I2– 1= I1, the term should bedropped. Again, from equation 2, we know the exercise value Et *

v(t*).

A practical exampleWe applied the above approach to an actual case and report the resultshere. Table 1 shows the input parameters, assuming constant mean andvolatility. And for this reserve, once the production begins, table 2 givesthe annual volumes. Table 3 gives the expected discounted cashflowswhen the drilling starts in each of the six decision years.

The total expected project value is $837.1 million. As expected, thisfigure exceeds any of the NPVs with predetermined starting dates. Theadditional value is the premium for the option of a flexible starting date(the option value). We also ran the case with volatility at half the levelassumed above while keeping all other parameters unchanged – the totalexpected project value dropped to $797.1 million. There is a not insignif-icant $40-million-dollar fall in project value as a result of the lowervolatility.

A note on hedgingWe have calculated an option value that accounts for the higher value ofbeing able to start drilling at an optimal time in the presence of marketvolatility. If we choose to hedge, we can monetise an option value. Howdo we hedge such a real option?

Once the drilling starts – that is, the option is exercised – equation twoshows that the exercise value is a function of expected spot prices in thefuture, which are equal to futures prices. Ideally, if there are futurescontracts for all these years, they can be straightforwardly hedged. Froma practical point of view, the New York Mercantile Exchange has onlyseen six years of crude and natural gas futures trading, with very little

2

1

1

1

(.)

I

i I

−

= +∑

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CALL FOR PAPERS

liquidity at two years and beyond. Hence, in our case, we have to useshort-term futures contracts to hedge longer-term price risks by ‘rollingthe hedge forward’.

For example, to hedge the price risk in the fourth year, we first go shorton contracts with delivery in the second year. Then, just before their expi-ration, close them out and go short on contracts with delivery in anothertwo years. And then, just before these new contracts’expiration, close themout and sell production in the spot market.

The downside of rolling hedges forward is that as we have incurredthe basis risk – from the price change in closing out contracts – the hedgewill be imperfect. This basis risk – like the volume risk and other unhedge-able risks – must be accommodated in the risk-adjusted drift. EPRM

Huagang ‘Hugh’ Li is director of research at Lukens Energy Group in Houston.

email: hli@lukensgroup.com

The author thanks Jeff Cliver, Hua Fang, Fred Hagemeyer, Jay Lukens,Rafael Mendible and Scott Smith

References

Cox, JC, Ingersoll, JE and Ross, SA , An intertemporal general equilibrium

model of asset prices, Econometrica, 53, pages 363–384, 1985

Dixit, AK and Pindyck, RS, Investment under Uncertainty, Princeton Univer-

sity Press, 1994

Longstaff, FA and Schwartz, ES, Valuing American options by simulation: a

simple least-squares approach, The Review of Financial Studies, 14, 1, pages

113–147, Spring 2001

Brennan, MJ and Schwartz, ES, A new approach to evaluating natural sesource

investments, Midland Corporate Finance Journal, pages 37–47, Spring 1985

Schwartz, ES, The stochastic behaviour of commodity prices: implications for

valuation and hedging, Journal of Finance, 52, 3, pages 923–973, 1997

Schwartz, ES and Trigeorgis, L, ed., Real Options and Investment under

Uncertainty – Classical Reading and Recent Contributions, MIT Press, 2001.

Start date Project net present value ($ millions)Immediately 781.9 Year 1 750.9 Year 2 704.1 Year 3 673.2 Year 4 638.8 Year 5 610.7

Source: author

TTaabbllee 33:: Expected cashflows