2012 Cambridge Business & Economics Conference
ISBN : 9780974211428
Valuation of Investments in Oil and Gas: A Real Options Approach
Vivian O. Okere, Providence College, Providence, Rhode Island, USA. 401-865-2671.
Zahra Amirhosseini, Islamic Azad University, Shahr-e-Qods Branch, Iran 989-121-883-239.
June 27-28, 2012
Cambridge, UK
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2012 Cambridge Business & Economics Conference
ISBN : 9780974211428
Valuation of Investments in Oil and Gas: A Real Options Approach
ABSTRACT
Real options approach is used to analyze the investment decision in oil and gas production. The
price of the option to invest is estimated using a binomial modeling and the backwards induction
methodology. The results indicate that the investor will exhibit a less risk-averse behavior when
the expected gain is equal to or exceeds the price of the option. Contrarily, the investor will be
more risk-averse if the expected gain is less than the price of the option.
Keywords: Binomial Modeling, Investments, Risk analysis and Valuation.
I
INTRODUCTION
Existing empirical analyses of corporate investments typically assume deterministic decisionmaking based on the expected net present value of returns. These models, however, may be
inadequate to either explain observed variations in corporate investments or provide a reliable
basis for projecting the effectiveness of future investing policies. The development of better
investment decision-making models become increasingly important as corporate investments
recognize that some investment decisions may not by fully rational. This paper addresses the real
options approach of valuing investments in oil and gas production under stochastic prices using
the binomial tree model.
The assumption that prices evolve according to a binomial process and that investments in oil
and gas are similar to investments in non-dividend paying assets permit us to state that an
investor is indifferent to risk and requires no additional compensation for risk. In other words,
the expected return on the investment is the risk free rate and the investment exists in a risk
neutral world. The implication is that the option (to invest) can be valued on the basis that
investor is risk neutral. Thus, the investor’s risk preference has no effect on the value of the
option (to invest) when it is expressed as a function of the price of the underlying asset, i.e., oil
and gas. As a result of the general principle of risk-neutral valuation used in option pricing, we
can with complete impunity assume the world is risk neutral because the resulting option prices
are correct in a risk neutral world as well as other worlds. This explains why the pricing formulas
of Black-Scholes (B-S) for European calls, c and p for puts on non-dividend paying stocks do not
involve the stock’s return,  .
c  S 0 N d1   Ye  rT N d 2 
p  Ye  rT N  d 2   S 0 N  d1 
where
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2012 Cambridge Business & Economics Conference
d1 
d2 
ln 

S0
ISBN : 9780974211428
2
   r    T 
Y    2  
 
 
 T
ln 

S0
2
   r    T 
Y    2  
 
 
 T
 d1   T
It is well known that an analytical solution does not exist when valuing an American-type call
option on a dividend-paying asset with positive exercise price (See Hull, 2005). Adapting the BS model to this paper, the variables S 0 is the price of the underlying asset, Y is the benchmark
representing the reference price of the underlying asset that the investor must earn to justify
future investments in oil and gas production; r is the risk free rate of return continuously
compounded, T is the expiration date,  is the volatility of the price of the underlying asset and
the function N x  is the cumulative probability function for a standardized normal variable.
II.
A GENERALIZED BINOMIAL MODEL ON RISK NEUTRAL INVESTMENTS
Let S 0 refer to the median price of the underlying asset and f is the option to invest in the
underlying asset. The option to invest will expire in time, T . During the life of the option, the
price of the underlying asset can move up from S 0 to a higher level, S u , and the payoff from the
option is f u . Conversely, the price of the underlying asset can move down from S 0 to a lower
price S d and the payoff from the option is f d . The inference is that u  1.0; d  1.0 . In other
words, the proportional increase in the price of the underlying asset when there is an up
movement is u  1.0; and the proportional decrease in the price of the underlying asset when
there is down movement is 1.0  d .
Hypothetically, if the price of the underlying asset goes up, the value of the portfolio at the end
of the option is
V  Su   f u
If the price of the underlying asset goes down, the value of the portfolio becomes
V  Sd   f d
The value of  that makes the portfolio riskless becomes
Su   f u  S d   f d
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2012 Cambridge Business & Economics Conference
ISBN : 9780974211428
fu  fd
(1)
Su  S d
Equation (1) shows that  is the ratio of the change in the price of the option to the change in
price of the underlying asset as we move between tree nodes. We denote the risk free interest rate
by r and the present value of the portfolio when the price of the underlying asset goes up
becomes:

S u   f u e  rT
Conversely, the present value of the portfolio if the price of the underlying asset goes down is
zero and the option is worthless:
S d   f d e rT
0
Now, suppose the cost of setting up the portfolio is
S0  f ;
It then follows that
S u   f u e  rT
 S 0   f u ; and


f  S 0  1  ue  rT  f u e  rT
If we substitute equation (1) for  and p 
e rT  d
, the equation above becomes
ud
f  e  rT  pf u  1  p  f d 
(2)
p is the probability of an up movement and 1  p is the probability of a down movement.
Assuming that the price of the underlying asset will evolve along a binomial tree from N= 0 to
N= 4 over four quarterly periods. Adapting Barberis (2009), we can assign values to S u and
S d based on the equations below.
1
T

2
 2
   
1
T

2
 2
   
Su   
Sd   
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1
T
1
T
1
2 T
1
2 T
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2012 Cambridge Business & Economics Conference
ISBN : 9780974211428
We set  , and  at  ,    1.1, 0.3 to represent the annual after tax median rate of return and
standard deviation respectively. Thus the price of the underlying asset goes up u =1.1619, and
down by d  0.8863.
In the pioneer research on prospect theory, Kahneman and Tversky (1979) and Tversky and
Kahneman (1992) stated that investors evaluate investment prospects according to a value
function centered on a reference value or target price. Fiegenbaum and Thomas (1988) suggested
that the median value is a better proxy for a firm’s reference value because it is unaffected by
extreme outliers. It is therefore easier for investment managers to justify investments if the firm
could earn a higher reference value. Benartzi and Thaler (1995) argued that loss-averse investors
will be reluctant to invest even if the risk premium is sizeable. Beginning with a median price of
$50, an up movement becomes $58 and down move is $44. If
r  0.05, T  0.25  1 / 4, f u  1.0 and f d =0.0 ; then
p
e 0.050.25  0.8863
 0.4582;
1.1619  0.8863
1  p  0.5418
and
f  e 0.050.25 0.4582  1.0  0.5418  0.0  0.4525
It should be emphasized that p which is the probability of an up movement in a risk neutral
world may be different in the real world. From above we obtain p  0.5418 when the expected
return on the underlying asset and the option is the risk free rate of 5%. However, in the real
world, the expected return on the underlying asset could be as high as 20% or more. Now, using
an expected return in the real world of 20% and q as the probability of an up movement in the
real world, it follows that:
58q  441  q   50e 0.20*0.25
q  0.5815
If the value of the option f is 0.4525 , we can through iteration estimate that the expected rate of
return in the real world could be as high as 100.32 percent using the equation
0.4525  0.5815e i0.25
Unfortunately, the discount rate of 100.32 percent may not be accurate because the option value
of 0.4525 may be unknown. Therefore, we can with complete impunity assume that the world is
risk neutral and as such the resulting option prices are correct in a risk neutral world as well as
other worlds.
Figure 1 below illustrates the tree of prices when the binomial model is used. At time zero, the
price of the underlying asset, S 0 , is known. At time t there are two possible prices, S u and S d ;
at time 2t there are three possible prices, S u , S u and S d and so on.
2
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2012 Cambridge Business & Economics Conference
ISBN : 9780974211428
Accordingly equation (2) becomes
f  e  rT  pf u  1  p  f d 
the expression for p becomes:
(3) and
p
e rT  d
ud
Repeated computations of equation (3) gives
(4)
f u  e  rT  pf uu  1  p  f ud 
(5)
f d  e  rT  pf ud  1  p  f dd 
(6)
Substituting equations (5) and (6) into (3), we get

f  e 2rT p 2 f uu  2 p1  p  f ud  1  p  f dd
2

(7)
Equation (7) is consistent with the principle of risk-neutral valuation discussed earlier. The
2
variables, p 2 , 2 p1  p  and 1  p  are probabilities that the upper, middle and lower nodes will
be reached. The option price is equal to its expected payoff in a risk neutral world discounted at
the risk free rate. As we add more steps to the binomial tree, the risk neutral valuation principle
continues to hold.
III
THE ALGEBRAIC APPROACH TO ESTIMATE OPTION VALUE.
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2012 Cambridge Business & Economics Conference
ISBN : 9780974211428
We assume that the option on fossil fuels such as for oil and gas could be likened to a nondividend- paying stock. Next, suppose that the life of an American call option is divided into N
subintervals of length it . We refer to the jth node at time it as the i, j  node, where 0 ≤ i ≤
N and
0 ≤ j ≤ i . We also define f i , j as the value of the option at the i, j  node. The value of the
underlying asset at the i, j  node is
Su d i j
j
for  j  0,....,1
The value of the option is known at time T. Since the value of the American call at its expiration
is max ST  Y , 0 , where S T is the median price of the oil per barrel at time T and Y is the
reference price that the investor must earn in order to continue to invest in future oil and gas
production. Then the value of the option can be expressed as

f N , j  max Suj d N  j  Y ,0

for  j  0,1,...., N 
The symbol p is the probability of moving from the i, j  node at time it to the
i  1, j  1 node at time i 1t and 1  p is the probability of moving from the i, j  node at
time it to the i  1, j  node at time i 1t . If there is an early exercise of the option, the value
of the option under risk neutral valuation assumption becomes


f i , j  e rt pf i 1, j 1  1  p  f i 1, j for 0 ≤ i ≤ N  1 and 0 ≤ j ≤ i .
The value of the option with an early exercise is


f i , j  max Suj d N  j  Y , e rt  pf i 1, j 1  1  p  f i 1, j 
We use backward induction to value the option to invest by starting at the end the tree (time T)
and solving the option values going backward. We note that an up movement in price followed
by a down movement leads to approximately the same price as a down movement followed by an
up price movement.
IV
BINOMIAL TREE FOR AN AMERICAN STYLE CALL ON A NON DIVIDEND
COMMODITY
Suppose the median price per barrel of oil is $50, the risk free rate is 5 percent, volatility is 30%
per annum and there is a four-month American call option on oil and gas investment which we
assume is a non-dividend paying asset. Using our notation,
S 0  $50, Y  $35, r  0.05,   0.30, and T  0.25
Earlier, the price of the underlying asset was estimated to go up, u =1.1619, and down,
d  0.8863. For illustration purposes, consider a quarterly American style call option on a nondividend paying asset like oil and gas and
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2012 Cambridge Business & Economics Conference
p
ISBN : 9780974211428
e rT  d e 0.05*0.25  0.8863

 0.4582; 1  p   0.5418
ud
1.1619  0.8863
The upper values in figure 2 represent the projected prices of the underlying commodity and the
lower values refer to the associated option prices. The option prices at the final nodes are
calculated as max ST  Y , 0 . Since the assumed reference (minimum acceptable) price depicting
the investor’s prior experience, Y  $35.00, the gain at node N is 40  35  5 and that is the same
as the option price. Subsequent gains (above the reference price of $35) are derived and option
prices at the penultimate nodes are calculated from the option prices at the final nodes. For
example, the gains at nodes D, G, H and I are respectively 32, 43, 24 and 10 and the option prices
are respectively 32.8, 43.5, 24.5 and 10.4 .
Based on the gains and option prices at each node, we check to see if an investor prefers to wait
for oil prices to further increase or if the investor is better-off with an early exercise and thus
invest in oil/gas production ‘now’. We observe that at nodes K , L, M and N the options are
exercised because the gains are equal to the option prices. Though the gains and option prices at
nodes D, G, H and I may not be significantly different from the option prices (the latter
consistently exceeding the gains by a small amount), the decision to exercise the option to invest
or wait may depend on the extent of the investor’s prior gains or losses and also the investor’s
aversion to potential losses emanating from oils spills and the accompanying environmental
damage. A similar argument could be made for investment decisions at nodes B and E. The
inferences at nodes A, C , F , J and O will be to delay the exercise of the option to invest because
the option prices are more than the gains above the reference price of $35.
Figure 2: Option prices and values at each node
t  0.25
Growth factor = e rT  1.0126
Probability of up movement = p  0.4582
Probability of down movement = 1  p  0.5418
Up step size, u  1.1619
Down step size, d  0.8863.
Discount factor per step= e  rT  e .05*.25  0.9876.
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The intuition is that the investor may not invest if the value gained on the investment is less than
the reference value, more so, if the investor incurred prior losses.
V.
CONCLUSION
This paper illustrates a numeric analysis for estimating the expected gains and losses and the
price of the option to invest when oil and gas prices follow a binomial tree process.
Theoretically, basic microeconomic reasoning will suggest that when the price of oil and gas is
relatively high, energy producing companies will increase deep water operations, and the supply
of the commodity will increase resulting in lower prices. Figure 2 shows that when the expected
gain is less than the price of the option, the investor (i.e., energy producing companies) will
exercise the option to wait and production will decrease. Conversely, when the expected gain is
equal or greater than the price of the option, the investor might opt to invest and produce oil and
gas. Therefore, the impetus to invest in new oil and gas production depends on the volatility of
returns. Our results further support the fact that investors in oil and gas evaluate the expected risk
and return of their investments.
VI.
REFERENCES
Barberis N., Xiong Wei. (2009). “What Drives the Disposition Effect? An Analysis of a LongStanding Preference-Based Explanation”.The Journal of Finance, LXIV, April (2), 751-784.
Fiegenbaum, A. and T. Howard. (1988).“Áttitudes Toward Risk and the Risk-Return Paradox:
Prospect Theory Explanations”. Academy of Management Journal. 31(1), 85-106.
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2012 Cambridge Business & Economics Conference
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Hull, J.C. (2011). Fundamentals of Futures and Options Markets. Seventh Edition. PearsonPrentice Hall
Kahneman, D. and A. Tversky. (1979). “Prospect Theory: An Analysis of Decision Under Risk”.
Econometrica, 47, March (2), 263-291.
Thaler, R.H., and E.J. Johnson. (1990). “Gambling with the House Money and Trying to Break
Even: The Effects of Prior Outcomes on Risky Choices”. Management Science, XXXVI, 643660.
Tversky, A. and D. Kahneman. (1992). “Advances in Prospect Theory: Cumulative
Representation of Uncertainty”. Journal of Risk and Uncertaint. V. 297-323.
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