2012Cambridge Business & Economics ConferenceISBN : 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.
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.
IINTRODUCTION
Existing empirical analyses of corporate investments typically assume deterministic decision-making 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 tostate 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,andfor puts on non-dividend paying stocks do not involve the stock’s return,
where
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 B-S model to this paper, the variables is the price of the underlying asset, 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, is theexpiration date, is the volatility of the price of the underlying asset and the function is the cumulative probability functionfor a standardized normal variable.
II.A Generalized binomial Model on Risk Neutral Investments
Let refer to the median price of the underlying asset and is the option to invest in the underlying asset. The option to invest will expire in time,. During the life of the option, the price of the underlying asset can move up fromto a higher level,, and the payoff from the option is. Conversely, the price of the underlying asset can movedown fromto a lower price and the payoff from the option is. The inference is that. In other words, the proportional increase in the price of the underlying asset when there is an up movement is and the proportional decrease in the price of the underlying asset when there is down movement is
Hypothetically, if the price of the underlying asset goesup, the value of the portfolio at the end of the option is
If the price of the underlying asset goes down, the value of the portfolio becomes
The value of that makes the portfolio riskless becomes
(1)
Equation (1) shows thatis 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 byand the present value of the portfolio when the price of the underlying asset goes up becomes:
Conversely, the present value of the portfolio if the price of the underlying asset goes down is zero and the option is worthless:
Now, suppose the cost of setting up the portfolio is
;
It then follows that
; and
If we substitute equation (1) forand,the equation above becomes
(2)
is the probability of an up movement and is the probability of a down movement. Assuming thatthe 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 and based on the equations below.
We set and at to represent the annual after tax median rate of return and standard deviation respectively. Thus the price of the underlying asset goes up =1.1619, and down by
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. Ifand=0.0; then
and
It should be emphasized that which is the probability of an up movement in a risk neutral world may be different in the real world.From above we obtainwhen 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 as the probability of an up movement in the real world, it follows that:
If the value of the optionis, 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
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, is known. At time there are two possible prices, and ; at time there are three possible prices, , and and so on.
Accordingly equation (2) becomes
(3) and
the expression for becomes:
(4)
Repeated computations of equation (3) gives
(5)
(6)
Substituting equations (5) and (6) into (3), we get
(7)
Equation (7) is consistent with the principle of risk-neutral valuation discussed earlier. The variables, andareprobabilities 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.
IIITHE ALGEBRAIC APPROACH TO ESTIMATE OPTION VALUE.
We assume that the option on fossil fuels such as for oil and gas could be likened to a non-dividend- paying stock. Next, suppose that the life of an American call option is divided into N subintervals of length. We refer to the node at time as the node, where ≤ ≤and
≤ ≤ . We also define as the value of the option at the node. The value of the underlying asset at the node is
The value of the option is known at time T. Since the value of the American call at its expiration is, where is the median price of the oil per barrel at time and is thereference 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
The symbol is theprobability of moving from the node at time to the node at time and is the probability of moving from the node at time to the node at time. If there is an early exercise of the option, the value of the option under risk neutral valuation assumption becomes
for ≤ ≤ and ≤ ≤ .
The value of the option with an early exercise is
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.
IVBINOMIAL 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,
Earlier, the price of the underlying asset was estimated to go up, =1.1619, and down, For illustration purposes, consider a quarterly American style call option on a non-dividend paying asset like oil and gas and
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. Since the assumed reference (minimum acceptable) price depicting the investor’s prior experience, the gain at node N is 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 are respectively and the option prices are respectively .
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 the options are exercised because the gains are equal to the option prices. Though the gains and option prices at nodes 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 The inferences at nodes 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
Growth factor =
Probability of up movement =
Probability of down movement =
Up step size,
Down step size,
Discount factor per step=
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 Long-Standing 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.
Hull, J.C. (2011). Fundamentals of Futures and Options Markets.Seventh Edition. Pearson-Prentice 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,643-660.
Tversky, A. and D. Kahneman. (1992). “Advances in Prospect Theory: Cumulative Representation of Uncertainty”. Journal of Risk and Uncertaint. V. 297-323.
June 27-28, 2012
Cambridge, UK 1