V ALUATION METHODS FOR R EAL O PTIONS

the project at that stage and further investments will not be made. In general, the exercise of an option early, will affect the value of the underlying asset, and thus the value of the options in the other phases of the project.

10.3.2.2 Rainbow options

Rainbow options are options where there is more than one source of uncertainty. There is a different volatility factor for each source of uncertainty. The valuation method is the same as with simple options, but now you get a quadrinomial tree instead of a binomial if there are two sources of uncertainty. This is illustrated in figure 10.3.2 (b). The assets can now take four values as you move from one time period to the other. Rainbow options can represent one or more of the options in a compounded option.

In terms of petroleum fields, there are two main uncertainties, oil price and reserve size. The second uncertainty, reserve size, will be resolved over time, and hence it is only in the early phases that there are multiple sources of uncertainty. This means that rainbow options could be well suited for this industry. However, as you have two volatilities the possible outcomes are twice as many. This makes a very complex method as in just three years there will be 64 different outcomes. In an industry with long time frames, this will get unmanageable very fast and consequently this type of option will not be used in the valuation of DETNOR.

Economics. It provides a closed-form solution for the equilibrium price of an option (Copeland and Antikarov, 2003). The model is based on the same way of thinking as the replicating portfolio, and the goal is to replicate the options cash flows with a combination of the price of the underlying asset and a risk-free asset. The equation is as follows:

(10.3) !_!=!!_!!(!_!)−!!^!!"!(!_!)

Where S0 is the price of the underlying, N(d) is the cumulative normal probability density function, X is the exercise price, r is the risk-free rate and t is the time to maturity (Hull, 2002).The formula can be explained more simply as the value of a replicating portfolio where N(d1) is the number of units necessary to form a mimicking portfolio, and the second part is the number of risk-free assets, each paying $1 at expiration. This is found by multiplying N(d2), which is the probability that the option will finish “in the money”, and Xe^-rt which is the exercise price at the time of maturity discounted back to present time (Copeland and Antikarov, 2003). The formula preforms very well in the real world. It has proven to be very flexible and can be used on not only stocks, but also foreign currencies, bonds and commodities (Brealey, Myers and Allen, 2011).

For optimal use of the model seven assumptions that must be upheld (Copeland and Antikarov, 2003):

1) It has to be a European option

2) There is only one source of uncertainty

3) The option is contingent on a single underlying risky asset 4) There are no dividends

5) The current marketplace and the stochastic process followed by the underlying asset is known

6) The variance of return of the underlying asset is constant through time 7) The exercise price is known and constant

Consequently, in order to be realistic in terms of real options, one or more of the Black-Scholes assumptions must be relaxed. Most investments or projects in the petroleum industry consist of compounded options and there is normally more than one source of uncertainty. As a result, the method is not very appropriate for the purpose of this thesis.

10.4.2 Monte Carlo simulations

A Monte Carlo simulation involves simulating thousands of paths of the underlying assets given the boundaries of uncertainty, which is defined by the volatility of the asset value. To conduct a Monte Carlo simulation, the following parameters needs to be identified: Current value of the underlying asset (s0), Volatility of the asset value (σ), exercise price (X), Option life (t), risk-free rate

corresponding to option life (r), and incremental time steps that will be considered over the option life (δt) (Kodukula and Papudesu, 2006).

The option life is split into a selected number of time steps, and thousands of simulations are made to find asset value at each step. At time 0 the asset value is at S0 and equal. The price can than go both up and down. This is calculated by finding the motion equation that fits best with the movements of the underlying asset. The simulation will give a varying value of the assets, which is calculated over the life of the option. At each time step, it is used to calculate the value of the option if the current price had incurred. When the simulations have run thousands of times, one will get a probability distribution over the value of the option (Hull, 2002).

The method includes multiple value drivers and it is flexible enough to capture many situations that may incur in real life. Monte Carlo simulations are especially useful in terms of European options, where there is a fixed exercise date. The method is not equally functional for American options, and especially in terms of staged sequential options. This is due to that decisions can be made at any stage, and when this has incurred a new path is formed. This will be a daunting task even for the fastest computer (Kodukula and Papudesu, 2006). As a result, Monte Carlo simulations will not be used directly to value the real options embedded in the firm’s petroleum licenses.

10.4.3 Binomial Trees

One of the most popular methods for determining option value includes constructing a binomial tree. The most used approach is the method developed by Cox, Ross, Rubenstein (1979) in their paper “Option pricing: a simplified approach”. This involves using a risk neutral method to valuing options (Hull, 2008).

The model is based on the assumption that the underlying asset can move either up (u) or down (d), where u > 1 and d < 1 over a time period (Δt). The u and d are determined by the underlying assets volatility, by the use of the following formulas:

(10.4) !=!!^{! ∆!}

(10.5) !=_!^!

The probability of an up-movement is denoted by p, and (1 – p) is the probability for a down-movement. p is calculated using the principle of risk-neutral valuation, which is based on the assumption that the world is risk-free. This assumes that the expected return of the underlying asset is the risk-free rate, and that future cash flows can be valued by discounting the expected

value at the risk-free rate. p is calculated using the following equation (Cox, Ross, Rubenstein, 1979):

(10.6) !=^!!∆!!!

!!!

The option value is found by using backward induction, which means starting at the end and working backwards. The option value is not decided by the expected price, but the current price that reflects future expectations. When valuing American options one has to check if it is preferable to exercise the options early or holding the option for an additional period. This is done by comparing the go-forward value to the intrinsic value of the option (Hull, 2008).

The use of a binominal tree is the only method of the common option valuation models that can value American options efficiently. It also handles complex options, like rainbow and sequential options well. This is due to the relative simplicity of the model. The simplicity is also the drawback of the model, as you are limited to low dimensions. It does not capture paths for the underlying assets, which leads to extreme values when using it over long time periods on high volatility assets.

Based on this overview, it seems that the binominal tree is the most suitable method to value the real options present in the upstream petroleum industry. This is mainly a result of this model being the only method able to value complex and American options. The binominal tree will hence be used to value the options identified in the next section of this thesis.