RO is completely different from DCF analysis, which is well established in the mining industry. The RO approach is a new paradigm that requires engineers and project managers to view uncertainties as opportunities for value creation; however, quantifying the values that can be derived from the available options can be a daunting task. The mathematical complexity involved in valuing RO has been the main hindrance in its adoption, as well as its application in solving problems related to real world projects. Like any other decision-making tool, RO uses available mathematical and decision analysis techniques. The binomial decision tree model is one technique used in valuing RO. As concluded by Brandao et al. (2005), the binomial decision tree model is the most popular, intuitive and transparent method used so far in RO analysis.
3.3.1 Assumptions of the Model
The best model for tracing the evolution of using key underlying variables to create project flexibility over a discrete time is the Cox, Ross and Rubinstein binomial tree model (Cox et al., 1979).This model was intended as a numerical procedure to solve the Black-Scholes equation.
There are two main assumptions underlying the binomial tree. The first assumption is that of a continuous random walk (Eq. 3.1).This assumption allows the tree to be modelled by a discrete random walk with the following properties 𝑡0= 0, t0= 0, 𝑡0 = ∆𝑡. The price of the underlying asset 𝑆 changes only
at discrete times 𝑡1 = ∆𝑡, 𝑡2= 2∆𝑡, 𝑡𝑛= 𝑛∆𝑡 = 𝑇, where 𝑇 is the expiration date of the option and ∆𝑡 = 𝑇
𝑛
REAL OPTION IN ACTION 54
If the price of the underlying asset is 𝑆𝑛,𝑖at 𝑖 and time 𝑡𝑛, then at the next time step it may take only one of two possible values:
Sn+1, i+1 = uSn, i>Sn, i or Sn+1, i+1 = dSn, i<Sn, i 3.1
This is equivalent to assuming there are only two returns possible at each time step u-1>0, d -1<0, where
u is the up movement and d is the down movement, and these two returns are the same for all time steps.
The probability, p of Sn,i moving up to Sn+1,i+1 = uSn, i is known. The same results for the probability q of Sn, i moving down to dSn+1, i, since p+q = 1 .
The second assumption underlying a binomial tree is that of risk-neutrality. This implies that the investor risk preferences are irrelevant to option valuation. This has two implications.
First, the expected return from all traded securities is the risk-free interest rate, rf. This means that the drift term 𝑢 in the stochastic differential equation for the asset return (Eq. 3.2)is replaced by the risk-free interest rate rf whenever it appears, and 𝑊𝑡 is a standard Wiener process:
𝑑𝑆𝑡
𝑆𝑡 = 𝑟𝑑𝑡 + 𝜎𝑑𝑊𝑡 3.2
Second, the option value 𝑉𝑛, at 𝑡𝑛= 𝑛∆𝑡 is its expected value 𝐸, at 𝑡𝑛+1= (𝑛 + 1)∆𝑡, discounted by the risk-free interest rate 𝑟𝑓:
𝑉𝑛 = 𝐸[exp(−𝑟𝑓∆𝑡)𝑉𝑛+1] 3.3
As per this approach, the probabilities p, q, and the returns u, d should reflect the important statistical properties of the continuous random walk, meaning they have to ensure that for t0 the underlying asset S follows the geometric Brownian motion (GBM). In short, the parameters p, q, u, d should give the correct values for the mean and the variance of the underlying asset, which is shown in Eq. (3.4)during a time interval ∆𝑡, where
2t
is the variance parameter and lnSn is the conditional distribution:𝑙𝑛𝑆𝑡𝑛+1≈ 𝑁(ln (𝑆𝑛+ (𝑏 −𝜎
2
2 ∆𝑡, 𝜎2∆𝑡) 3.4
Consequently, these parameters must solve the following equations:
𝑝 + 𝑞 = 1 3.5
𝐸 = 𝑝𝑙𝑛(𝑢𝑆𝑛) + 𝑞𝑙𝑛(𝑆𝑛) = ln(𝑆𝑛) + (𝑏 +𝜎2
REAL OPTION IN ACTION 55
𝑃(𝑙𝑛(𝑢𝑆𝑛) − 𝐸) + 𝑞(𝑙𝑛(𝑑𝑆𝑛) − 𝐸)2 = 𝜎2∆𝑡 3.7
Substituting q = p -1 in Eqs. (3.6) and (3.7), there are three unknown parameters and two non-linear equations to solve. To obtain a unique solution, a supplementary restriction for the parameters is needed. Variable b is the option carry cost and is equal to rf. Cox et al. (1979) chose the restriction ud =1, as it
drastically simplifies the tree. At time point tn there are only i = 1, …, n +1 possible nodes and:
𝑆𝑛,1 = 𝑢𝑛𝑑𝑛 − 𝑖𝑆0, 3.8
Where S0 is the asset price in t0. Solving the Eqs. (3.6) – (3.8)for p, u and d and neglecting the terms smaller
than ∆𝑡 results in:
d
u
t
d
t
u
t
b
p
)
,
exp(
),
exp(
),
1
2
1
(
2
1
2
1
2
3.9 Therefore:for continuously compound growth
𝑝 =𝑒
𝑟∆𝑡− 𝑢
𝑢 − 𝑑
3.10 for annually compounded growth
𝑝 =1 + 𝑟 − 𝑑 𝑢 − 𝑑
The time steps are of equal length, so that the risk-neutral probability p (as calculated by Eq. 3.9) is the same at each node. The option price Vn,i = V(Sn,i, tn), at node i and time tn is the expected pay-off at tn+1 discounted at the risk-free interest rate:
𝑉𝑛,𝑖= 𝐸exp(−𝑟𝑓∆𝑡)[𝑝𝑉𝑛+1,,𝑖+1+(1−𝑝)𝑉𝑛,𝑖+1] 3.11
Simply, the value of the uncertain variables, the uncertain price value in the next period, Vt+
t
is equal tothe value of this period Vt multiplied by the continuous growth rate r for an interval Δt. The growth rate r is a random variable that is normally distributed with constant expected growth (r҄ ) and constant standard deviation σ:
𝑉𝑡+∆𝑡= 𝑉𝑡𝑒𝑟∆𝑡 3.12
𝐸(𝑉∆𝑡) = 𝑉0𝑒𝑟∆𝑡 = 𝑝𝑢𝑉0+ (1 − 𝑝)𝑑𝑉0 3.13
REAL OPTION IN ACTION 56
At the end of the tree, the option price is known. It equals the option value at expiration. For a call option, this is:
VN,I = max(0, SN,I – K), k = 0,….,n 3.15
Where I is the capital investment and K is the exercise price. The option values at each node, Vn,i, i = 0,…,n and n = N – 1,…,0, will then be determined recursively by working backwards through the tree.
Finally, the binomial decision tree model, which is the proposed method for RO analysis in this study, will be applied following a modified four-step process (Fig.3.1) developed by Copeland & Antikarov
(2003), and advanced by Brandao et al. (2005). An alternative approach which would also have been used instead of the decision tree analysis is a stochastic simulation process such as Monte Carlo simulation. This method would have produced pretty much similar results but with less intuition. The Black – Scholes option model would also have been used but its application was limited as this method requires options that are European style which can only be exercised at a specified date. The proposed binomial decision tree model is summarised in Fig. 3.1 below.
Fig. 3.1, Binomial model application steps (source: Copeland & Antikarov, 2003).