Stochastic processes

It is formally defined as a process that is described by the change of some random variable over time, which may be either discrete or continuous. During this time, events may be happening at various points along the path that may affect the ultimate value of the process (Alao & Oloni, 2015).

2.8.5.1 Brownian Motion for modelling prices

This random stochastic process generates the course of each movement which creates a probability space for all the possible outcomes. The Brownian motion, however, is a deterministic system, which means that the nodes generated can be mathematically calculated.

Let 𝛺 be a set of possible outcomes from an experiment or uncertain event (𝜔₁, 𝜔₂, … ), and let’s take a random variable 𝛸, say the price of the commodity, which is a function from the 𝛺 of the possible outcomes, and let it be a real number ℝ from the set. An algebra𝑖𝑐 𝛸 needs to be generated from set, and all the subsets of 𝛸 are referred to as events which are the outcomes of a particular experiment where 𝛸: 𝛺 → ℝ.

Therefore, the events of 𝛸 are those for which a probability can be given that they will occur. Probability is a measure on 𝛸, and it is the chance of the event occurring or not.

𝑃(∅) = 0, 𝑃(𝛺) = 1 2.37

P (disjoint events) = ∑𝑃(each event)

The Brownian and all the other stochastics processes briefly discussed in the next subsections are conceptually probability-based risk assessment tools which quantify risk (Jablonowski et al., 2017).

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Therefore, they cannot be independently used to value flexibility without the incorporation of the RO methodology.

2.8.5.2 Wiener process

This process is a form of the Brownian motion stochastic process Wt which is characterised by the following three facts:

W0 = 0

Wt is almost certainly continuous (in another words it has a continuous sample path)

Wt has an independent increment with a distribution Wt-Ws~N(0, t-s). Thus, the summary equation for the Wiener process is

𝑑𝑥 = 𝑎𝑑𝑡 + 𝑏𝑑𝑊(𝑡) 2.38

Where 𝑎 and 𝑏 are constants and the 𝑑𝑥 = 𝑎𝑑𝑡 can be integrated to 𝑥 = 𝑥₀+ 𝑎𝑡, where 𝑥₀ is the initial value and if the time period is 𝑇, then the variable is increased by 𝑎𝑡. 𝑏𝑑𝑧 accounts for the noise or variability to the path followed by 𝑥. The amount of this noise or variability is 𝑏 times the Wiener process.

2.8.5.3 Geometric Brownian Motion (GBM) and mean-reverting process (MRP)

The models used for generating realisations of the market and economic variables are the GBM and MRP models (Dimitrakopoulos & Abdelsabour, 2007; Mun, 2006).Some researchers in the past used one-factor GBM models to reduce the dimension of the problem to one. This simplification implies that there is a perfect correlation between two completely independent variables such as metal price and the grade of metal. However, such an assumption is erroneous as the grade is an internal variable which is usually project specific and the price is an external variable (Ajak & Topal, 2015; Dimitrakopoulos & Abdelsabour, 2007). Eq. (2.39)shows the GBM model.

∆S

𝑆 = 𝜇∆𝑡 + 𝜎𝜀√∆𝑡

2.39

Where;

∆𝑆 is the change in the price of risky asset or stock price.

µ is the expected rate of return.

σ√∆t is the stochastic component.

ɛ is the normal distribution.

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𝑑𝑆

𝑆 = 𝑘(𝑢𝜇 − 𝑙𝑛𝑆)𝑑𝑡 + 𝜎𝑑𝑧

2.40

Where k is the reversion speed at which the log of a price reverts to a long-term equilibrium log price 𝜇.

2.8.5.4 Itō’s process for computing option value

An Ito process is a generalised form of the Wiener process in which the parameters a and b are functions of the value of the underlying variable 𝑥 and 𝑡. These are both the expected drift and the volatility that can change over time. An Ito process with many dimensions is represented by:

𝑥_𝑡 = 𝑥₀ + ∫ 𝑎𝑡 _𝑠

0

𝑑𝑠 + ∫ 𝑏𝑡 _𝑠

0

𝑑𝑊_𝑠 2.41

Where W is a m-dimensional standard Brownian motion and 𝑎 and 𝑏 are n-dimensional and 𝑎(𝑛 ∗ 𝑚) - dimensional 𝐹_𝑡 adapted processes, respectively. Eq. (2.38) and the n-dimensional stochastic differential equation forms Eq. (2.42).

𝑑𝑋_𝑡 = 𝑎(𝑋_𝑡+ 𝑡)𝑑𝑡 + 𝑏(𝑋_𝑡+ 𝑡)𝑑𝑊_𝑡 ; 𝑋₀ = 𝑥 2.42 Thus, Eq. (2.42) can be represented as:

𝑥_𝑡 = 𝑥₀ + ∫ 𝑎(𝑋_𝑠 , 𝑠) 𝑡 0 𝑑𝑠 + ∫ 𝑏(𝑋_𝑠 , 𝑠) 𝑡 0 𝑑𝑊_𝑠 2.43

If the value of a variable 𝑥 follows the Itō process, the variable 𝑥 has a drift rate of 𝑎 with a variance rate of 𝑏2. Itō’s lemma shows that a function 𝐺 of 𝑥 and 𝑡 follows the process shown in Eq. (_2.44).

𝑑𝐺 = (𝛿𝐺_𝛿𝑥𝑎 +𝛿𝐺 𝛿𝑡 + 1 2𝛿 𝛿2_𝐺 𝑥2 𝑏2) 𝑑𝑡 + 𝛿𝐺 𝛿𝑥𝑏𝑑𝑧 2.44

Therefore, G also follows an Itō process with the drift rate of: 𝛿𝐺 𝛿𝑥𝑎 + 𝛿𝐺 𝛿𝑡 + 1 2 𝛿2_𝐺 𝛿𝑥2𝑏2 2.45

moreover, the variance of:

(𝛿𝐺 𝛿𝑥𝑏)

2

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A complete study of the Itō’s process is beyond the scope of this thesis. Thus, this lemma is not derived as it has been widely described in many calculus books. However, this section will only deal with its application as it will be applied to the analysis of real options later in Chapter 4.

2.8.5.5 Application of Itō’s lemma on a forward contract

The formula used in the calculation of a forward contract price is as follows:

𝐹₀= 𝑆₀𝑒𝑟𝑇 _2.47

Where 𝐹 is the forward price as time passes, 𝑆 is the current stock price, 𝑡 is the time when the forward contract is being exercised before maturity and 𝑎 < 𝑇, 𝑟 is the rate of return and 𝑒 is the natural number which is constant. Thus,

𝐹₀ = 𝑆₀𝑒𝑟(𝑇−𝑡) _2.48

For instance, if 𝑆₀ = $50, 𝑟=0.05 and 𝑇= 1 year, then 𝐹₀ will be equal to $52.55 2.8.5.6 Partial derivatives of forward price (F)

Assuming that Eq. (2.39)gives the process for the stock price 𝑆. The Itō process can then be used to determine the process for 𝐹.

𝛿𝐹 𝛿𝑆= 𝑒𝑟(𝑇−𝑡), 𝛿2_𝐹 𝛿𝑆2= 0, 𝛿𝐹 𝛿𝑡= −𝑟𝑆𝑒𝑟(𝑇−𝑡) 2.49

Substitute 𝐺 function with 𝐹, 𝑎 with 𝑢𝑆, and b with 𝜎𝑆. The new equation will appear as follows:

𝑑𝐹 = (𝛿𝐹_𝛿𝑥µS +𝛿𝐹 𝛿𝑡 + 1 2𝛿 𝛿2_𝐹 𝑥2 (σS)2) 𝑑𝑡 + 𝛿𝐹 𝛿𝑥σS𝑑𝑧 2.50

However, it is already shown that:

𝛿𝐹 𝛿𝑆= 𝑒 𝑟(𝑇−𝑡)_, 𝛿2𝐹 𝛿𝑆2= 0, 𝛿𝐹 𝛿𝑡= −𝑟𝑆𝑒 𝑟(𝑇−𝑡) 2.51 Therefore substituting this derivative into the equation produces the following:

𝑑𝐹 = (𝑒𝑟(𝑇−𝑡)_{µS − 𝑟𝑆𝑒}𝑟(𝑇−𝑡)_{+ 0(σS)}2_{)𝑑𝑡 + 𝑒}𝑟(𝑇−𝑡)_{σS𝑑𝑧 2.52}

Since 𝐹₀= 𝑆₀𝑒𝑟(𝑇−𝑡)_{, substituting F for 𝑆}

0𝑒𝑟(𝑇−𝑡) produces

𝑑𝐹 = (𝑢 − 𝑟)𝐹𝑑𝑡 + 𝜎𝐹𝑑𝑧 2.53

Similar to the stock price 𝑆, the forward price 𝐹 follows the GBM. This has an expected growth rate of 𝑢 − 𝑟 rather than 𝑢. The growth rate of 𝐹 is the excess return of the risky asset with a risk-free rate.

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Assuming variable 𝑆 follows an Itō process which contains a non-stochastic and a stochastic component, then the following statement is true about any function 𝐺(𝑆, 𝑡), that is a function of 𝑆 and 𝑡.

𝑑G(S, t) = (µ𝑆𝑡 𝛿𝐺 𝛿𝑠 + 𝛿𝐺 𝛿𝑡 + 1 2𝜎2𝑆2 𝛿2_𝐺 𝛿𝑆2𝑏2) 𝑑𝑡 + 𝜎𝑆𝑡 𝛿𝐺 𝛿𝑆𝑑𝑊𝑡 2.54 2.8.5.7 Lognormal distribution

This is a continuous probability distribution of an uncertain variable such as commodity price whose logarithm of the returns is normally distributed (National Institute of Standards and Technology, 2012).A variable modelled as lognormal is a multiplicative product of many independent random variables and each of which is positive with a probability density function shown in Eq. (2.55).

𝑓_𝑥(x; µ, σ) =_{𝑥𝜎√2𝜋}1 𝑒−(𝑙𝑛𝑥−𝜇)22𝜎2, , 𝑥 > 0 and 𝑋 = 𝑒𝑢+𝜎𝑧 2.55

In a lognormal distribution 𝑋, the parameters denoted by 𝑢 and 𝜎 are, respectively, the mean and standard deviation of the variable’s natural logarithm and 𝑧 is the standard normal variable (z-score). By definition, the variable’s logarithm is normally distributed.

This method used a normal distribution of the gains from the risky asset. In another words, the prices of the risky asset are lognormally distributed. Lognormal distributions are much more tailed to the right than the ordinary continuous normal distribution (bell-shaped distribution). Prices of the asset which meet this distribution usually range between zero and infinity but, never go below zero. This can be simply put as saying that stock prices will always stay above zero.

There are a few cases where the prices of the risk asset do not meet the normal distribution criteria and become common when changes are rapid and frequent. Such departures from lognormal distributions are measured by coefficients of skewness and kurtosis of the values.

2.8.5.8 Application of Itō’s lemma to lognormal process

The aim here is to produce an equation of 𝑌_𝑡 = 𝐹(𝑆_𝑡) such that 𝑌_𝑡 will not contain any references to 𝑆_𝑡. Now the equation 𝑌_𝑡 = 𝑙𝑜𝑔(𝑆_𝑡) will be used, but it should be noted that 𝑌_𝑡 is not a function of 𝑡 but 𝑆_𝑡.

𝑑𝑌𝑡 = [0 + 1 𝑆𝑡𝜇𝑡𝑆𝑡+ 1 2( 1 𝑆_𝑡2) 𝜎𝑡2𝑆𝑡2] 𝑑𝑡 + 1 𝑆𝑡𝜎𝑡𝑆𝑡𝑑𝑊𝑡 2.56 𝑑𝑌_𝑡 = (𝜇𝑡+ 𝜎_𝑡2 2) 𝑑𝑡 + 𝜎𝑡𝑑𝑊𝑡 2.57

LITERATURE REVIEW 36 𝑌_𝑡 = ∫ (𝜇𝑡+ 𝜎_𝑡2 2) 𝑡 0 𝑑𝑡 + ∫ 𝜎𝑡 _𝑡𝑑𝑊_𝑡 0 2.58 However, it is already established that 𝑌𝑡 = log(𝑆𝑡 )

Therefore, 𝑌_𝑡 = log(𝑆₀ + ∫ 𝜇𝑡 _𝑢 0 𝑆_𝑢 𝑑_𝑢 + ∫ 𝜎𝑡 _𝑢 0 𝑆_𝑢 𝑑𝑤_𝑢 ) 2.59

and it is also true that,

𝑆𝑡 = 𝑒(𝑌𝑡 ) 2.60 𝑆_𝑡 = 𝑆₀ 𝑒(∫ (𝜇𝑢 − 𝜎 2 2 )𝑑𝑢 + ∫ 𝜎₀𝑡 𝑢 𝑆𝑢 𝑑𝑤𝑢 𝑡 0 ) 2.61

It should be highlighted that this equation is a Geometric Brownian Motion if the 𝑢 and σ are constants and the 𝑌_𝑡 = log(𝑆_𝑡 ) is normally distributed and the 𝑆_𝑡 is lognormal.

As discussed previously, the application of stochastic processes alone assesses risk but, does not value flexibility. They are commonly utilised in scenario planning where alternatives are assessed, and the decision is made on the scenario with an optimal value. Therefore, there is no room for future decisions as the future is assumed to mimic the simulated path. As a consequence, these processes are backed up with RO which can create the managerial flexibility (Ajak & Topal, 2015; Santos et al., 2014; Guj, 2011).