Section III: Risk allocation in large-scale infrastructure projects
Chapter 3. Real Option Methodology
3.4 The approaches to solving real option problems
3.4.1 The binomial lattice and the binomial decision tree
Cox‘s (1979) binomial lattice is a well-known discrete-time model representation of the behaviour of asset prices. A binomial model is used to approximate the continuous time model of a stochastic differential equation, i.e., the geometric Brownian motion (GBM). The binomial model assumes that the value of the underlying asset, e.g., the project‘s cash flow, follows a binomial distribution. The binomial distribution is the diagram of a different possible path that the stock (underlying asset) price might follow over the life of the option (Hull, 2009). The binomial model is a useful technique for pricing an option with simply discrete mathematics and a discrete formula (Cox et al., 1979). The method is accurate, remarkably robust and frequently used for valuing financial and real options (Hahn, 2005). This model involves constructing a path, which is called a binomial tree. The asset price (s) can move up (u) or down (d) by a certain amount and probability. Figure 3.3 shows a binomial lattice with three time steps.
Figure 3.3: Example of the binomial lattice with three time steps
Real option model
Continuous time model
Analytical (example: the Black-Scholes model)
Numerical (examples: simulation, finite difference)
S0ud S0 S0u S0u2 S0u3 S0u2d S0d S0d2 S0ud2 S0d3 Discrete time model
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In a binomial lattice, the value of an option is computed under a risk-neutral probability equation. In a risk-neutral world, investors are indifferent about risk, so they do not require compensation for risk. The expected return on all securities is a risk-free interest rate, which is based on the assumption that one can construct a risk-free hedged portfolio. For example, a risk-free hedged portfolio could be constructed with the option to invest and a short position in a project. Also, the binomial lattice uses the notion of arbitrage in which the risk-adjusted probabilities can be computed if there is no arbitrage opportunity. This method adjusts the probability that leads to pay-outs and then discounts the cash flows by the risk-free interest rate, as risk is already adjusted at the cash flow. Using risk-neutral valuation and a no- arbitrage argument makes it more convenient to value options. It can be argued that this approach is beneficially grounded for valuing derivatives but it is highly susceptible to estimating the future value of derivatives. The probability ―p‖ is used in the risk-neutral condition, in which risks have been accounted for. This method is very useful as it does not need to estimate project-specific discount rates at different nodes along the binomial. The neutral probability ―p‖ is calculated by the following equation (Mun, 2006):
= a risk-neutral probability where
rf
= the risk-free interest rate (%)Δt = the time step interval u = the up factor
= √
d = the down factor
= √
In the binomial lattice, the stock price changes in a small time interval (Δt), either by an uptick or a downtick. The observation time (t) starts from t0 and increases with multiples of Δt. The changes in stock prices are independent of each other. When the time step (Δt) is small, the binomial lattice provides a good approximation of an option‘s value. The procedure starts with a forward movement of an underlying value (St) and determines an option‘s value at its expiration date. Then, using risk-adjusted probability, the option value can be
determined by working backward with the lattice. The binomial model computes option values by assuming risk-neutral probability, so that we can discount project cash flows at the risk-free interest rate of return.
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The binomial lattice is preferred for a real option analysis because it provides more flexibility for computing various option types, especially for exotic options, which are difficult to calculate using the closed-form analytical model, as it is quite a complex mathematical equation. The model is simple for implementing and easy for explanation. The advantage of the binomial model is its flexibility in analysing complex options such as American options or dividend-paying European options (Scaramozzino, 2010). Those option values can also be determined by solving the numerical method. The main advantage of binomial lattice over other methods (e.g., Black-Scholes model) is that it can be used to accurately price the American options. The Black-Scholes model cannot be used to accurately price American- style options as it only calculates the option price at one point while it is possible to check at every point in an option path for the possibility of early exercise in the binomial model. It also has an advantage because the mathematical formula is relatively easy compared to other methods. Moreover, the calculation is more accurate as the real market development can be inserted in the binomial model; therefore, the calculation can link with the actual market development.
The traditional valuation method normally uses WACC as a discount rate for an individual project without options. WACC is the hurdle rate or discount rate for evaluating projects, which is calculated using the following formula:
WACC = (E/V) * Re) + [((D/V) * Rd)*(1-T)] where
E = Market value of the company‘s equity D = Market value of the company‘s debt
V = Total market value of the company (E + D) Re = Cost of equity
Rd = Cost of debt T= Tax Rate
The disadvantage of WACC is its rigid assumptions without flexibility in the method of evaluation of new projects. The impractical assumption of an unchanged capital structure does not happen all the time. However, the existence of managerial flexibility of real options changes the risk of the project if the manager chooses to exercise real options to increase project value. Then WACC would not be the appropriate discount rate for the project with options. The binomial lattice overcomes the shortcoming of WACC by using the risk-free
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interest rate as the discount rate to calculate the option payoff. However, the main limitation of the binomial model is its relatively slow speed. The higher accuracy of the lattice model however comes at a cost. This method is more time-consuming than the other closed-form methods such as the Black-Scholes model.
Traditional option pricing methods require certain assumptions such as the complete market, which implies that there are marketable securities or portfolios of securities whose payoffs replicate the project‘s payoff in all states and periods (Brandao and Dyer, 2005). This assumption is important in the field of continuous-time real option valuation. Although this assumption is required for options on financial assets, for most of the real asset projects, no such replicating portfolio of securities exists and markets are said to be incomplete (Brandao and Dyer, 2005). With this limitation of the continuous-time real option, Copeland and Vladimir (2001) proposed an alternative discrete-time method (i.e., the binomial lattice model) based on the assumption that the present value of the project without options is the best unbiased estimator of the market value of the project. With this assumption, this option can be valued with traditional option pricing methods including the binomial lattice method.
The main assumptions of the binomial model (Cox et al., 1979) are that i) the constant risk- free interest rate is applied; ii) investors can borrow or lend as much as they want, and short selling is allowed; iii) there are no taxes, transaction costs or margin requirements, and no cash dividends are paid during the life of the option; iv) there are no arbitrage opportunities; v) stocks and options are traded in the perfect market with the same underlying uncertainties; and vi) the evaluation process is stationary over time.
The binomial model is the cornerstone of the option theory and it is the most widely used method for pricing the option and the real option. Although it can be argued that many assumptions seem restrictive, the binomial lattice has proven to be a solid and rigorous method for pricing both financial options and real options.
Table 3.4 lists previous research studies that apply the binomial lattice to real options.
Authors Title Conclusion
Brandao and Dyer (2005)
Decision Analysis and Real Options: A Discrete Time
The authors demonstrated that a binomial tree based on a binomial
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Authors Title Conclusion
Approach to Real Valuation lattice can be used to evaluate real options. The authors used a
hypothetical project to demonstrate that a binomial tree can be used to value real options, e.g., to abandon options. The authors conclude that the binomial tree method is
computationally simpler and more intuitive than the traditional method. However, the authors argued that the use of the binomial tree for valuing projects with real options was not applicable for all situations. Trigeorgis (1993a) The Nature of Option
Interactions and the Valuation of Investments with Multiple Real Options
This paper uses the binomial lattice to illustrate the interaction among multiple options (the option to defer, abandon, contract or expand, invest and switch use). The results show the incremental value of multiple options on individual options. Furthermore, multiple options tend to preserve a number of the familiar option‘s properties. The results of the study show that when adding an option to the other option, the incremental value is generally less than its value in isolation, and this value decreases if more options are added.
Trigeorgis (1993b) Real Options and
Interactions with Financial Flexibility
The author applied the binomial lattice to value several real options and to illustrate how option
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Authors Title Conclusion
interaction among options. The author illustrated how to value the various types of real options—e.g., option to defer, option to expand and option to abandon—in the investment capital. The author then extended real option analysis to the context of venture capital. The mix of the equity holders‘ option to default and the debtor‘s option to abandon the project was examined. The author concluded that option flexibility allows better financing terms in later stages of project financing and is therefore clearly more valuable than a passive alternative of financing terms that were irrevocably committed at the beginning of a project.
Table 3.4: Previous research studies using the binomial model on real option problems
Cox et al (1979) were the first researchers who applied the binomial lattice to approximate the underlying stochastic process and then calculate the option value through the use of risk- neutral pricing techniques. Their approximations to the underlying stochastic processes relied only on simple algebra and are therefore more transparent and computationally efficient. The binomial model has advantages over the Black-Scholes model in that it can be used to accurately price American options. With the binomial model it is possible to calculate the option value at every point in an option‘s life.
Although this method is widely used in pricing options, it has been argued that the binomial model is more cumbersome as the number of time periods increases. The increase in time periods requires intensive labour, especially handling problems involving multiple uncertainties, ―path-dependent‖ uncertainties, and complex options (Tsui, 2005). The binomial model has the same limitations as those of the option pricing model. The method
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makes it difficult to find a replicating portfolio and consequently hinders the risk-neutral valuation. It also has practical problems with risk-neutral valuation when the inferred option pricing parameters are not applicable to the real world. For example, the probability of the real project‘s success or failure is different in the real and the risk-neutral world.
Emmanuel et al (2014) argued that the binomial model sometimes fails to value the managerial flexibility in many types of projects. The model is also difficult to adapt to more complex situations. In summary, Emmanuel et al (2014) argued that the binomial model is suitable and more accurate for pricing options with early exercise opportunities and that it is relatively easy to implement. However it can be quite difficult to adapt to more complex situations. Emmanuel et al (2014) argued that the model is much more capable of pricing early exercise because it considers the cash flow at each time period rather than just the cash flows at expiration.
Copeland and Vladimir (2001) were among the pioneers to apply the binomial lattice model to evaluate the real option problem. Their method assumed that the present value of the cash flows of the project without flexibility (traditional NPV) is the best unbiased estimator of the market value of the project. The traditional NPV is used as the value of the underlying asset for an option pricing model called the marketed asset disclaimer (MAD). In this method, the value of the project with flexibility is the value of the project without flexibility plus the value of the embedded options. Therefore, in the MAD approach, it requires an additivity argument, which can be proven in Williams (1938) or Schall (1972). The MAD approach proposes an alternative method to solve real option valuation problems based on the no arbitrage principle when the underlying asset is not traded in the capital market. Copeland and Vladimir (2001) proposed to build a binomial lattice model, with a binomial approximation to a geometric Brownian motion (gBm), to estimate the project value and real option value. However, it has been argued by Brandao and Dyer (2005) that the use of the MAD assumption to create a complete market for an asset that is usually not traded in the market may lead to significant errors. Moreover, it has been argued that the present value of the project is not readily
observable but only approximated. Therefore, different analysts may receive various values of the underlying asset and recommend different exercise strategies.
For many researchers in infrastructure evaluations, the binomial model is the simplest method for real option pricing. The mathematics of the model is relatively easy to understand and it is not difficult to implement. The binomial model was recommended by Pichayapan et al (2003)
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for practical use in real option analysis. Compared with other methods, the binomial lattice is found to be most suitable for practitioners due to its lower complexity and efficiency in calculation (Pichayapan et al, 2003). The binomial model is used by many researchers i.e., Pichayapan et al (2003) and Charoenpornpattana et al (2003), in expressway evaluation for illustration to practitioners in real application. Infrastructure projects gain benefit from the use of the binomial model method for option valuation instead of traditional analysis, as the model can deal with projects with high risk and great uncertainty.