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Weekly UpdatesINTRODUCTION
Recently, managers have addressed the uncertainty in various capital budgeting decisions by applying an options analysis to their evaluations of the project (Amram and Kulatilaka (1999); Copeland and Antikarov (2003); Dixit and Pindyck (1994); Luenberger (1998); Park (2006); and Trigeorgis (1996)). This evaluation technique, real options analysis, provides an opportunity to improve strategic investment decisions in an uncertain environment, but the real options valuation concept requires some adjustments in order to be useful in management decisions (Van Putten and MacMillan 2004). One of the most critical issues in real options is deciding the timing of investment or divestment in the project during the life of the option. Recent research shows that failing to exercise real options on time reduces the project's value much less than predicted. However, the question of whether real option holders exercise their options optimally has not been extensively researched. Therefore, the possibility of early action also needs to be considered in order to make real options more useful.
After deciding to retain the real option, the investors are required to decide the timing for exercising or divesting the option. Because of the irreversibility of the investments, deciding on the optimal investment timing is one of the most important factors in the real options valuation model. In the financial options model, the timing to exercise is defined as the point at which the value of immediate exercise is higher than that of holding the option to its expiration date. However, research has indicated that early exercise is never optimal on a non-dividend-paying stock in the financial call option theory, which is applied to most real options valuation models (Hull 2005).
Copeland and Tufano (2004) stated that defining the optimal exercise timing of the real option is essential in order for real options to work well in the real world. They suggested that failing to exercise real options on time reduces the value of the projects much less than predicted. However, in spite of its importance to decision-makers, there has not been adequate research into selecting the optimal timing for real options. Brennan and Schwartz (1985) developed an evaluation model wherein they set stochastic output prices in order to decide the optimal investment timing for continuing or abandoning a mining project. Moel and Tufano (2002) studied for opening and closing decisions of gold mines through real options. McDonald and Siegel (1986) studied how to optimally time the investment in an irreversible project when the benefit and cost of the project follow geometric Brownian motion. They used the simulation technique to show that risk-averse investors should wait until benefits are twice the investment costs.
Yaksick (1996) suggested a method to compute an expected exercise timing of a perpetual American option. Shackleton and Wojakowski (2002) derived a numerical expression for computing the expected return and for finding the optimal timing for the exercise of real options by using the risk-adjusted stopping time method, which is based on the actual probability distribution of payoff times. Rhys et al. (2002) summarized the recent developments in this area, reporting that only a few studies have been conducted to analyze the problem of timing but concluding that some progress is being made in the research.
A new decision rule, based on the binomial lattice valuation model, is presented here to determine the optimal timing of irreversible investments. The expected profits under the new decision rule are compared to the traditional rule in order to demonstrate the advantages of this new approach, and a simulation technique is used to present the benefit of new early exercise rule in real options. The fundamental of the simulation for validating the new decision rule is generating path-dependent project values. After simulating the required amount of the project value path, the expected future values of the two cases are calculated. Then paired t-tests are conducted to observe the differences between the traditional decision rule and the new decision rule.
The remainder of this research is organized as follows. The next section reviews the general early exercise rule of options and discusses the new decision model. The following section substantiates the new approach by comparing the expected values of projects using the current options decision rule with those using the new decision rule. A simulation technique is used to generate the expected future project values. Then the new decision model is applied to a deferral option and an abandonment option, followed by a summary of the research as well as its conclusions.
DEVELOPING A NEW DECISION RULE OF REAL OPTIONS
Although an early exercise of a financial call option is known to be never optimal, the same logic does not apply to the real options. First, real options do not have the same characteristics as financial options. One of the most important differences is that in the real options framework, the investment is irreversible once the project is undertaken, and the invested budget is generally not tradable. Therefore, deriving the optimal exercise timing of real options requires a unique approach compare to that of financial options. The process of developing the new decision rule concentrates on the total expected future profit by taking early actions.
Early Exercise in Financial Options
Before relating the details of the new decision model, it is necessary to review the early exercise decision of the financial option pricing. An American option can be exercised at any time during its life. The early exercise of the American option is decided by comparing the value of waiting to the payoff of early exercise. A binomial lattice valuation model that was originated by Cox et al. (1979) is applied to calculate the early decision points.
Figure 1 illustrates the procedures for deciding the early exercise in node [??] by a binomial lattice approach. The initial stock price [S.sub.0] moves to one of two values, [S.sub.0]u and [S.sub.0]d, during the first time interval. The two values also will move to two possible directions, "up" or "down," during the next time interval, and so on. The parameter u represents an "up" movement and d a "down" movement during a time interval [DELTA]t. The other parameters in the lattice are p, which represents the probability that the stock price takes an "up" movement: 1 - p, which is the probability that the stock price moves "down": and [gamma], the risk-free rate of the model. "K" represents the strike price of the options. Then, the processes for deciding on early exercise at node [??] are
1. Compute the option value OV[u.sup.2] by waiting.
OV[u.sup.2] = [p(OV[u.sup.3]) + (1 - p)(OV[u.sup.2]d)] x [e.sup.-r[DELTA]t]
2. Compute the immediate payoff.
OV[u.sup.2]: = max([S.sub.0][u.sup.2] - K, 0), for a call option
OV[u.sup.2] = max(K - [S.sub.0][u.sup.2], 0), for a put option
3. Select the highest value of step 1 and step 2 as the option value of the node [??]. If the value computed by the immediate payoff option from step 2 is higher than that of waiting, which is defined from step 1, early exercise is preferred in the node. However, the empirical test demonstrates that early exercise is never optimal for American call options of the no-dividend-paying stock, while early exercise may be possible in an American put option.
[FIGURE 1 OMITTED]
Proposed Decision Model
The new idea for determining the early investment points of the real option is based on the opportunity cost concept, which is defined by comparing the expected future option value with the expected future profit earned by early action. Two important assumptions are necessary for the development of the new decision model.
* The first assumption is that once the option is exercised, the projected profit is immediately realized and is available for other investment purposes.
* The second assumption is that the investment in the other projects will earn the risk-adjusted rate of return of the company, compounded continuously. These two assumptions are widely used in the engineering economics analysis of the capital investment decision.
Before explaining the method of making decision in detail, it is necessary to define some notations.
Notations
[ILLUSTRATION OMITTED]
T: Option life
[DELTA]t: Length of the option period
T/[DELTA]t = N: Number of time period during the option life
n: Time node, n = 0, 1, 2, ..., N
m: Value states in each time node. m = 0, 1, 2, ..., n
[V.sub.nm] : Estimated project value at mth highest value of time n.
[OV.sub.nm]: Option value at node nm.
For the call style option model, since we initially considered a deferral option model, the option values at the end nodes are defined as [OV.sub.Nm] = [V.sub.Nm]--I and the values at the other time nodes are defined as:
[OV.sub.nm] = [OV.sub.(n+1)m] x p + [OV.sub.(n+1)(m+1)] x (1 - p)] x [e.sup.-r[DELTA]t], n = 0, 1, 2, ..., N - 1, m = 0, 1, 2, ..., n
where p: risk-neutral probability and r: risk-free interest rate.
For a put-style real option such as an abandonment option, the option values at the very last time nodes are determined by [OV.sub.Nm] = I - [V.sub.Nm] and the value of the other time nodes are determined by
[OV.sub.nm] = [OV.sub.(n+1)m] x p + [OV.sub.(n+1)(m+1)] x (1 - p)] x [e.sup.-r[DELTA]t], n = 0, 1, 2, ..., N - 1, m = 0, 1, 2, ..., n
where P [V.sub.nm]: Estimated early exercise profit at node nm.
For the call-style, P[V.sub.nm] = [V.sub.nm] - I, n = 0, 1, 2, ..., N - 1,
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