Evaluation of real options with information costs.

ABSTRACT

This paper presents a simple framework for the analysis, valuation and simulation of several real options in the presence of shadow costs of incomplete information. Information costs can be viewed as sunk costs in the spirit of Merton's (1987) model of capital market equilibrium with incomplete information. We incorporate these sunk costs in standard discounted cash flow techniques and present the basic concepts of real options. The justification of information costs in real projects is based on the observation that R&D needs to be done before investment decisions. These costs account for all the expenses needed to be informed about an investment opportunity and the management of projects. This analysis extends the models in Bellalah (1999, 2001) for the valuation of real options within information uncertainty. We present valuation procedures and simulations for the values of common real options in the presence of shadow costs of incomplete information.

JEL Classification: G12; G13; G14; G31

Keywords: Asset pricing; Option pricing; Information and market efficiency; Capital budgeting; Investment policy

I. INTRODUCTION

A company's value creation is determined by resource allocation and the proper evaluation of investment alternatives. Managers make capital investments to create future growth for shareholders. Investments lead to patents or technologies, which open up new growth possibilities. In general, managers use the basic investment techniques as the capital asset pricing model (CAPM), the cost of capital and the discount cash flow techniques, DCF. In investment valuation, organisations also use quantitative approaches such as net present value (NPV), decision tree analysis (DTA), payback time, or scenario/simulation which do not account for intangible factors such as future competitive advantage, future opportunities, managerial flexibility, the strategic value of the investment, etc. This is because the expected outcomes are not easy to forecast and the variability of investment returns may be extremely high. New techniques for capital budgeting incorporate real options, active management, and strategic interactions between investment and financing decisions. (1)

Information plays a central role in the capital budgeting process and in investment and financing decisions. Edwards and Wagner (1999) study the role of information in capturing the research advantage and how to incorporate information into the decision process of active investment management. They show that implementation costs make sense only when weighed against the benefit of enhanced performance. They recognise that the most valuable commodity in the market is information that reduces uncertainty. In this spirit, trading cost information is part of the research that gives a manager active advantage. Edwards and Wagner (1999) show that managers must measure and develop confidence in the value of their research and then incorporate feedback from the market.

Merton (1987) adopts most of the assumptions of the original CAPM and relaxes the assumption of equal information across investors. He assumes that investors only hold securities of which they are aware. In his model, the expected returns increase with systematic risk, firm-specific risk, and relative market value. The expected returns decrease, however, with relative size of the firm's investor base, referred to in Merton's model as the "degree of investor recognition". The intuition behind Merton's model is that investors consider only a part of the opportunity set, that full diversification is not possible, and that firm specific risk is priced in equilibrium. The main distinction between Merton's model and the standard CAPM is that investors invest only in the securities about which they are "aware". This assumption is referred to as incomplete information. The more general implication is that securities markets are segmented. The intuition behind this result is that the absence of a firm-specific risk component in the CAPM comes about because such risk can be eliminated (through diversification) and is not priced. It appears from Merton's model that the effect of incomplete information on expected returns is greater than the highest specific risk of the firm and the highest weight of the asset in the investor's portfolio. The effect of Merton's non-market risk factors on expected returns depend on whether the asset is widely held or not. (2)

Kadlec and McConnell (1994) document the effect of share value on the NYSE and report the results of a joint test of Merton's (1987) investor recognition factor and Amihud and Mendelson's (1986) liquidity factor as explanations of the listing effect. The cross-sectional regressions provide support for investor recognition as a source of value from exchange listing. The regressions support Merton's model. The results also provide support for superior liquidity as a source of value from exchange listing. They provide support to Amihud and Mendelson (1986) model.

Foerster and Karolyi (1999) construct an empirical proxy for the shadow cost of incomplete information for each firm, using the methodology in Kadlec and McConnell (1994). The investor recognition hypothesis of Merton suggests that abnormal returns may be due to the changes in the shareholder base, adjusted by the stock's residual variance and relative size. The results obtained by Foerster and Karolyi (1999) are supportive of the Merton (1987) hypothesis and consistent with Kadlec and McConnell (1994).

Coval and Moskowitz (1999) document the economic significance of geography and attempt to uncover the effect of distance on portfolio choice. They find that local equity preference is strongly related to firm size, leverage and output tradability. Their results suggest an information-based explanation for local equity. This is consistent with the findings in Kang and Stulz (1997) who find that foreign investors underweigh small, highly levered firms, and firms that do not have significant exports. These results may be a response to severe information asymmetries associated with these firms.

Brennan and Cao (1997) develop a model of international equity portfolio investment flows which is based on the differences in informational endowments between foreign and domestic investors. The authors show that when domestic investors possess a cumulative information advantage over foreign investors about their domestic market, investors tend to purchase (sell) foreign assets in periods when the return on foreign assets is high (low).

Stulz (1999) examines the effect of globalisation on the cost of equity capital and argues that this cost decreases because of globalisation. The empirical evidence gives support to the theoretical prediction that globalisation decreases the cost of capital. He gives strong theoretical arguments justifying why the cost of capital should fall when markets become more open to foreign investors. Following Merton (1987), Stulz (1999) assumes that some investors do not hold some securities because they do not know about them. He provides a model in which this assumption amounts to attributing the home bias to ignorance or a non-modelled behavioural bias. This leads Stulz (1999) to show that the impact of globalisation on the cost of capital depends heavily on the extent of the home bias. However, the empirical evidence in Stulz (1999) shows that the effect of globalisation on the cost of capital is rather small because of the home bias.

Merton's (1987) model shows that asset returns are an increasing function of their beta risk, residual risk, and a decreasing function of the available information for these assets. Amihud and Mendelson (1988) consider several observed corporate policies that can be viewed as increasing the liquidity of investments. Their suggested policies include going public, instituting limited liabilities on equity claims, listing on organised exchanges, distributing ownership among many shareholders, etc. Since the transmission of this information is costly as in Merton's model, Amihud and Mendelson (1988) show how managers can balance the costs against the added value from the higher liquidity of the claims of the firm.

The above literature reveals the importance of information costs in the pricing of financial and real assets. Using this framework, Bellalah and Jacquillat (1995) and Bellalah (1999) develop simple models for the pricing of financial options in the presence of information costs. A similar analysis can be extended to real options using the same methods as in Bellalah (2001). Our work extends the standard capital budgeting techniques by accounting for the dynamic dimension of existing theories. The main objective is to analyse numerically the real option approach in capital budgeting investment decisions and compare this approach to the traditional NPV. This limits the study to only one stochastic underlying variable: the cash inflows. (3)

This paper is organised as follows. Section 1 reminds the use of traditional capital budgeting models. It incorporates also information costs in standard discounted cash flow techniques. Section 2 analyses the basic concepts and specific features of real options. Section 3 deals with valuation and simulation of real options. It suggests an extension and an adaptation of Black-Scholes (1973) model, Merton (1998) model and the binomial approach for the valuation of real options by accounting for the effects of incomplete information. Two cases are analysed: the case when the underlying asset is observable and the case when it is neither observable nor continuously traded. Simulation results are proposed to show the impact of information costs on real options values. (4)

II. TRADITIONAL MODELS AND REAL OPTIONS

A. Traditional Capital Budgeting Models and Information Costs Investment decisions are often made with reference to standard discounted cash flow techniques, (DCF analysis). The most common capital budgeting models used by corporations involve either the basic net present value (NPV), Scenario/Simulation, or Decision Tree Analysis (DTA).

The NPV is the sum of the expected future cash flows minus the initial costs of investments. This method seems to give better results than the accounting rate of return (ARR), the profitability index (PI), the internal rate of return (IRR), the modified internal rate of return (MIRR), and the payback method. However, this method ignores flexibility, assumes that the investment either falls into a reversible or an irreversible category, and that managers are given unbiased expected cash flows. For ease of exposition, the following notations are used:[E.sub.P] ([CF.sub.t]): expected cash flow; R: risk adjusted discount rate; r: risk-free discount rate; [bar.C][[bar.F].sub.t]: certainty equivalent cash flow; [I.sub.0]: investment outlay at time 0; T: time to maturity; and [[lambda].sub.s], ([[lambda].sub.c]): information cost regarding the firm's cash flows (and the real option).

In the presence of information costs, the NPV can be written as:

NPV = [T.summation over (t=1)] [E.sub.P]([CF.sub.t])/(1 + R + [[lambda.sub.s]]) - [I.sub.0] [T.summation over (t=1)] [bar.C][[bar.F].sub.t])/[(1 + R + [[lambda.sub.s]]).sup.t] - [I.sub.0]

It is important to note that the information cost appears as an additional discount rate in the discounting of risky streams. This is the main intuition in Merton's (1987) model. In fact, this cost reflects the additional return required by investors to get compensated for their investments in information. An investor does not invest in a real project if he does not know about that project. The process of information acquisition has a cost that must be accounted for in the computation of the present value of cash flows. If the manager pays 2 million in the process of information acquisition and the investment is equal to 100 million, then he must require at least 2/100 or 2% as an additional return above the riskless interest rate. Hence, instead of a discount rate r, a new discount rate equal to (r + 2 %) must be used as a rough approximation in this case.

Several managers rely on sensitivity analysis using high, low, or medium scenarios to bind the uncertainty. This method tends to show the impact on NPV and its sensitivity to each variable. Then the resulting NPV values are recorded. It assumes that other variables are constant in scenario base of their expected values. This technique recognises the existence of uncertainty but does not capture the flexibility due to "uncertainty" and offers little managerial guidance in investment decision process. In this analysis, information costs can be easily introduced in the simulation of the present values of risky streams in the same way as we have done for the calculation of the NPV.

The Monte Carlo simulation is not biased when modelling cash flows and deciding on the values for the relevant variables and correlation. For each variable, a probability distribution is designated and the cash flows are simulated discretely. Then, they are used to calculate the NPV. However, the serial dependency is complex to quantify. The NPV distribution given by the simulation is also hard to interpret economically (Trigeorgis, 1990 and 1993). This method is useful in the calculation of projects under uncertainty, even though, it has its proper limits. Information costs can also be easily integrated in this analysis in the discounting of the risky streams.

The Decision Tree Analysis approach takes into account later decisions and incorporates some of the manager's flexibility into the valuation process. Investments are divided into a series of sub-investments that will be undertaken at different stages. The implementation of these investments in the future will depend upon some future event, thus enabling managers to decide whether to invest further or not. This process can not be implemented without additional information. This leads necessarily to information costs in the spirit of Merton (1987).

B. Analysis of Real Options

During the last decade real options have been given increasing interest by corporate practitioners in industries where the projects are costly and uncertain. Companies allocate resources for existing businesses or new ventures, and managers decide whether to invest now, to do nothing or to wait. When valuing investment decisions, the options to abandon or to defer, the options to expand or to switch are embedded into the project. These implicit options occur naturally or may be planned at some flexibility. (5)

Investment decision-making seems to be justified as a way to account for flexibility and can be thought of in terms of real options (Dixit and Pindyck, 1994). Option pricing theory evaluates the firm as its operating options were managed optimally, without future information on optimal choices to be made. A distinction must be made between real assets, (which have a market value) and real options, (which consider the opportunities to purchase future real assets on favourable terms). (6)

Investment is defined in financial economics as the act of incurring an immediate cost in the expectation of future rewards (Dixit and Pindyck, 1995). The initial outlay is a payment for a right with no obligation to undertake a project. Real options give the right to receive a future cash flow from the investment cost. This is equivalent to a standard call option on a real asset. Using the option theory, the company can be viewed as a future possibility where an investor pays a premium for the right to buy a specific stock at a known exercise price at a certain time in the future. The investment amount is then the strike price, allowing the investor to capture the value of the underlying project (Trigeorgis, 1990 and 1993). A real option strategy forces managers to compare every opportunity arising from existing investments with the full range of opportunities open to them. It promotes strategic leverage and encourages managers to exploit situations where investment can keep their company in the game. The strategy reduces the upside as well as the downside risk, and empowers managers to defer the investment opportunity without increasing the exercise price.

Real options can be used by managers with a basic understanding of option pricing models and tools. As they are important in strategic and financial analysis, they can be a complement to the standard NPV valuation. The NPV ignores the value of flexibility and creates a static picture of existing investments and opportunities. The traditional techniques treat opportunities as a "now or never" investment even if many investments can be deferred in the future without loosing their value.

There is a large scope for applications of option pricing techniques for valuation of an entire firm. (7) A real option confers flexibilities to its holder as the option to invest, to wait, to divest, etc. (8) These options can be economically important. The decision about when to invest is analogous to the decision about when to exercise an American call. The sensitivity of the value of the firm to these possibilities makes a real option valuation method better than the standard NPV. This is because an ordinary NPV valuation predicts future cash flows according to today's information. By using the real option's approach, the value of a company corresponds to the value of a portfolio of operating options yielding a stream of future cash flows. This portfolio can be seen as a portfolio of financial options on those future cash flows.

There are totally irreversible investments (where the whole investment cost is lost at the end of the operating phase), and partially irreversible investments (whose value can be partially recovered). Irreversibility can also arise from government regulations which make investments irreversible. An irreversible investment opportunity is like a standard call option even if the asset can be sold to another investor. Two types of uncertainty are present in capital investments: economic uncertainty and technical uncertainty each with a positive increase effect on the value of a real option. The economic uncertainty is correlated with the actual exogenous movement of the economy: interest rate, inflation, industry prices, etc. This uncertainty could be reduced by waiting for new information before making the final investment. The technical uncertainty is the uncertainty in the project itself. It is endogenous to the decision process and is affected by management. For example, the uncertainty in the outcome of a R&D project can only be reduced with an actual step by step investment, until the future technical uncertainty is resolved (Dixit and Pindyck, 1994).

The analogy between financial and real options also has its limitations. There are three factors that make a real option different from a financial option: the proprietary state, the complex characteristics, and non tradability of real options (Kester, 1993). Firstly, all financial options are proprietary, and the holder decides when the option should be exercised. Real options would present a proprietary characteristic when the company has a unique and exclusive know how in a technological process or has access to a patent. In general, investment opportunities with barriers to entry serve as proprietary real options. This is not the case when investment opportunities are shared by competitors and other participants. Secondly, most financial options are derived from the underlying asset. Some real options have more complex characteristics. They give the holder the right not only to receive the gross present value of the future cash flows from the investment, but also investment opportunities in the future. In this case, the option becomes compounded and written on many another options. Thirdly, when compared to the financial options markets, the real options markets are imperfect and only some proprietary real options can be traded with high transaction costs and few participants (Trigeorgis, 1990 and 1993). Shared real options cannot be tradable on the market since they are already a public good for the whole industry.

III. VALUATION AND SIMULATION OF REAL OPTIONS IN THE PRESENCE OF INCOMPLETE INFORMATION

A. Valuation Procedure in the Presence of Information Costs in a Continuous-time Setting

The valuation of financial options is based on the fact that an option can be replicated by a portfolio of traded securities. Since this equivalence is not dependent on risk attitudes, the value of the expected future payoffs can be derived from a risk-neutral approach and discounted at the risk-free interest rate. This concept can also be applied to real options, even if they are not traded in financial markets. The fundamental assumption is that a non traded project has the value that it would have had if it were traded in the financial markets (Smith and Nau, 1994).

Trigeorgis (1990, 1993) shows that in the DCF analysis the discount rate is received by identifying a twin security for each project. The twin security has the same risk characteristics as the specific project and is traded in financial markets. In this context, the option analogy could use the same twin security to replicate a non-arbitrage portfolio. Given the price of the project's twin security, management can, in principle, replicate the returns to a real option by purchasing a certain number of shares while financing the purchase partly by borrowing at the risk-free rate. This makes the application of risk neutral valuation techniques for traded and non traded assets possible. The derivation of the standard formulas for option pricing in the presence of information costs appears in Bellalah (1999, 2001).

1. A General Derivation of the Values of Real Options

The use of option valuation techniques in the valuation of real assets is based on some important assumptions. (9) In general, individual values of real options are non-additive and the combined value could be complex to compute. Kulatilaka (1993) shows that the combined value of interacting options could either be higher or lower than the sum of the individual values. The combined value is dependent on the type of options, the degree of separation, the degree of being "in the money", and the order of the options involved. Trigeorgis (1990, 1993) describes the interaction between options as basically additive. This is the case when the interacting options are of different types, i.e. calls and puts. He gives an example on the interaction between the option to abandon (which is equivalent to a put) and the growth option (which is equivalent to a call). He shows that these two options are additive because they are of different types.

a. Real option inputs

Because the value of a real option is determined by seven parameters, exploiting proactive flexibility is a question of pulling one or more parameters. To extend the Black & Scholes (1973) model and the binomial model to a context taking into account the presence of shadow costs of incomplete information, seven input parameters are required: expected cash inflows and cash outflows, the annual cost (or value lost over the duration of the option), the risk-free interest rate, the level of uncertainty, changes in the duration, and information costs.

Gross present value of the project, V, is the value of the expected cash flows to be received from the investment. It is considered significant without the investments. A higher present value of expected operating cash inflows can be achieved by increasing revenues, raising the price earned, producing more, or by generating compound business opportunities. The economic uncertainty is assumed to influence the gross present value and thus make it follow a geometric Brownian motion with a random part determined by the standard Wiener process dz(t).

The capital investments to be made, I, is the present value of the fixed costs over the lifetime of the investment. It is equivalent to the exercise price of a financial option. Here, we suppose certain capital investments. The reduction of the expected operating cash outflows can be achieved by leveraging economies of scale or by leveraging economies of scope in partnership.

The dividends, (10) [delta], are sums paid regularly to stockholders. This could be the costs incurred to preserve the option by keeping the opportunity alive, or the cash flows lost to competitors that go ahead and invest in another opportunity. The cost of waiting could be high if an early entrant were to seize the initiative. The dividends are correspondingly high, thus reducing the option value of waiting and the value lost to competitors can be reduced by discouraging them from exercising their options. This is the case for example in locking up key customers or lobbying for regulatory.

The risk-free interest rate, r, corresponds to the interest rate for a risk-free bond with the same expiration date as the project. Expected increase in the interest rate raises the option value, despite its negative effect on NPV (reduces the PV of the exercise price). (11)

The volatility, [sigma], is the standard deviation of the growth rate of the value of future cash inflows. This is the crucial difference from NPV analysis. When uncertainty of expected cash flows rises it increases the value of flexibility. For a project it could be a little more complex to find the correct volatility when compared to financial options.

Time to maturity, T, corresponds to the time left until the opportunity disappears. It depends on technology (products life cycle), competitive advantages (intensity of competition), and contracts (patents, leases, licences). The time to maturity, is subjectively defined by management as the time it takes for competitors to exploit the same opportunity. (12) An increase in the opportunity's time raises the option's value because it increases the total uncertainty. The company might be able to extend its option by extending exclusive raw material supply contracts, locking up distribution channels, etc.

The information costs, [lambda], are the costs engaged by investors to get informed about the projects and their real options. We make a distinction between information costs related to the underlying project cash flows, [[lambda].sub.V], and information costs related to each implicit real option, [[lambda].sub.C].

b. Valuation of real options when the underlying asset is observable under incomplete information

Consider the following dynamics of the project's value:

DV/V = [mu]dt + [sigma]dz (1)

where [mu] and [sigma] refer to the instantaneous rate of return and the standard deviation of the project, and dz is a geometric Brownian motion. Let X be the price of a dynamic portfolio of assets perfectly correlated with V:

dX/X = [alpha] dt + [sigma] dz (2)

where [alpha] stands for the expected return from owning a completed project. Let [delta] = [alpha] - [mu]. In this context, [delta] represents an opportunity cost of delaying investment. If [delta] is zero, then there is no opportunity cost to keeping the option alive. Hence, the value of [delta] must be positive. Let G(V) be the value of the firm's option to invest. Using Merton's (1987) model, Bellalah and Jacquillat (1995) and Bellalah (1999, 2001) obtain option prices in the context of incomplete information.

Consider a portfolio: long an option which is worth G(V) and go short [G.sub.V] units of the project. The value of this portfolio is:

P = G - [G.sub.V]V (3)

Since the short position includes [G.sub.V] units of the project, it requires the paying out of an amount [delta] [G.sub.V]V. The total return for this portfolio over a short interval of time, dt, is:

dP = dG - [G.sub.V] dV - [delta] [G.sub.V] V dt (4)

Since there are information costs embedded in the option and its underlying assets, the return must be equal to (r + [[lambda].sub.V]) for the project and (r + [[lambda].sub.C]) for the option, where [[lambda].sub.V] and [[lambda].sub.C] refer respectively to the information costs on the project and the option. In this context:

dP = (r + [[lambda].sub.C]) G dt - (r + [[lambda].sub.V]) [G.sub.V] V dt (5)

Assuming that a hedged position is constructed and since the application of Ito's lemma, the value of dG is:

dG =1/2 [[sigma].sup.2] [V.sup.2] [G.sub.vv]dt + [mu]V [G.sub.V] dt + [sigma]V [G.sub.v]dz + [G.sub.t]dt (6)

Substituting dV and dG, given respectively by relations (1) and (6), in equation (4), we get after simplification:

dP = (1/2 [[sigma].sup.2] [V.sup.2] [G.sub.vv] - [delta] V [G.sub.v] + [G.sub.t])dt (7)

When the time to maturity of the option is finite, this equation becomes:

1/2 [G.sub.vv][[sigma].sup.2][V.sup.2] + (r + [[lambda].sub.V] - [delta]) V [G.sub.V] - (r + [[lambda].sub.C]) G + [G.sub.t] = 0(8)

For the valuation of standard calls, under the following condition:

G = max (V - I, 0) (9)

The call value is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This formula constitutes an adaptation of the Black and Scholes (1973) model to valuing the real options in the context of incomplete information. (13)

c. Valuation of real options when the underlying asset is neither observable nor continuously traded under incomplete information

Using the same analysis as in Merton (1998) and following the same approach as above, the equivalent of equation (28) in Merton (1998) is:

1/2 [v.sup.2][V.sup.2] [G.sub.vv] + (r + [[lambda].sub.V] - [delta])V [G.sub.v] - (r + [[lambda].sub.C])G + [G.sub.t] = 0 (11)

where [v.sup.2] is the variance of the V-Fund portfolio in Merton (1998). This equation can be solved under the following condition:

G (V,T) = E[h(VY)]

where Y is a log-normally distributed random variable with E(Y) = 1 and variance of ln(Y) is equal t[??][[??].sup.2]T and E(.) is the expectation operator over the distribution of Y.

The solution to this equation when h(V) = max(V - I, 0) is given by:

G = V exp(([[lambda].sub.V] - [[lambda].sub.C]) T) N([d.sub.11]) - I exp(-(r + [[lambda].sub.C]) T) N([d.sub.11] - [square root ([gamma])]) (12)

with:

[d.sub.11] = [ln (V/1) + (r + [[lambda].sub.V])T + [gamma]/2]/[square root of ([gamma])], [gamma] = [v.sup.2]T + [[theta].sup.2] T.

When compared to formula (10), this formula allows understanding the effect of the underlying asset price to not be observable. The main difference in the option pricing formula with and without continuous observation of the underlying asset is that the variance of the underlying does not go to zero around the maturity date because of the "jump" event at expiration. This formula can be applied when the underlying asset is neither continuously traded nor continuously observable. This is a simple generalization of formula (27) in Merton (1998) to account for the effects of incomplete information.

2. The Value of the Option to Invest

The value of the option to invest under incomplete information can be computed using the following equation:

1/2 [[sigma].sup.2] [V.sup.2] [G.sub.vv] + (r + [[lambda].sub.V] - [delta]) V [G.sub.V] - (r + [[lambda].sub.C]) G = 0 (13)

This equation for the value of G(V) must satisfy the following conditions:

G(0) = 0, G([V.sup.*]) = [V.sup.*] - I, [G.sub.V] ([V.sup.*]) = 1

The value [V.sup.*] is the price at which it is optimal to invest. At that time, the firm receives the difference [V.sup.*] - I. Following Bellalah (2001), the solution to the differential equation is:

G(V) = a[V.sup.[beta]] (14)

where:

[beta] = 1/2 - t + [[lambda].sub.V] - [delta]/[[sigma]sup.2] + [[[(r + [[lambda].sub.V] - [delta 1]/[[sigma].sup.2] - 1/2).sup.] + 2 (r + [[lambda].sub.C])/[[sigma].sup.2]].sup.0.5]

and

a = [V.sup.*] - I/[V.sup.*][beta], [V.sup.*] = [beta]I/[beta] - 1

Table 1 simulates the value of the investment opportunity G(V), given by equation (14), as a function of the project value, V, in the presence of information costs, [lambda]. r is the interest rate, [delta] is the opportunity cost of delaying project or a constant payout rate, I denotes the cost of investment or investment expenditure, [sigma] stands for the volatility, [[lambda].sub.C] (respectively [[lambda].sub.V]) represents the information cost related to G(V) (respectively V). It is assumed that r = 5.5 %, [delta] = 6 %, I = 500, and [sigma] = 45 %. All things being equal, a larger project value can be associated with a greater value of the option to invest. In the presence of the shadow costs of incomplete information regarding project value, the option to invest value increases. In the case where information costs concern the option value, option to expand value drops instead of increasing. It is of interest to note that the negative effect due to incomplete option value information and the positive effect due to incomplete project value information are compensated. But, on the whole, the presence of two types of information costs increases the option to expand value compared to its level in the complete information case.

3. The Value of the Option to Expand

The management can expand the project if economic or technical conditions are favourable. An option to expand is a call option to acquire an additional part to the initial project, where the cost to expand is the exercise price. This managerial flexibility has a value and the cost of expanding could be reduced if flexibility is built into the project at an early stage. The value of this option in the presence of shadow costs of incomplete information can be computed using the following formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

with:

[d.sub.1] = [ln(V/I) + (r + [[lambda].sub.V] - [delta] + 1/2 [[sigma].sup.2])T]/[sigma][square root of (T)], [d.sub.2] = [d.sub.1] - [sigma][square root of (T)]

Table 2 simulates the value of the time to expand option with and without information costs for the option and its underlying asset. It is assumed that I = 500, [delta] = 6%, r = 5.5%, [sigma] = 45%, and T = 12. This Table shows that the high project values generate an increase in the value of the option to expand. In the presence of the shadow costs of incomplete information regarding project value, the option value increases. In the case where information costs concern the option value, option to expand value decreases. Finally, when we take into account the information costs on both the underlying project and the option, the option to expand value increases.

4. The Value of the Option to Contract

The option to contract has a positive value if market conditions turn weaker than originally expected. In this case, management can then reduce the scale of operations and thus saving part of the planned investment outlays. This analogous to a put option on part of the initial project, with exercise price equal to the potential cost savings. Following Trigeorgis, this option may be particularly valuable in the case of new-product introductions in uncertain markets. The value of the option to contract can be simulated using the following formula.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

with:

[d.sub.1] [ln(V/I) + (r + [[lambda].sub.V] - [delta] + 1/2 [[sigma].sup.2])T]/[sigma][square root of (T)], [d.sub.2] = [d.sub.1] - [sigma] [square root of (T)]

Table 3 indicates the value of the option to contract in the presence of information costs for the option and its underlying asset. We investigate the opportunity to contract the scale of the project by 5%, (14) saving an amount of 100. It is assumed that [delta] = 6%, r = 5.5%, [sigma] = 45%, and T = 12. The exercise price is the same for all of table 3 and is equal to 100. In this context, the firm has the right, but not the obligation, to undertake the put option at an exercise price corresponding to 100. Table 3 also describes the option to contract values in the case of information uncertainty. As expected, the high project values (i.e. favourable market conditions) generate a decrease in the value of the option to contract: the option to contract has a positive value if market conditions are unfavourable.

5. The Value of the Option to Abandon

Management can abandon current project and resale value of capital equipment. If prices suffer a sustainable decline or the operation does poorly for some other reason, management may have a valuable option to abandon the project in exchange for its salvage value. The option to abandon a project provides partial insurance against failure. The option to abandon can be valued as an American put option on the project's current value, with an exercise price corresponding to the salvage or best alternative use value. Table 4 simulates the values of the option to abandon for different levels of information costs regarding the option and its underlying asset. I is the value received on abandonment and T is the number of years until abandon (years). We use the equation (16) and assume that I = 150, [delta] = 5%, r = 5%, [sigma] = 40%, and T = 10. In the same spirit as for the option to contract, this Table shows that the value of the option to abandon the project increases when market conditions decline severely (that is, when the value of the project is weak).

6. The Value of the Option to Switch and the Growth Option

In general, investment is a link of interrelated projects opening future growth opportunities. The growth option provides the company with a possibility to make a follow-on investment in the future. It is analogous to a call option. The option to grow is used when an initial investment is required for further development. The project can be considered as a link in a chain of related projects and may serve as a springboard for future project generations. But unless the firm makes that initial investment, subsequent generations will not be feasible. Kester (1984) recognised the importance of the real growth option on firms and argued that the growth option constituted can account for more than half of the market value for most of the companies. The value of the growth option can be computed using formula (15). Table 5 simulates the values of the growth option as a function of project value and information costs. It is assumed that I = 30, [delta] = 5%, r = 7%, and [sigma] = 35%.

Table 5 simulates the values of the growth option for different levels of information costs. The high project values generate an increase in the value of the growth option. In the presence of the shadow costs of incomplete information regarding project value, the option value increases. In the case where information costs concern the option value, the growth option value drops instead of increasing. And the presence of two types of information costs increases the growth option value compared to its level in the complete information case.

The managerial flexibility to be able to shut-down and restart operations can be valuable if prices are such that cash revenues are not sufficient to cover variable operating costs. It might be better not to operate temporarily. If prices rise sufficiently, operations can be restarted. Thus, operations in each year may be seen as a call option to acquire that year's cash revenues by paying the variable costs of operating as a strike price. It is equivalent to the firm having a portfolio of call and put options. For example, being able to temporarily shut down a project is equivalent to a put option and restarting operations when the project has been down and become up is equivalent to a call option.

B. The Valuation Procedure in a Discrete-time Setting in the Presence of Information Costs: The Value of the Time-to-build Option

Few investments in practice are a single up-front outlay. However, most investments are sequential and staged into several investments. This creates valuable options to default at any given stage. The completion of one stage gives the right, but not the obligation, to undertake the next stage and the options that this stage provides. The staged investment can be viewed as a series of compound options. In this case, the valuation process can be computed discretely. The project is a perpetual cash flow with a fixed capital outlay. There are points when the project has a positive NPV, but we are better off not taking it because the option to undertake the project in the future is more valuable. Since the investment is irreversible, when we take the project, we destroy the value of waiting. It is possible in this context to extend the standard binomial model to account for the effects of information costs. When generating the binomial tree for the underlying asset, we must account for the information cost of the investment opportunity. When we work backwards, we must account for the information cost regarding the option.

In most investments opportunities, management holds an option to defer the life time of investment and see if the cash outflow meets the product price. Some projects could increase in value when new information is available and uncertainty decreased with more favourable conditions. (15) The value of waiting to invest or the option to defer can be seen as an American call option on the gross present value of the future expected cash flows (Trigeorgis (1990, 1993). Using a risk-neutral approach, we adapt the binomial model to account for the effects of incomplete information. The valuation procedure can be described in the following steps. Assuming that the state variable of the project value is the price P of the output, and the project generates a unit per year, the gross present value is:

V = P/r = (17)

The up multiplier, u, and down multiplier, d, are calculated by using formulas in Cox, Ross and Rubinstein (1979):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](18)

where T is the number of years to expiration and N is the number of binomial periods. These multipliers are used to calculate the future gross value V in the nodes of the binomial tree. The risk neutral probability for the up and down branches is calculated as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

The discount factor at each node is:

([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

The binomial tree should be constructed in such way that it can incorporate the investments needed. We count backward from the end and calculate, in every node, the value by using the binomial formula for one period and subtracting the value of the investment. To consider this in the binomial tree, the value at each node should be the maximum of the value of the project in this node and zero. The calculation of the value in each node should continue in this backward calculation until the value of the firm finally reaches the present time.

Figure 1 simulates the values of the time to build option in the complete information case. It is assumed that the state variable of the project value P = 300, the present value of the cash-flows from the project V = 3000, initial investment I = 800, number of years to maturity T = 6, volatility [sigma] = 40%, risk-free interest rate r = 10%, cash-flow rate generated by the project [delta] =10%, number of binomial periods N = 6, up multiplier u = 1.49182, down multiplier d = 0.67032, up probability p = 0.40131, down probability 1 - p = 0.59869, discount factor 0.90484, and information costs [[lambda].sub.V] = [[lambda].sub.C] = 0%. Columns of the binomial tree have four elements in each node: state variable, project value, NPV (if project is undertaken), and option value.

Figure 2 simulates the values of the time to build option in the presence of information costs for the option and its underlying asset: [[lambda].sub.V] = [[lambda].sub.C] = 1%. It is assumed that the state variable of the project value P = 300, present value of the cash-flows from the project V = 3000, initial investment I = 800, number of years to maturity T = 6, volatility [sigma] = 40%, risk-free interest rate r = 10%, cash-flow rate generated by the project [delta] = 10%, number of binomial periods N = 6, up multiplier u = 1.49182, down multiplier d = 0.67032, up probability p = 0.41355, down probability 1 - p = 0.58645, and discount factor 0.89583.

Coherently with our findings in the continuous time setting, time to build option value increases as information costs rise. When information uncertainty concerns the underlying project, [[lambda].sub.V] = 1%, option value increases (1339.34). In the case where information costs are related to the option value, [[lambda].sub.C] = 1%, the growth option value drops instead of increasing (1168.23). The presence of two types of information costs increases the time to build option value compared to its level in the complete information case (1261.34 for [[lambda].sub.V] = [[lambda].sub.C] = 1%, compared to 1240.47 when [[lambda].sub.V] = [[lambda].sub.V] = 0%). Our results show that when information uncertainty of expected cash flows rises it increases the value of flexibility. This gives support to the well-known relation between the general uncertainty and the value of flexibility.

III. CONCLUSION

This paper reviews the main concepts in real options and extends the literature for the valuation of real options in the presence of information costs. These options are fundamental in the valuation process of investments and capital budgeting. However, they are valued in a standard framework ignoring the role of information costs in investment decisions. Information costs play a central role in the capital budgeting process, since managers do not invest in projects they do not know about. When money is engaged in research and development, in project analysis and valuation, it is natural to require a return that accounts for these expenses. Therefore, information costs or shadow costs of incomplete information represent an appropriate component of the discount rate in investment decisions.

We introduce information costs in the spirit of Merton (1987) and Bellalah (1999, 2001) in the capital budgeting process and real options valuation. We suggest a general derivation for the valuation of real options when the underlying asset is observable and when it is not observable. This provides a generalisation of the Black-Scholes (1973) formula, the Merton (1998) formula and the binomial approach which accounts for the effects of incomplete information. We examine the valuation of the option to invest, the option to expand, the option to contract, the option to abandon, the switch option and the growth option, the option to shut down and restart, the option to defer, and the time to build option in the presence of information costs. Simulation results are provided using reasonable values for information costs. Our analysis can be extended to other types of real options. In particular, it can be applied to compound real options and "exotic" real options. The analysis can also be tested using real data. It is also possible to extend our study to account for stochastic volatility of cash flows.

BIBLIOGRAPHY 

Agmon, T., 1991, "Capital Budgeting and the Utilization of Full Information: Performance Evaluation and the Exercise of Real Options", Managerial Finance, vol. 17, N. 2/3, 42-50.

Amihud, Y., and H. Mendelson, 1986, "Asset Pricing and the Bid-Ask Spread", Journal of Financial Economics, 17, 223-249.

Amihud, Y., and H. Mendelson, 1988, "Liquidity and Asset Pricing: Financial Management Implications", Financial Management, Spring, 5-15.

Bellalah, M., and B. Jacquillat, 1995, "Option Valuation with Information Costs: Theory and Tests", Financial Review, August, 617-635.

Bellalah, M., 1999, "The Valuation of Futures and Commodity Options with Information Costs", Journal of Futures Markets, September.

Bellalah, M., 2001, "Irreversibility, Sunk Costs and Investment Under Incomplete Information", R&D Management.

Berger, P.G., E. Ofek, and I. Swary, 1996, "Investor Valuation of the Abandonment Option" Journal of Financial Economics, 42, 257-287.

Black, F., and M. Scholes, 1973, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, N. 81, 673-659.

Brennan, M., and H. Cao, 1997, "International Portfolio Investment Flows", Journal of Finance, Vol. LII, N. 5, December, 1851-1879.

Brennan, M.J., and E. Schwartz, 1985, "Evaluating Natural Resource Investments", Journal of Business.

Coval, J., and T.F. Moskowitz, 1999, "Home Bias at Home: Local Equity Preference in Domestic Portfolios", Working Paper, University of Michigan.

Cox, J.C., and S.A. Ross, and M. Rubinstein, 1979, "Option Pricing: A Simplified Approach", Journal of Financial Economics, 7, N. 3, 229-263.

Cox, J.C., and S.A. Ross, 1976, "A Survey of Some New Results in Financial Option Pricing Theory", The Journal of Finance, XXXI, N. 2, May.

Dentskevich, P., and G. Salkin, 1991, "Valuation of Real Projects Using Option Pricing Techniques", OMEGA International Journal of Management Science, Vol. 19, N. 4, 207-222.

Dixit, A.K., 1992, "Investment and Hysteresis", Journal of Economic Perspectives, vol. 6, N. 1, Winter, 107-132.

Dixit, A.K., 1995, "Irreversible investment with uncertainty and scale economies", Journal of Economic Dynamics and Control, N. 19, 327-350.

Dixit, A.K., and R.S. Pindyck, 1994, Investment under Uncertainty, Princeton University Press.

Dixit, A.K., and R.S. Pindyck, 1995, "The Options Approach to Capital Investment", Harvard Business Review, May-June, 105-115.

Dixit, A., R.S. Pindyck, and S. Sodal, 1999, "A Markup Interpretation of Optimal Investment Rules", Economic Journal, Vol. 109, N. 455, April, 179-189.

Edwards, M., and H. Wagner, 1999, "Capturing the Research Advantage", Journal of Portfolio Management, Spring, 18-24.

Faulkner, T.W., 1996, "Applying 'Options Thinking' to R&D Valuation", Research and Technology Management, May-June.

Foerster, S., and G. Karolyi, 1999, "The Effect of Market Segmentation and Investor Recognition on Asset Prices: Evidence from Foreign Stocks Listing in the United States", Journal of Finance, 49,981-1014.

Ingersoll, J.E., and S.A. Ross, 1992, "Waiting to Invest: Investment and Uncertainty", Journal of Business, Vol. 65, N. I, 1-30.

Kadlec, G., and J. McConnell, 1994, "The Effect of Market Segmentation and Illiquidity on Asset Prices: Evidence from Exchange Listings", Journal of Finance, 49, 611-635.

Kang, J., and R. Stulz, 1997, "Why Is There A Home Bias? An Analysis of Foreign Portfolio Equity in Japan", Journal of Financial Economics, 46, 3-28.

Kester, W.C., 1984, "Today's Options for Tomorrow's Growth", Harvard Business Review, March-April, 153-160.

Kester, W.C., 1993, "Turning Growth Options into Real Assets", Capital budgeting under uncertainty, 187-207.

Kogut, B., 1991, "Joint Ventures and the Option to Expand and Acquire", Management Science, Vol. 37, N. 1, January.

Kogut, B., and N. Kulatilaka, 1994a, "Operating Flexibility, Global Manufacturing, and the Option Value of a Multinational Network", Management Science, Vol. 40, N. 1, January.

Kogut, B., and N. Kulatilaka, 1994b, "Options Thinking and Platform Investments: Investing in Opportunity", California Management Review, Vol. 36, N. 2, Winter.

Kulatilaka, N., 1993, "The Value of Flexibility: The Case of a Dual-Fuel Industrial Steam Boiler", Financial Management, Vol. 22, N. 3, Autumn, 271-280.

Lee, C.J., 1988, "Capital Budgeting under Uncertainty: The Issue of Optimal Timing", Journal of Business Finance and Accounting, 15(2) Summer.

McDonald, R., and D. Siegel, 1984, "Option Pricing when the Underlying Assets Earns A Below-Equilibrium Rate of Return: A Note", Journal of Finance.

McDonald, R., and D. Siegel, 1986, "The Value of Waiting to Invest", Quarterly Journal of Economics.

Merton, R., 1987, "A Simple Model of Capital Market Equilibrium with Incomplete Information", Journal of Finance, Vol. 42, N. 3, July, 483-510.

Merton, R.C., 1998, "Applications of Option Pricing Theory: Twenty Five Years Later", American Economic Review, Vol. 88, N. 3, 323-345.

Myers, S.C., 1977, "Determinants of Corporate Borrowing", Journal of Financial Economics, Volume 5, N. 2, November, 147-175.

Myers, S.C., 1984, "Finance Theory and Financial Strategy", Interfaces, 14:1, Jan.-Feb., 126-137.

Myers, S.C., and S. Majd, 1990, "Abandonment Value and Project Life", Advances in Futures and Options Research, Vol.4, 1-21.

Newton, D., 1996, "Opting for the Right Value for R&D", The Financial Times, June, 28.

Paddock, J., D. Siegel, and J. Smith, 1988, "Option Valuation of Claims on Real Assets: The Case of Offshore Petroleum Leases", Quarterly Journal of Economics.

Pindyck, R.S., 1991, "Irreversibility, Uncertainty, and Investment", Journal of Economic Literature, Vol. XXIX, Sept., 1100-1148.

Smith, J.E., and R.F. Nau, 1994, "Valuing Risky Projects: Option Pricing Theory and Decision Analysis", Management Science, Vol. 41, N. 5, May.

Stulz, R., 1999, "Globalization of Equity Markets and the Cost of Capital", Working Paper, February, 923-934.

Trigeorgis, L., 1990, "A Real Option Application in Natural-Resource Investments", Advances in Futures and Options Research, Vol. 4, 153-156.

Trigeorgis, L., 1993 "The Nature of Options Interactions and the Valuation of Investments with Multiple Real Options", Journal of Financial and Quantitative Analysis, Vol. 28, N. 1, March, 1-20.

ENDNOTES

(1.) For a survey of these techniques, the reader can refer to Smith and Nau (1994), Lee (1988), Agmon (1991) among others.

(2.) Merton's model may be stated as follows: [R.sub.V] - r = [[beta].sub.v] [[R.sub.m] - r] + [[lambda].sub.V] - [[beta].sub.v] [[lambda].sub.m] where [R.sub.V]: the equilibrium expected return on an asset V, [R.sub.m]: the equilibrium expected return on the market portfolio, r: one plus the riskless rate of interest, [[beta].sub.v] = cov([R.sub.v]/[R.sub.m])/var([R.sub.m]), [[lambda].sub.V]: the equilibrium aggregate "shadow cost" for the asset V. It is of the same dimension as the expected rate of return on this asset V, [[lambda].sub.m]: the weighted average shadow cost of incomplete information over all assets.

(3.) For a survey of the literature on real options, the reader can refer to Trigeorgis (1990, 1993), Pindyck (1991), Padock, Siegel and Smith (1988), Newton (1996), Myers (1984), Myers and Majd (1990) among others.

(4.) We concentrate our analysis in the option to invest, the option to expand, the option to contract, the option to abandon, the option to switch and the growth option, the option to shut down and restart operations, the option to defer, and the time to build option. These real options are studied in different contexts by Kogut (1991), Kogut and Kulatilaka (1994a, b), Mac Donald and Siegel (1984, 1986), Brennan and Schwartz (1985), Berger, Ofek and Swary (1996) among others. Several other real options exist, but the same analysis applies.

(5.) These options appear in the work of Dentskevich and Salkin (1991), Dixit (1992, 1995), Dixit and Pindyck (1994, 1995), Faulkner (1996) and Ingersoll and Ross (1992) among others.

(6.) Myers (1977) shows that the value of a firm is the combined value of the assets already in use and the present value of the future investment opportunities.

(7.) A typical example is firms in the oil and gas exploration and production business. Other examples include power stations and pharmaceutical companies. See for example Paddock, Siegel and Smith (1988).

(8.) Two types of flexibility are present in the project: internal and external flexibility. The internal flexibility corresponds to the manager's flexibility to modify the project. This can include expansion, alteration, abandonment, etc. The external flexibility corresponds to the growth option which gives the possibility to perform another project.

(9.) For a survey of the literature on standard options and exotic options pricing, the reader can refer to Cox, Ross and Rubinstein (1979), Cox and Ross (1976), Black and Scholes (1973), among others.

(10.) Or the lost value in time.

(11.) Dixit & Pindyck (1994, 1995) argue that the risk free interest rate is useful for three types of real economic problems. Dixit, Pindyck and Sodal (1999) use an exogenous discount rate for incomplete markets analysis. Firstly, in complete markets, by changing the probability measure, any stochastic process can be transformed to a risk-neutral one. Secondly, economic applications assume that firms are risk-neutral even when investors and stockholders are risk-averse. Thirdly, there is no correlation between the market portfolio and macroeconomic shocks.

(12.) Dixit and Pindyck (1994) explain that the time to maturity is defined by the expiration of the patent. After the expiration, the firm loses the opportunity to gain a competitive advantage due to the patent.

(13.) By using the model of Merton (1987), Bellalah and Jacquillat (1995) and Bellalah (2001) deduce the financial option value in the context of incomplete information.

(14.) We assume here that the firm has the possibility to abandon 5% of the investment value.

(15.) Ingersoll and Ross (1992) showed that the option to defer is reversible and more valuable when there is high economic uncertainty and long investment horizons.

* An earlier version of this article was presented at the Second International Conference on Banking and Finance (ICBF)--University of Utara Malaysia, at Entretiens de la Finance--AFFI, Paris, at the Second International Finance Conference (IFC2)--Tunisia, at the 52nd Annual Meeting of the Midwest Finance Association--Saint Louis, and at the 9th Annual Conference on Computing in Economics and Finance--University of Washington at Seattle. We would like to thank Professors Patrick Roger, Gordon Sick, Madhu Kalimipalli, and Hatem Ben Ameur for helpful comments. Any remaining errors are ours.

Inass El Farissi (a), Jean-Michel Sahut (c) and Mondher Bellalah (c)

(a) Assistant Professor of Finance, University of Picardie, France

(b) Professor of Finance, Groupe Sup de Amiens & CEREGE--University of Poitiers, France [e-mail]

(c) Professor of Finance, Thema--University of Cergy, France

Table 1
Investment opportunity value G(V)

                                G(V)

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 0.00%    [[lambda].sub.C] = 0.00%

300              100,56                      115,37
350              125,35                      142,11
400              151,71                      170,23
450              179,53                      199,62
500              208,71                      230,19
550              239,17                      261,86
600              270,85                      294,56
650              303,68                      328,23
700              337,61                      362,83

                                G(V)

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 1.00%    [[lambda].sub.C] = 1.00%

300               92,03                      104,56
350              115,65                      129,88
400              140,97                      156,72
450              167,86                      184,97
500              196,24                      214,52
550              226,02                      245,30
600              257,14                      277,25
650              289,54                      310,30
700              323,16                      344,40

Table 2
Time to expand option values

                                Call Values

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 0.00%    [[lambda].sub.C] = 0.00%

300             63.86224                    76.42005
350             80.37755                    95.78343
400             97.67687                   115.99436
450            115.62982                   136.90720
500            134,13617                   158,41084
550            153,11718                   180,41857
600            172,50962                   202,86120
650            192,26205                   225,68269
700            212,33196                   248,83699

                                Call Values

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 1.00%    [[lambda].sub.C] = 1.00%

300             56.64072                    67.77850
350             71.28849                    84.95228
400             86.63161                   102.87777
450            102.55445                   121.42580
500            118,96811                   140,49781
550            135,80275                   160,01692
600            153,00231                   179,92175
650            170,52115                   200,16259
700            188,32156                   220,69861

Table 3
Option to contract values

                           Option Values

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 0.00%    [[lambda].sub.C] = 0.00%

1               51.66080                    51.65769
50              50.49407                    50.34767
100             49.38165                    49.11073
150             48.34655                    47.96912
200             47.37949                    46.90969
250             46.47192                    45.92108
300             45.61670                    44.99408
350             44.80790                    44.12124
400             44.04060                    43.29644
450             43.31066                    42.51459
500             42.61457                    41.77144
550             41.94931                    41.06335

                           Option Values

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 1.00%    [[lambda].sub.C] = 1.00%

1               45.81902                    45.81626
50              44.78422                    44.65438
100             43.79760                    43.55731
150             42.87954                    42.54479
200             42.02183                    41.60516
250             41.21689                    40.72834
300             40.45838                    39.90617
350             39.74104                    39.13203
400             39.06051                    38.40050
450             38.41311                    37.70706
500             37.79573                    37.04794
550             37.20570                    36.41993

Table 4
Option to abandon values

                             Put Values

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 0.00%    [[lambda].sub.C] = 0.00%

1               90.37313                    90.30938
50              66.92000                    65.07766
100             52.58641                    50.27457
150             43.02524                    40.62699
200             36.15249                    33.80373
250             30.96604                    28.71878
300             26.91483                    24.78730
350             23.66723                    21.66288
400             21.01038                    19.12596
450             18.80079                    17.03008
500             16.93798                    15.27362
550             15.34938                    13.78377

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 0.00%    [[lambda].sub.C] = 1.00%

1               81.77299                    81.71530
50              60.55172                    58.88470
100             47.58215                    45.49031
150             38.93085                    36.76082
200             32.71213                    30.58688
250             28.01923                    25.98583
300             24.35355                    22.42848
350             21.41499                    19.60139
400             19.01098                    17.30589
450             17.01166                    15.40946
500             15.32612                    13.82014
550             13.88869                    12.47207

Table 5
Growth option prices

                             Call Values

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 0.00%    [[lambda].sub.C] = 0.00%

1               0.00950                     0.01411
25              6.27361                     7.79132
50              16.31713                    19.78988
75              27.22501                    32.66339
100             38.48884                    45.88367
125             49.93540                    59.27752
150             61.48820                    72.77038
175             73.10801                    86.32475
200             84.77264                    99.91972
225             96.46860                   113.54277
250            108.18723                   127.18593
275            119.92274                   140.84393

                             Call Values

        [[lambda].sub.V] = 0.00%    [[lambda].sub.V] = 1.00%
V       [[lambda].sub.C] = 0.00%    [[lambda].sub.C] = 1.00%

1                0.00818                     0.01215
25               5.39975                     6.70605
50              14.04428                    17.03331
75              23.43279                    28.11364
100             33.12765                    39.49244
125             42.97980                    51.02064
150             52.92338                    62.63405
175             62.92464                    74.30040
200             72.96449                    86.00170
225             83.03129                    97.72716
250             93.11761                   109.46994
275            103.21846                   121.22550

Figure 1:
Time to build binomial tree using binomial approach

                                           996.03508
                                          9960.35077
                                          9160.35077
                                          6786.15476
                             667.66228
                            6676.62279
                            5876.62279
                            3939.21806
               447.54741                   447.54741
              4475.47409                  4475.47409
              3675.47409                  3675.47409
              2238.35236                  2722.85818
 300.00000                   300.00000
3000.00000                  3000.00000
2200.00000                  2200.00000
1240.47432                  1491.43732
               201.09601                   201.09601
              2010.96014                  2010.96014
              1210.96014                  1210.96014
               789.48966                   927.99068
                             134.79869
                            1347.98689
                             547.98689
                             457.64901
                                            90.65826
                                           903.58264
                                           103.58264
                                           222.76435

                             3306.95291
                            33069.52914
                            32269.52914
                            32269.52914
               2216.71683
              22167.16830
              21367.16830
              19333.81339
 1485.90973                  1485.90973
14859.09727                 14859.09727
14059.09727                 14059.09727
11510.61530                 14059.09727
               996.03508
              9960.35077
              9160.35077
              8288.62814
 667.66228                   667.66228
6676.62279                  6676.62279
5876.62279                  5876.62279
4811.37180                  5876.62279
               447.54741
              4475.47409
              3675.47409
              3325.70649
 300.00000                   300.00000
3000.00000                  3000.00000
2200.00000                  2200.00000
1801.20766                  2200.00000
                20109601
              2010.96014
              1210.96014
              1095.72204
 134.79869                   134.79869
1347.98689                  1347.98689
 547.98689                   547.98689
 505.67512                   547.98689
                90.35826
               903.58264
               103.58264
               198.98633
  60.56896                    60.56896
 605.68955                   605.68955
-194.31045                  -194.31045
  72.25640                     0.00000
                40.60058
               406.00585
              -393.99415
                 0.00000
                              27.21539
                             272.15386
                            -527.84614
                               0.00000

Figure 2:
Time to build binomial price and standard NPV

                                            996.03508
                                           9960.35077
                                           9160.35077
                                           6803.67035
                              667.66228
                             6676.62279
                             5876.62279
                             3960.22496
                447.54741                   447.54741
               4475.47409                  4475.47409
               3675.47409                  3675.47409
               2260.72925                  2740.37377
 300.00000                   300.00000
3000.00000                  3000.00000
2200.00000                  2200.00000
1261.33910                  1510.53373
                201.09601                   201.09601
               2010.96014                  2010.96014
               1210.96014                  1210.96014
                806.69474                   942.79299
                              134.79869
                             1347.98689
                              547.98689
                              470.31867
                                            90.35826
                                           903.58264
                                           103.58264
                                           230.39831

                             3306.95291
                            33069.52914
                            32269.52914
                            32269.52914
               2216.71683
              22167.16830
              21367.16830
              19341.01602
 1485.90973                  1485.90973
14859.09727                 14859.09727
14059.09727                 14059.09727
11523.58486                 14059.09727
                996.03508
               9960.35077
               9160.35077
               8295.83076
  667.66228                   667.66228
 6676.62279                  6676.62279
 5876.62279                  5876.62279
 4824.34136                  5876.62279
                447.54741
               4475.47409
               3675.47409
               3325.70649
 300.00000                   300.00000
3000.00000                  3000.00000
2200.00000                  2200.00000
1814.17722                  2200.00000
                 20109601
               2010.96014
               1210.96014
               1102.92467
  134.79869                   134.79869
 1347.98689                  1347.98689
  547.98689                   547.98689
  515.25467                   547.98689
                 90.35826
                903.58264
                103.58264
                203.01204
   60.56896                    60.56896
  605.68955                   605.68955
 -194.31045                  -194.31045
   75.20963                     0.00000
                 40.60058
                406.00585
               -393.99415
                  0.00000
                               27.21539
                              272.15386
                             -527.84614
                                0.00000