...investment decisions are irreversible. However, the net present value rule in standard capital budgeting overlooks the value of manager's discretion to respond to uncertain outcomes and to alter its course of action. Many firms with significant growth potential involve staged investments, starting with investments in intellectual assets such as an untested product or a new technology and followed by investments in production and marketing. Contingent on the results of the initial stage and the eventual reception to the market, the initial investment provides the firm the option to invest further in production to generate net cash flows from sales over the economic life of the product. The option to alter the real investment decision offers the analogy with financial options and opens door to option-pricing techniques in capital budgeting practice.

The risk characteristics in real investments can be recognized into two categories: (1) market risks and (2) non-market risks or private risks. Market risk is faced by all firms operating in the industry and reflects the risk that the size of the market or the price and cost of the new product will be higher or lower than expected. With market risks alone the market can become complete. Asset pricing in a complete market is well established. The value of an investment asset (a focus asset) is obtained from the prices of other traded assets (basis assets). Continuous trading in the underlying assets makes it possible to replicate the asset's payoffs, and the present value of the focus asset must equal the price of the replicating portfolio in the absence of arbitrage opportunities. For example, a gold-mining firm observes uncertain gold prices that result from fluctuating supply and demand of gold. The risk and return profile of the investment opportunity in a gold mine can be tracked by a portfolio of gold-mining stocks and riskless bonds. With no-arbitrage the price of the tracking portfolio is expected to align with the value of the gold mine. The value of the focus asset does not require specifying investors' risk preference and is obtained by discounting at the risk-free rates the asset's expected payoff computed with risk-neutral probabilities, which are the probabilities that correctly price assets in a risk-neutral economy.

However, some risk characteristics cannot be replicated by trading securities available in the capital market. Those are the private risks inherent in many real investment decisions. For example, a drug firm has to pass through a number of R&D stages, from development and clinical trials to FDA approval before the new drug is marketed. Each and every one of these stages exhibits unique technical risks that cannot be replicated with available securities in the capital markets. In a related application, an oil firm recognizes risks related to the amount of oil reserves in the ground before the results of an exploratory oil well are confirmed. Since these risks are separate from the risk of oil price movements, the oil firm cannot hedge using oil priced-indexed securities, such as oil futures and oil stocks. In the existence of private risks the market now becomes incomplete. Merton (1987) provides theoretic references on incomplete markets and Merton (1998) applies the option-pricing theory to real option valuation. More recently, Kaufman and Mattar (2003) illustrate including private risks to the classical portfolio allocation problem in incomplete markets, while Staum (2007) broadly covers all major approaches to pricing derivative securities in incomplete markets.

Exact replication of the focus assets' payoffs is not readily available in incomplete markets because the private risk factors that affect an asset's payoffs are not represented by traded assets. The no-arbitrage option pricing techniques in incomplete markets cannot provide a unique real option price and only offer wide bounds, which are called the no-arbitrage bounds. E1 Karoui and Quenez (1995) provide a dynamic programming algorithm to compute the no-arbitrage bounds. However, the bounds provided by the no-arbitrage approach used in complete markets are too wide to be useful on the value of the option because there are infinite elements in the set of admissible martingale measures that accurately price the option, the focus asset. Holding the option becomes inherently risky because the portfolio of basis assets does not perfectly hedge the real option. Although marginal indifference pricing (Staum, 2004) provides a unique price based on expected utility in the absence of exact replication, the parameter values of a chosen utility function are subject to misspecification.

Observing weaknesses of the no-arbitrage bounds and the expected utility researchers pursue a compromise between the two by good deal bounds: see Cochrane and Saa-Requejo (2000), Carr et al. (2001), and Bernardo and Ledoit (2000). Constructing price bounds in incomplete markets is also closely related to imposing restrictions to the pricing kernel. More specifically, Cochrane and Saa-Requejo (2000) derive tighter bounds on option prices by imposing restrictions on Sharpe ratios or, equivalently, on the volatilities of the pricing kernel. Cases with high Sharpe ratios provide near-arbitrage opportunities and cannot last because investors bid to buy assets with high Sharpe ratios. One of the problems with this method is that a Sharpe ratio can be very low for arbitrage opportunities and cannot be reduced further with a typical threshold. In addition, it can easily construct bounds that conflict with risk-averse preference. In a related study, Bernardo and Ledoit (2000) rule out investment opportunities whose attractiveness to a representative investor exceeds a specified threshold. Their measure of attractiveness is the ratio of the expectation of the positive parts of the payoff (gain) on an investment to the expectation of the negative parts of the payoff (loss). By varying the gain-loss ratio, they are able to accommodate any bounds, from unique values to no-arbitrage bounds. In addition to setting an arbitrary gain-loss ratio, this approach easily tolerates the narrow bounds that conflict with risk preference implied by the benchmark pricing kernels.

Pyo (2007) suggests a way of narrowing the bounds that does not impose exogenous restrictions as in the literature. Instead, using the expected utility and the pricing kernel, Pyo extracts additional information on the investor's risk preference from the initial estimation of the preference parameters. Pyo observes maximum deviations around a benchmark price and takes a minimum of the two in the sense that the minimax deviation implicitly exhibits more confidence on a pricing implication on that side than on the other side. Minimax bounds presented by Pyo are shown to be simple in derivation, consistent with risk-aversion, and efficient in tightness of the bounds.

While Pyo (2007) illustrates advantages of minimax bounds on call option values in the absence of dynamic rebalancing, we apply the bounds to real options with private risks. Furthermore, Pyo does not have to calibrate utility functions because Pyo uses the connection between the Black-Scholes formula and exponential utility function as shown by Rubinstein (1976). Since real options embedded in capital budgeting decisions display characteristics different from call options on stock, this article presents a framework of constructing minimax bounds around a real option price by specifically calibrating utility functions in equilibrium settings. To make the results robust, we utilize a versatile utility function--HARA (hyperbolic absolute risk aversion) class utility function--which exhibits various patterns of risk aversion: increasing, decreasing, or constant. Although expressions for the minimax bounds are not neat, the framework applies the minimax bounds to real options and presents practical ways of deriving narrow bounds in most cases of utility functions.

We use a numerical example to price a real option and to illustrate how the minimax deviation suggested by Pyo (2007) is compared to those two approaches in the literature in the case of a real option embedded in a capital budgeting problem. It is not necessary to set up an arbitrary threshold for the bounds and the obtained bounds are consistent with risk preference reflected in a benchmark price. More importantly, the bounds are really tight around a benchmark price as opposed to those constructed with an arbitrary threshold in the literature.

The article is organized as follows. First, we derive the value of an option to defer a capital investment decision in a dynamically incomplete market, making explicit assumptions about a representative investor's attitudes toward risk. We show that when pricing is embedded in a utility maximization framework, a unique martingale measure emerges that reflects the investor's willingness to pay for cash across states and time. Next, we derive the corresponding risk-adjusted probabilities and note that the expressions for the risk-adjusted probabilities in incomplete markets closely resemble the expressions for the risk-neutral probabilities that would be obtained in complete markets. We show that there is a linear relationship between changes in the investor's coefficient of absolute risk aversion and changes in the values of the risk-adjusted probabilities. More important, changing the absolute risk aversion leads to very small changes in risk-adjusted probabilities. This is useful because it implies that less precise information about the investor's risk preferences, measured by its coefficient of absolute risk aversion, does not translate proportionally into less precise valuations. Then we show that the precise location of the option value depends on the risk-adjusted probabilities for the risks not priced in the market and the no-arbitrage bounds. We then develop the methodology to tighten the bounds on the values of the option. Comparison with other methods is presented, followed by a numerical example and the conclusion.

THE VALUE OF A REAL OPTION IN INCOMPLETE MARKETS

In the next three sections we apply the minimax approach to the value of a real option in incomplete markets constructing narrow bounds of the real option value. We illustrate the approach with an option to defer a real investment decision.

A Risky Project

Consider a risky project requiring the following alternative capital outlays, depending whether the investment occurs now, [I.sub.N], or is deferred one period, [I.sub.D] :

[I.sub.N] = [[I.sub.0], [I.sub.1] x I([V.sub.M] [greater than or equal to] [W.sub.M])] if invested immediately, or [I.sub.D] = {0, [[I.sub.0](1 + [r.sub.f]) + [I.sub.1]] x I [[V.sub.M] [greater than or equal to] [W.sub.M] - [I.sub.0](1 + [r.sub.f]) - [I.sub.1]} if deferred.

where the rate [r.sub.f] denotes the constant risk free rate of interest, and I {[V.sub.M] [greater than or equal to] [W.sub.M]} is an indicator function with [V.sub.M], the value of the project, and [W.sub.M], the selling price of the project at time 1. The M has two states, {u, d}, where u indicates the up state in the binomial tree for variable M, and d the down state. The [I.sub.D] assumes that the [I.sub.0] grows at a risk-free rate if the investment is deferred. The [I.sub.1] remains the same for both [I.sub.N] and [I.sub.D] because it is required for both at the same time if the investment is taken.

The project faces two different types of risks. The first risk is realized at time 1. This risk can be traded in the securities markets by means of a twin security whose payoffs are perfectly correlated with that source of risk. Accordingly, we call...

NOTE: All illustrations and photos have been removed from this article.



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