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Valuation and Information Acquisition Policy for Claims Written on Noisy Real Assets.

Timothy J. Riddiough [*]

We study contingent claims written on real assets, whose values are observed with noise, and the acquisition of information to improve irreversible exercise decisions. We determine the conditional expected asset value and show that it can depend on historical observed values. In a noisy setting, claim values are calculated by simply adjusting the asset value and variance inputs and applying standard valuation procedures for pricing European and America options. Noise tends to slow the rate of information arrival, reduce contingent claim value, and provide incentives to purposefully acquire additional information. These incentives are illustrated for the case of secured risky debt. The value of acquired information increases when the option holder is indifferent between exercise alternatives, and decreases as one choice increasingly dominates the other. Opportunities to repeatedly acquire information reduce over- or underinvestment in information.

In contrast to exchange-traded financial asset markets, real asset markets are often decentralized. Unique physical, contractual-relational, or locational characteristics of real assets underlie market decentralization and can introduce noise into the asset valuation process. If the exact current value of the real asset value is uncertain, then both contingent claim valuations and exercise decisions must reflect the imprecise value estimate. Also, to reduce the likelihood of error and increase the value of a contingent claim, the claimholder might want to acquire additional information to more precisely determine an asset's value.

One example is the research and development of new products. The precise current value of the yet undeveloped new product is uncertain, and the option to develop the product can be interpreted as a pure growth option. Other examples include corporate assets such as mines, factories, real estate, and intellectual property. Since these assets trade infrequently, value estimates are usually imprecise.

An important contingent claim that depends on real asset value is secured risky debt financing (e.g., Titman and Torous, 1989). Exercise decisions for such debt must reflect the inexact measurement of the underlying asset value and motivate the claimholder to acquire additional information.

In this paper, we study the effects of noise on contingent claims, and we examine claimholders' incentives to acquire the information to improve irreversible exercise decisions. [1] In our analysis, the exact real asset value cannot be determined directly from publicly available information. Instead, it must be inferred from noisy observations. We show that in the absence of purposeful information acquisition (IA), claimholders base their valuation and exercise decisions on the conditional expected asset value and its variance ("revealed variance"). We apply the optimal filtering techniques of Liptser and Shiryayev (1978) to derive these distributional measures for a variety of asset value and noise specifications.

Our analysis shows that noise generally slows the rate of information arrival and reduces the value of contingent claims. Noise also affects option exercise policy on many types of claims, since the value of waiting to resolve additional uncertainty typically is less than the no-noise case. We illustrate and quantify these effects by giving an example of a firm's perpetual option to develop an investment project. In general, claims values are determined by making simple adjustments to the asset value and variance inputs before applying standard valuation procedures. This example shows that the noisy investment option loses significant value relative to the noiseless case.

The potential for an error in option exercise gives claimholders an incentive to acquire additional information on the underlying asset value. A claimholder must first determine how much, if any, information to acquire. The claimholder might also be able to repeatedly gather and process information and must determine an optimal sequence of IA.

To focus on these issues, we examine IA for the case of a borrower who holds the default option on a secured risky debt contract. We show that the claimholder acquires information to reduce errors at the time of option exercise (the loan repayment date) if the acquisition of information is sufficiently inexpensive and if the expected asset value is close to the option's exercise price (the contracted loan payoff). When there are multiple opportunities to gather information, it is optimal for the claimholder to acquire information in smaller increments. The claimholder does this to reduce the potential of ex post over- and underinvestment in IA.

The paper is organized as follows. Section I discusses earlier studies. In Section II, we use optimal filtering techniques to determine the distributional parameters for valuing contingent claims and we examine a perpetual investment option as an example. Section III contains an application of a secured risky debt contract in which we determine the optimal levels and sequence of IA. Section IV discusses exercise and IA policy for other types of claims such as compound, American, and strategic options. Section V summarizes and concludes.

I. Background

Earlier work examines valuation and optimal exercise policy for contingent claims when asset value noise exists. Merton (1998) values a call option when the underlying asset value is not directly observable, but there is a portfolio of traded assets (with maximum correlation to the nontraded asset) that provides a value estimate and allows for dynamic replication. Flesaker (1991) derives the expectation of an underlying lognormal asset value conditional on the noisy signal and shows that the presence of noise reduces the value of the claim and alters exercise policy. He does so in a restricted setting with a multiplicative lognormal noise term and no initial noise. Neither of these papers allows for the purposeful acquisition of additional information to improve the precision of the value estimate.

Others have analyzed security valuation and portfolio selection issues when information is either imperfectly observed or differentially distributed across investors. In a setting in which expected returns on investments can only be estimated with error, Gennotte (1986) uses nonlinear filtering theory to derive the optimal estimator of stochastic expected returns. He obtains a separation result (i.e., first, he estimates parameters, and then he selects the portfolio conditional on these parameters.) Dothan and Feldman (1986) obtain similar results when the information structure is conditionally Gaussian. Merton (1987) focuses on pricing effects and portfolio selection issues when the quality of information is the same for all securities, but the information is differentially distributed across investors.

We extend this earlier work by determining the relevant distributional parameters when there is both initial and accumulating noise and for cases where the underlying real asset and noise process follow a variety of different specifications. This approach highlights the effects of noise on the valuation and optimal exercise policy for contingent claims in a number of potential economic settings. We then apply these results to two common corporate real options: 1) the valuation of firm's option to develop an investment project and 2) a secured risky debt contract. We further extend the existing literature by considering the purposeful acquisition of costly information that can be used to increase the precision of the conditional expected value estimate.

Howard's (1965, 1966) early work on information theory recognizes the value of information in a setting in which a bidder for a project faces uncertainties in both project cost and competitor bids. These studies show that the value of information is the difference between expected profits without information and expected profits from exogenous perfect information. In a decision-tree framework, Howard (1967) outlines the process for determining the optimal level of costly one-time experimentation. In our model, the level of noise reduction and the number of times information is acquired are endogenously determined by the contingent claimholder in a real-options setting.

II. Optimal Filtering Results and Special Cases

Classical models of exchange-traded options use prices from informationally efficient markets for the underlying asset. These markets aggregate diverse information into a price that reflects the full, current information available about the asset. In contrast, the assets that underlie many real options trade infrequently, if at all, so there is very little information available on these assets. As a result, real optionholders must form estimates of underlying asset value based on their own (generally incomplete) information set.

To distinguish between these two information scenarios, we use the phrase "full information" (Fl) to denote the classical setting in which the underlying asset trades in active markets that costlessly aggregate diverse information sets. We use the phrase "incomplete information" (II) to denote the model in which the value of the current underlying asset must be determined from a limited information set. Our general setting considers a real asset whose FI value cannot be perfectly observed. Section A presents the expected FI value conditional on the observable information. Section B discusses two important variances that measure the uncertainty in the current estimated value and the dispersion of future estimated values. In addition, we develop the relation between these two variances and the variance of the FL asset value. The values defined in these two sections are inputs for standard option models that can be used to value options on noisy real assets.

Three special cases are developed in Section C. The first case has an outside source of noise that obscures the Fl value. The second case has no outside noise but only a portion of the asset value can be observed. In both cases, the different noise sources generate distinct conditional expected values, but uncertainty about the FI value increases through time. In the third case, a mean-reverting drift is added to the stochastic noise component. The drift models a tendency of past errors to dissipate, which competes with the increasing uncertainty from the variance. Depending on the model parameters, either effect could dominate. Further, since the noise and the FL value have different distributions, the entire history of observed values is useful in disentangling noise from FI value.

A. Asset Value, Observed Value, and the Conditional Expected Value

We consider a setting in which perfect information about the current FI asset value for some real asset is unavailable to all market participants (i.e., the FI value cannot be directly observed or perfectly inferred by payoffs of any existing assets). The implications of this assumption are that asset markets are incomplete in an Arrow-Debreu sense and that cash flows generated by the real asset cannot be used to perfectly determine the FI asset value.

To determine the best estimate of its real asset value, the firm uses publicly available information that contains imperfect, or noisy signals of FI real asset value. For example, the continuous release of relevant macroeconomic data or current information on the market activity of roughly comparable real assets could provide partial information as to the FI asset value.

Let X (t) be the FI value, Y (t) be noise that hinders perfect observation of the FI value, and Z (t) = f (X (t), Y (t)) be a freely available noisy signal of FI value (available for t [greater than or equal to] 0). The true value and the observed value change through time according to,

dX (t) = [[a.sub.0] + [a.sub.1] X (t)]dt + [b.sub.1]d[W.sub.1] + [b.sub.2]d[W.sub.2] (1)

dZ (t) = [[A.sub.0] + [A.sub.1] X (t)]dt + [B.sub.1]d[W.sub.1] + [B.sub.2]d[W.sub.2] (2)

where d[W.sub.1] and d[W.sub.2] are increments of uncorrelated standard Wiener processes, and all [a.sub.i], [b.sub.i], [A.sub.i], and [B.sub.i] can be functions of time and the observed value.

Although the series of noisy observations cannot completely determine the FL value at any point in time, it can be used to form an expectation of the FI value conditional on the available information. We use I(t) to denote the information at time t, which includes the current and historical observed values. Let,

m(t) = E[X(t)/I(t)] (3)

represent the expected value conditional on the available information.

B. The FI Variance, Residual Variance, and Revealed Variance

We define the measure,

[gamma](t) = Var[X(t)/I(t)] (4)

as the variance of the estimated value around the true value. This value is the residual variance that remains after the optimal estimation of the FI value. There might also be noise at the initial observation date. That is, the initial noise level, [gamma](0) = [[[sigma].sup.2].sub.y0], might be positive.

We are particularly interested in the effects of noise on contingent claim valuation and exercise policy. We simplify our analysis by assuming that investors are risk-neutral. This assumption implies that contingent claim values are equal to the expected cash flows discounted at the riskfree rate of interest. This is true even if a riskless hedge portfolio does not exist.

When a firm makes an exercise decision on a real option, it uses the best available estimate of current value, m(t), to make an II optimal decision. Because there is some residual variance, the firm might end up exercising options when the unobservable Fl value is out-of-the-money or not exercising (European) options when the FI value is in-the-money. Thus, the expected payoff to the option prior to exercise is based on m(t) even though the realized option payoff might be based on the true asset value.

Consider the example of a firm facing a capital budgeting decision about launching a new product line (i.e., a growth option). As part of the process of determining the value of the project, the firm estimates demand for the new product using the available information. Given the demand estimate, the firm decides whether the expected cash flows from the product are sufficient to warrant paying the costs of launching the product. If the firm's estimate of demand is higher than the actual demand, then the firm could launch a product line that generates realized cash flows too low to justify the launch costs. Unfortunately, once the cash flows are realized, the irreversible launch costs have been spent. The firm has a slightly different problem if its estimate of demand is lower than the actual demand. Without launching the product, the firm does not realize that its demand estimate is low and continues to postpone investment.

In addition to the uncertainty in the current Fl value of the project, the firm's ability to accurately infer future expected asset values influences the exercise policy and the option value. As new information arrives, the firm updates its estimate of the conditional expected asset value. The variance of future conditional expected values from the initial date of the contingent claim to its terminal date (i.e., the "revealed variance") is a statistic that summarizes the arrival rate of publicly available information that the firm can act on when making exercise decisions. The total revealed variance from times to t (where 0 [less than or equal to] s [less than or equal to] t ) is:

v(s,t) = Var[m(t)/I(s)](S)]

In general, v(s, t) will differ from the total FL variance over the same time horizon, [V.sub.FI] (s,t).

To see the importance of the revealed variance, we consider two extreme scenarios. In the first, the value signal series is highly informative. In the second, the value signal series is extremely noisy. If future signals are highly informative, then the conditional expected value will closely track the FL value, and the revealed variance will be nearly equal to the FI variance. Since the conditional expected value behaves similarly to the FI value, the firm's option exercise policy and valuation will differ little from the Fl case. Alternatively, if new signals are extremely noisy (and thus provide little useful new information about asset value), the firm's new estimate of value will be similar to old estimates of value and revealed variance will be near zero. As a result, there is far less value in waiting to resolve uncertainty. It is optimal to exercise these options earlier, and the time premium of the options will be lower than the FI case. Thus, the variance due to the arrival of new information (i.e. , the revealed variance) drives the firm's exercise decisions and option value, not the variance of the future Fl value.

The revealed variance (v ), the residual variance ([gamma]), the FI variance ([v.sup.FI] ), and the initial noise variance ([[[sigma].sup.2].sub.y0]) are related. The appendix provides a proof for the following relation: V(0,t) = [V.sup.FI] (0,t) - ([gamma](t) - [[[sigma].sup.2].sub.y0]) (6)

That is, the total revealed variance, V(0,t), is the total FI variance, [V.sup.FI] (0,t), less the accumulated residual noise, [gamma](t) - [[[sigma].sup.2].sub.y0].

For a given setting in which residual noise grows as time passes (i.e., [gamma](t) [greater than] [[[sigma].sup.2].sub.y0]), the revealed variance will be less than the FI variance. Therefore, the options will be less valuable when there is noise than they are in the FI case. When residual noise decreases over time (i.e., [gamma](t) [less than] [[[sigma].sup.2].sub.y0]), the revealed variance will be greater than the FI variance, and options on a noisy asset will be worth more than in the FI case. This increase in value results because optionholders not only benefit from the variance of the FI value, but also, as time passes, from more precise estimates of the FI value. Finally, if there is a constant level of residual variance, then the revealed variance equals the FI variance and the option values are the same with or without noise. [2]

C. Special Cases

We use three special cases to investigate the effects of different noise specifications. Given the dynamics for the FI value and the observed value, Liptser and Shiryayev (1978) provide differential equations for the conditional expected value and the residual variance (see Appendix). Solutions to these differential equations give the firm a best estimate of current asset value and a measure of the accuracy of the estimate. In all our cases, the FI value is normally distributed and the observed value is the sum of the FL value and the noise term (i.e., Z(t) = X(t) + Y(t)). [3] Given the solutions for the normal variables, a transformation can be applied to generate the solution for the product of lognormal variables.

1. Observation Hindered by Accumulating Noise

In this first case, additional noise accumulates linearly over time at the rate of a [[[sigma].sup.2].sub.y], which, along with the initial uncertainty, [[[sigma].sup.2].sub.y0], causes the observed value to differ from the FI value. Equation T.1 in Table I states the dynamics. The noise has an expected value of zero and is persistent in the sense that future changes in noise levels are completely independent of past values. Black (1986) suggests that noise is cumulative if there is no way to acquire and trade on proprietary information on the FI asset value. Therefore, this case best describes projects in which it is difficult to infer value from other assets and in which the continuous acquisition of more precise information of FI asset value is prohibitively expensive.

The solutions for m(t) and [gamma](t) are provided in Equation T.2 of Table I, where:

[[rho].sup.2] = [[[sigma].sup.2].sub.x]/[[[sigma].sup.2].sub.x] + [[[sigma].sup.2].sub.y] (7)

The conditional expected value is the arithmetic weighted average of the time zero conditional expected value, Z(0) + [[micro].sub.x]t and the current observed value, Z(t). The weight for the arithmetic average is the FI variance to total accumulating variance ratio, [[rho].sup.2].

For any [[rho].sup.2] [less than]1 ([[[sigma].sup.2].sub.y] [greater than] 0), we use the initial observation to determine m(t). However, we need only the initial observation and the most current observation to determine the conditional expected value. The original observation has no accumulating noise, but does not contain current information about the FI value. The most recent observation contains the most current information about the FI value but also has the most accumulated noise. The weighting scheme places more weight on the current value when there is less accumulating noise. More weight is put on the time zero conditional expected value when more accumulating noise makes the current observed value less precise.

Observe that, although [[rho].sup.2] and m(t) depend on the variance rate at which noise accumulates, [[[sigma].sup.2].sub.y] they do not depend on the level of initial noise, [[[sigma].sup.2].sub.y0] Without being able to acquire additional information, the initial noise has a symmetric, or neutral, effect on the conditional expected value.

A relevant alternative expression for the conditional expected value is:

M(t) =Z(0) [[micro].sub.x]t(1-1/t[[[integral].sup.t].sub.0][[rho].sup.2]ds)+[[[i ntegral].sup.t].sub.0][[rho].sup.2]dZ(s) (8)

The conditional expected value is the original observed value scaled up by two terms. The first is the total drift, [[micro].sub.x]t, times one minus the average [[rho].sup.2]. The second term is the cumulative weighted change in the observed value, where the weight is [[rho].sup.2]. This expression clarifies the role for intermediate value signals in the determination of the conditional expected value. Since [[rho].sup.2] is constant in this case, the final integral in Equation 8 is [[rho].sup.2] (Z(t) - Z (0)), demonstrating that Z(t) and Z(0) are the only values that are relevant in determining the conditional expected value. However, when [rho] is time varying (see Section 3), the entire path of value signals is used to determine the expected asset value.

The residual variance, [gamma](t) in Equation T.2 of Table I, is the sum of the initial variance, [[[sigma].sup.2].sub.y0], and further errors which accumulate at the rate of [[rho].sup.2][[[sigma].sup.2].sub.y]. The revealed variance rate of m(t) is [[rho].sup.2][[[sigma].sup.2].sub.x] Thus, by using m(t) as the value of the underlying asset and 2 [[rho].sup.2][[[sigma].sup.2].sub.x] as the variance rate of the asset, we can value European calls and puts by using the formulas developed in Brennan (1979). For any option with a convex payoff function, option value decreases with increasing rates of accumulating noise.

When the observed value, the FI value, and the noise are all lognormally distributed and Z(t) = X(t) Y(t), the resulting conditional expected value is:

M(t)=[(Z(0)[e.sup.[[micro].sub.x]t]).sup.1-[[rho].sup.2]][(Z(t)[e.sup .1/2[[[sigma].sup.2].sub.y]t]).sup.[[rho].sup.2]] (9)

This quantity is the geometric weighted average of the time zero conditional expected value of Z(t) (i.e., Z (0)[e.sup.[[micro].sub.x]t]) and the convexity-adjusted current observed value, Z (t)[e.sup.1/2[[[sigma].sup.2].sub.y]t]. The important variances are,

Residual Variance: [gamma](t) = [[rho].sup.2][[[sigma].sup.2].sub.y]t + [[[sigma].sup.2].sub.y0] (10)

Revealed Variance: V(0,t) = [[rho].sup.2][[[sigma].sup.2].sub.x]t (11)

which are the same as in the normal case. To value a European (American) call or put option written on a noisy asset in this setting, one would simply use the Black-Scholes formula (standard numerical techniques) in which the underlying asset value is m(t) and the variance rate is [[rho].sup.2][[[sigma].sup.2].sub.x]. That is, adjusting for the noise in the underlying asset price reduces to just altering the asset value and variance inputs for standard valuation procedures.

2. Partial Observation

Equation T.3 in Table I presents an alternative specification when X(t), Y(t), and Z(t) are normally distributed. Again, the noise term has an expected value of zero, and the observed value is the sum of the FI value and the noise. Unlike the observed values in the previous section, which were continuously hindered by outside noise, the dynamics in Equation T.3 of Table I allow for perfect observation of some portion of the FI variance, [[[sigma].sup.2].sub.[x.sub.1]], and complete unobservability for the remaining portion of FI variance, [[[sigma].sup.2].sub.[x.sub.2]] where [[[sigma].sup.2].sub.[x.sub.1]] + [[[sigma].sup.2].sub.[x.sub.2]] = [[[sigma].sup.2].sub.x].

Equation T.4 in Table I gives the solution for these dynamics. With partial observation, the conditional expected value is the most recently observed value. Historical observations provide no incremental value to investors, since the current observation accurately shows a portion of current asset value. That is, historical observations are stale values of the current revealed component and therefore provide no further information on the current unrevealed component.

We note that standard European (American) puts and calls can be valued by using the Brennan formula (standard numerical techniques) with underlying asset value equal to m(t) and variance rate equal to the revealed variance rate, [[[sigma].sup.2].sub.[x.sub.1]] given the normality assumption. Holding the volatility of the true value constant, option value decreases as noise volatility increases.

3. Mean Reverting Noise

In the final case (Panel C of Table I), the Fl value is normally distributed, the noise term is mean reverting, and the observed value is the sum of the FI value and the noise term. When the observed value is below (above) the FI value, the noise term has a positive (negative) drift. Thus, past errors tend to dissipate, where [kappa] is the rate of dissipation. This case suggests that the actions of certain agents might keep observed values from wandering too far from their fundamental values (see Black, 1986). Purposeful research or the application of an IA technology could be responsible for this mean reverting tendency. However, we do not model the precise structure underlying this phenomenon in this specification.

The conditional expected value in Equation T.7 of Table I is the initial observation, Z(0), plus three terms. The first term is the unconditional drift of the observed value times one minus the average [[zeta].sup.2](s). (When [kappa]=0, the weight [[zeta].sup.2](t) is equal to the weight from the first two special cases, [[rho].sup.2].) The second term is a weighted sum of increments of the observed value. These first two terms are similar to the terms in Equation 8 for the normal case, except that the weights, [[zeta].sup.2](s), are now time dependent. The third term is related to the past conditional noise terms (we note that E[Y(s)\I(s)] = Z(s) - m(s)). Since the observed value is the sum of a constant mean FI value and a mean reverting error, previous conditional noise plays a role in determining the current conditional expected value.

Previous realizations of the observed asset value can provide valuable information about the size of the current estimation error, because the drift of the observed value is equal to the drift of the FI value when the error is zero, but is larger (smaller) than the FI drift when the error is negative (positive). If the trajectory of the observed value path deviates sufficiently from m, then what appears to be part of the error component could be reclassified later as part of the FI value component. Persistent positive (negative) levels of conditional noise suggest that past estimates of the FI value were too low (too high), implying that the current estimate of value should be increased (decreased) to correct for ex post estimation error. Hence, the entire time path of realized observed values is informative in this version of noisy real asset values.

We also note that, in contrast with previous results, both the initial level of noise, [[[sigma].sup.2].sub.y0] and all intermediate values of [gamma](s) are relevant in the determination of noisy asset and contingent claim values (as we see in the calculation of [gamma](t)). For example, a high initial [[[sigma].sup.2].sub.y0], combined with a strong tendency for noise to mean revert suggests that the arrival of information is highly valuable relative to initial conditions. Therefore, the initial observation should be increasingly de-emphasized when forming later conditional expected values.

There are some important differences in the residual variance and, hence, option valuation relative to the two earlier special cases. First, for any level of initial noise, the residual variance asymptotes to c as t [right arrow] [infinity]. Second, if [[[sigma].sup.2].sub.y0] = c , then the residual variance is constant. If residual variance is constant, then the first three terms of the conditional expected value in Equation T.7 of Table I simplify to:

(Z(0) + [[micro].sub.x]t)(1 - [rho]) + [rho]Z(t) (12)

This expression differs from the conditional expected value in Equation T.2 of Table I only in that the weight for the weighted average is [rho] instead of [[rho].sup.2]. That is, mean reverting noise lends a larger weight to current observations and a smaller weight to the original observation. Further, since the residual variance is constant, the revealed variance is equal to the FI variance, so II option values equal FI option values. Third, if the initial noise is very high [[[sigma].sup.2].sub.y0] [greater than] c, then the residual variance is monotonically decreasing, and the total revealed variance is greater than the total FI variance. This leads to II option values that are higher than FI option values. (Alternatively, when the initial noise is low (i.e., smaller than c), the residual variance is an increasing function of time, which implies that II option values will be lower than FI option values.)

D. Example

To illustrate the effects of noise on real options, we consider the case of a firm faced with a perpetual investment decision. (See Dixit and Pindyck, 1994, for further detail on the perpetual investment option.) Assume noise accumulates and that the FI project value and noise are both lognormally distributed as in Section C.1. Then the conditional expected discounted project value, m, is lognormally distributed with a drift of [[micro].sub.x] and an instantaneous standard deviation rate of [rho][[sigma].sub.x] (see Equations 9-11). The value of the perpetual investment project, C(m), and the critical investment threshold, m, are,

C(m) = I/[lambda] - 1 [(m/[m.sup.*]).sup.[lambda]] (13)

[m.sup.*] = I([lambda]/[lambda] - 1) (14)


[lambda]=1/2[[rho].sup.2][[[sigma].sup.2].sub.x] - [[micro].sub.x] + [square root][([[micro].sub.x]-1/2[[rho].sup.2][[[sigma].sup.2].sub.x]).sup.2 ] + 2r[[rho].sup.2][[[sigma].sup.2].sub.x]/[[rho].sup.2] [[[sigma].sup.2].sub.x] (15)

Equations 13 and 14 show the effects of noise on the investment decision.

Figure I displays the effects of noise on the critical investment thresholds of three projects that differ in the Fl volatility rate, [[sigma].sub.x]. When noise volatility is zero, the critical thresholds are those from the noiseless Dixit and Pindyck (1994) case. As noise volatility ([[sigma].sub.x]) increases, the critical threshold decreases and approaches the traditional NPV threshold. [4]

The difference in investment criteria also has a significant impact on the real option value. Figure II displays the impact on project value for the same three projects when the investment option is at-the-money. For a wide range of noise levels, the investment value remains larger than in the traditional NPV case but loses significant value relative to the noiseless case. In general, the lower the current estimate of project value, the larger the impact noise has on the percent loss in option value. Thus, if the investment option is out-of-the-money (in-the-money), then the percentage loss in option value is larger (smaller) than that in Figure II.

In summary, the firm's incomplete information can be used in decision-making, but the option exercise policy and real option value are significantly different from the FI case. The lack of full information about the underlying asset value introduces the potential for errors in exercise policy and typically causes a decrease in option value. This, in turn, creates an incentive to acquire additional information on the FI asset value.

III. Debt Contracting and the Incentives to Acquire Information

In this section, we explore incentives to acquire information, at a cost, by examining the case in which equityholders use secured discount debt to finance an investment in an imperfectly observable real asset. We choose this setting because it illustrates IA (information acquisition) incentives in a contingent claims context, and because it directly relates to the literature on debt contracting and financial intermediation.

A. Imperfect Observability Without Costly IA

We examine a setting in which outside debt finances the ownership of a noisy real asset. At time zero, equityholders issue noncallable, nonrecourse discount debt with face value, F, and maturity date T [greater than] 0. This debt is secured by the non-dividend-paying asset. When the debt is issued, the equityholder and lender both believe the asset value is m(0) = Z(0), which might differ from its FI value, X(0). We assume that the initial uncertainty in the FI value of the asset is [gamma](0) = [[[sigma].sup.2].sub.y0]. After the debt is issued, additional uncertainty accrues for the FI asset value. We also assume that at time t [greater than or equal to] 0, the observed signal, Z(t), is the product of a lognormally distributed FI asset value, X(t), and a lognormally distributed noise process, Y(t), as discussed at the end of Section II.C.1 (see Equations 9-11).

To provide a benchmark for further analysis and discussion, we consider optimal exercise policy when costly IA is infeasible. We assume a contract provision prevents asset sale prior to debt payoff. The debt can be repaid by using the firm's financial slack or by raising additional equity capital. In this case, the equityholder will default on the debt at time T if and only if m(T) [less than] F. Debt value is the Merton (1974) debt value with an underlying asset price equal to the conditional expected value, m, and variance rate equal to the revealed variance rate, [[rho].sub.2][[[sigma].sup.2].sub.x]. (The debt value formula is given in the Appendix.) If the equityholder cannot acquire additional information, then the exercise policy and debt value depend on the conditional expected value but not on the initial level of noise.

Figure III shows the yield to maturity on the secured debt as a function of noise volatility, [[sigma].sub.y]. An increase in noise volatility reduces the information content of the value signal and introduces the potential for borrower error in repayment and default decisions. Either of these errors is detrimental to the borrower (and beneficial for the lender). Thus, noise reduces the required debt yield.

Figure III shows that the promised yield can be sensitive even to small amounts of noise that might obscure real asset observability. Empirical studies on the level of noise volatility for real estate, which is often financed with secured debt, find that valuation error is, on average, approximately 10%. Many other assets can be used as collateral for secured debt. These assets, including any inventory that does not have active secondary markets, specialized factories or mines, [5] receivables, royalties, and other types of intellectual property (including patents, trademarks, and copyrights), might have much higher noise levels due to asset uniqueness and the lack of an active market for the assets. Other assets, such as automobiles, trucks, inventory with active markets, railroad cars, computer equipment, and heavy construction equipment are more commodity-like, have relatively active secondary markets, and, hence, might have noise levels comparable to or even lower than real estate noise levels.

We define a Type 1 error as II optimally defaulting on the debt when it is not FI optimal to do so (i.e., m(T) [less than]F [less than]X(T)). We define a Type 2 error as II optimally paying off the debt when it is FL optimal to default (i.e., X(T) [less than] F [less than] m(T)). (See Appendix for formulas for expected losses.)

The difference between debt value under FI and II is equal to the discounted present value of Type 1 and Type 2 errors. (These potential errors, through their effect on debt value, also determine the yields that are shown in Figure III.) This difference is also equal to the discounted expected benefits of IA in which both the borrower and lender learn the FI asset value. The existence of these errors suggests that the equityholder might have an incentive to acquire costly information in an effort to reduce or eliminate errors in exercise decisions.

B. Costly Acquisition of Information

Before making an irreversible default or repayment decision, the borrower might want to acquire additional information to more precisely determine the FI asset value. We develop our results in this section under the assumption that asset value information is acquired by and paid for by the borrower, who has no subsequent opportunity to resell the information or otherwise engage in its strategic redeployment.

1. Single-Opportunity IA

We assume that the equityholder can pay to acquire additional information that will lower the current residual variance by a factor of [beta] [epsilon] [0,1]. The newly acquired information can be combined with any existing information to form a new, larger, post-IA information set, [I.sup.IA] ([beta], T). Given the new information, the equityholder forms a new conditional expected value,

[[m.sup.IA].sub.I] ([beta], T) E[X(T)\[I.sup.IA] ([beta], T)] (16)

in which the conditioning takes place by using the costly new information. This value estimate is more accurate, in the sense that residual uncertainty has been reduced by [beta],

[[gamma].sup.IA] ([beta], T) = (1 - [beta])[gamma](T) = (1 - [beta])([[rho].sup.2] [[[sigma].sup.2].sub.y]t + [[[sigma].sup.2].sub.y0]) (17)

Thus, [beta] is the level of IA precision, where [beta] = 0 corresponds to a completely uninformative signal (IA does not reduce residual uncertainty at all), and [beta] = 1 corresponds to a fully revealing signal (IA completely reveals the current FI value).

As noted above, when asset value information is incomplete, it is the borrower who is motivated to gather additional information prior to exercising an irreversible repayment or default option. In this setting, the borrower acquires information only at the terminal date. There are three reasons for this outcome. First, residual uncertainty continues to build after IA. Thus, the early acquisition of information means the equityholder will have paid for a level of precision, [beta] but will have greater residual uncertainty by the time he uses the information for the terminal default decision. Second, if the equityholder waits until the debt matures, then the full set of observed values (available only at the terminal date) could suggest that an alternative precision is optimal, or that IA is not necessary at all. Finally, even if gathering IA is a certainty, the equityholder prefers to postpone the IA cost.

The expected benefits to discretionary IA is an option value in which the total variance of the option equals the variance reduction due to IA. When m(T) [less than] F and the equityholder does not acquire costly information, the equityholder defaults on the debt. In this case, the advantage of IA is that the better-informed equityholder will not default if the revised estimate of asset value, [[m.sup.IA].sub.I] ([beta],T), exceeds F, thus reducing Type I errors. The expected benefits (at exercise) from IA are [[m.sup.IA].sub.I] ([beta], T) - F when [[m.sup.IA].sub.I] ([beta], T)[greater than] F. Thus, acquiring information corresponds to a call option to "repurchase" the assets (with pre-IA expected value of m(T)) for the face value of the loan, F.

When m(T) [greater than] F and no costly information is acquired, the equityholder makes the final payment on the debt. In this case, acquiring information creates an opportunity for the equityholder to put the asset for F. The expected benefits (at exercise) of reducing Type 2 errors when [[m.sup.IA].sub.1] ([beta],T) [less than] F are F - [[m.sup.IA].sub.1] ([beta],T).

IA and subsequent default on the debt can result in a discontinuous drop in estimated debt value. Sudden changes in debt value are consistent with evidence on significant declines in corporate bond prices (e.g., Altman, 1989, and Duffie and Lando, 1998) and with sudden drops in commercial real estate loan values once borrower default and foreclosure has occurred (e.g., Ciochetti and Riddiough, 1999).

The option premium for IA is analogous to the option time premium in the classical FI setting. In the classical setting, the time premium is due to the passive (and relatively slow) arrival of new information on the underlying asset value. With [tau] time left until the option exercise date, the total uncertainty to be resolved equals [tau][[[sigma].sup.2].sub.x]. In our case, costly information arrives discretely at the option exercise date and corresponds to the reduction of uncertainty of the current asset value by [beta] ([[rho].sup.2][[[sigma].sup.2].sub.y]T + [[[sigma].sup.2].sub.y0]).

For a given precision, [beta], the expected benefit of single-opportunity IA is:

B(m,[beta],T) = {m[phi]([e.sub.1](m,[beta],T))-F[phi]([e.sub.2](m.[beta],T)) if 0[less than]m[less than]f and 0[less than][beta][less than or equal to]1 F[phi](-[e.sub.2](m,[beta]T))-m[phi](-[e.sub.1](m,[beta]T)) if m [greater than or equal to] F and 0 [less than] [beta][less than or equal to]1 (18) 0 if [beta] = 0 or m = 0

with [e.sub.i] as defined in Equation A. 10.

We note that the benefits to IA are greatest at the point at which the equityholder is indifferent between debt payoff and default (i.e., when m(T) = F) and that benefits decline monotonically as \m(T) - Fincreases. In other words, the acquisition of additional information is most valuable when the equityholder is indifferent between exercise alternatives, and decreases as the better choice increasingly dominates other alternatives.

Next, we examine the cost of acquiring information. Let the cost function be,

[C.sub.1]([beta]) = [eta][[beta].sup.[alpha]] (19)

where [eta] and [alpha] are positive constants. We assume that the cost function increases in precision with convexity depending on [alpha] and that it is independent of expected asset value.

Some level of IA will take place when benefits of IA exceed the costs. Assuming that IA is feasible for some m(T) (i.e., there is a [beta] such that B(F,[beta],T) - [C.sub.1]([beta])[greater than] 0), there will be a region containing F in which it is optimal for the equityholder to acquire additional information.

Figure IV illustrates the net benefits to IA across various conditional expected asset values, m(T). The figure compares four different levels of precision and demonstrates that the optimal level of precision varies across different conditional expected values. For example, when m(T) is near F = 100, an intermediate precision, [beta] = 0.6, is best (given only these four [beta]s). As conditional expected values move away from face value in either direction, less precision is optimal (i.e., [beta]= 0.4 is optimal). Eventually, it is no longer profitable to acquire any information. We note that there are high ([beta] = 0.8) and low ([beta] = 0.2) levels of precision that are never optimal. Thus, different conditional expected values will yield different optimal precision levels, but not all precision levels are optimal.

If the equityholder can choose any [beta] [epsilon] [0,1], the expression for the optimal, [beta] is:

[[beta].sup.*](m(T),T) = arg [max.sub.[beta][epsilon][0,1]][B(m(T), [beta],T)- [C.sub.1]([beta])] (20)

If the optimal precision is not a corner solution, then [[beta].sup.*] must equate marginal benefits of IA to marginal costs, and we can use implicit differentiation to calculate the partial derivatives of optimal precision with respect to [[[sigma].sup.2].sub.y], [[[sigma].sup.2].sub.y0], and [[[sigma].sup.2].sub.x]. (See the Appendix for these derivatives).

We study the likely case in which FI asset volatility is larger than the accumulating noise volatility. For shorter-term options, the optimal precision level is most sensitive to changes in initial noise level and least sensitive to changes in the FI asset variance rate. For longerterm options, noise accumulates over a long enough time so that optimal precision is most sensitive to changes in the accumulating noise variance rate.

In Figure V, Panels A, B, and C graphically display these comparative static results for the optimal IA precision, [[beta].sup.*]. Figure V, Panel A, shows that as the standard deviation of the noise process increases, so does the optimal level of IA since the benefits to IA increase with the amount of noise. The increase in IA precision is quite dramatic. For example, at m(T) = F, the optimal precision increases from 0.445 to 0.731 as the accumulating noise volatility rate goes from 0.05 to 0.15.

In Figure V, Panel B, optimal precision is not as sensitive to changes in the initial noise volatility for the intermediate-term option. For m(T) = F, optimal precision increases from 0.573 to 0.643 as initial noise volatility goes from 0.05 to 0.15. When noise volatility is less than FI asset volatility, as in Figure V, Panel C, optimal precision is least sensitive to changes in FI volatility. For m(T) = F, the optimal level of IA increases only slightly (from 0.582 to 0.614) as FI asset volatility increases from 0.2 to 0.6. We note that because benefits to IA increase with increases in volatility, the range of conditional expected values for which it is optimal to gather information also increases as any of the volatilities increase.

We develop IA incentives under the assumption that the optionholder has no opportunity to resell or otherwise strategically redeploy the costly information. Depending on whether or not the optionholder can verify the acquired information, there are several ways to think about this assumption. First, if we suppose that the post-IA knowledge of asset value can be credibly conveyed, there may be strong incentives for the borrower to freely share this information with the lender. This sharing incentive follows because the absence of information-sharing introduces the potential for costly replication of IA by lenders who are required to sell the asset, or by outside investors who wish to acquire the asset from the lender upon default. These potential costs will be reflected in higher financing costs for the borrower.

A credible information-sharing commitment could involve a third-party IA specialist whose cost schedule is known in advance and who agrees to share the results of IA with the lender, should default occur. This scenario is similar to that in Diamond (1985), who states that voluntary information disclosure helps avoid costly duplication of effort by outside investors. Diamond and Verrecchia (1991) extend this idea to show that disclosure reduces information asymmetries to increase asset demand from large investors.

2. Multiple-Opportunity IA

One-time IA may be a reasonable policy when there is only a short time in which to acquire information to improve an irreversible exercise decision. However, there are many other circumstances in which multiple occurrences of IA are both possible and optimal, even when IA has been undertaken many times in the past.

The intuition behind multiple-time IA is that there will be an incentive to acquire information in smaller increments in an attempt to reduce ex post over- or underinvestment information. To see this, consider a one-time IA opportunity that is available to the equityholder. If m(T) is near the exercise boundary, then significant IA might reduce the probability of error. But, with multiple-time IA, the equityholder might initially choose lower precision (and hence lower cost) IA. If the revised [[m.sup.IA].sub.1] ([beta],T) turns out to be significantly different from F, then the equityholder will be confident in making an exercise decision with no additional costly IA. However, if [[m.sup.IA].sub.1] ([beta],T) is still near F, the equityholder might aggressively acquire additional information. In this way, he can target higher, more costly levels of IA only when he most needs to.

We present a case in which two IA opportunities are available to the equityholder. We denote the precision realized from the first IA opportunity as [[beta].sub.1] the precision realized from the second IA opportunity as [[beta].sub.2], and the revised expected asset value from the first IA opportunity as [[m.sup.IA].sub.1] ([[beta].sub.1],T). The second precision, [[beta].sub.2], is the cumulative (total) precision, which suggests that [[beta].sub.1] [less than or equal to] [[beta].sub.2] [less than or equal to] 1 and that higher precision leads to finer information sets, ([I.sup.IA]([[beta].sub.1],T) [subset or equal to] [I.sup.IA] ([[beta].sub.2],T)).

We assume that the information sets are invariant to the order in which information arrives. For example, acquiring information once at a precision of [beta] = 0.7 generates the same information set as a sequence in which the initial precision is [[beta].sub.1] = 0.4 , and the cumulative precision is [[beta].sub.2] = 0.7.

To develop the optimal multiple-opportunity IA policy, we begin by analyzing the second round of IA. The variance that is reduced from acquiring information a second time is the difference in the residual variance in the first and second rounds of IA:

[[gamma].sup.IA]([[beta].sub.1],T)-[[gamma].sup.IA]([[beta].sub.2],T) =([[beta].sub.2] - [[beta].sub.1])([[rho].sup.2][[[sigma].sup.2].sub.y]T + [[[sigma].sup.2].sub.y0]) (21)

If first-time IA occurs, calculating the optimal IA region for the second time will be similar to the one-time case except that [[m.sup.IA].sub.1] ([[beta].sub.1],T) is the revised conditional expected asset value, and precision level increases from [[beta].sub.1] to [[beta].sub.2]. The benefits from the second round of IA are the benefits from the incremental level of precision, B([[m.sup.IA].sub.1]([[beta].sub.1],T),[[beta].sub.2] - [[beta].sub.1],T).

We assume that the cost function for increasing precision from [[beta].sub.1] to [[beta].sub.2] is:

C([[beta].sub.1],[[beta].sub.2])={[eta][[[beta].sup.[alpha]].sub.2] - [theta][eta][[[beta].sup.[alpha]].sub.1] if [[beta].sub.2] [greater than] [[beta].sub.1]

0 otherwise (22)

The parameter [theta] [epsilon] [0,1], indicates the magnitude of second-time cost savings derived from first-time IA. When [theta] = 1, second-time IA costs are based solely on the incremental amount of precision obtained. When [theta] = 0, first-time IA costs must be duplicated for any second-time IA. Intermediate values of [theta] indicate that partial duplication of cost occurs in the second round. The optimal precision level for the second round of IA (given a precision from the first round) maximizes the net benefits, B([[m.sup.IA].sub.1] ([[beta].sub.1],T),[[beta].sub.2] - [[beta].sub.1],T)- C([[beta].sub.1],[[beta].sub.2]). Thus, the optimal precision level, [[[beta].sup.*].sub.2] ([[m.sup.IA].sub.1]([[beta].sub.1],T),[[beta].sub.1]), depends on the precision and the conditional expected value that results from the first-round IA. If a second round of IA is not optimal, then [[[beta].sup.*].sub.2]([[m.sup.IA].sub.1]([[beta].sub.1],T),[[beta].s ub.1]) = [[beta].sub.1].

Determining the optimal precision for the first IA opportunity is complicated; the precision from the first round affects optimal precision for the second round, and the benefits from first round IA include the option value associated with a second-time IA opportunity. The value of the option to acquire second-time information is the expected net incremental benefits obtained from the second round of IA for a given first-round precision, [[beta].sub.1]. Therefore, the total net benefits from a first-round IA are the direct net benefits from the first round, plus the expected incremental net benefits from the second round. [6] The optimal first round precision, [[beta].sup.*].sub.1], maximizes the total net benefits.

Figure VI, Panel A, displays endogenously determined optimal precision levels under a two-time [A approach with [theta]=0 (i.e., first-time costs must be duplicated on second-time IA). We show precision levels in this figure prior to the initial acquisition of information and compare optimal precisions, given two-time IA, to those obtained in the one-time IA case. Figure VI also shows the expected increment to first-round IA, E\[[[beta].sup.*].sub.2] - [[[beta].sup.*].sub.1]\[I.sup.IA] (0,T)\, which equals the difference between the expected cumulative precision from second-round IA and the optimal precision level from first-round IA.

The total expected precision with multiple-time IA in Figure VI is less than the optimal precision realized from one-time IA. Multiple-opportunity IA allows information to be gathered in smaller increments, thus reducing the potential for ex post overinvestment in information. As [[[beta].sup.*].sub.1] moves away from its maximum value, the expected second-round incremental IA increases and total expected IA decreases. The incremental second-round IA is more intense than first-round IA for some m(T) in the IA range that are sufficiently far from F. Thus, an advantage of multiple-time IA is that the equityholder can perform a low precision test for the first-time IA when m(T) is at some distance from F. If the outcome of the test, [[m.sup.IA].sub.1] ([[beta].sup.1], T), is still sufficiently far from F, no further IA is warranted. However, if [[m.sup.IA].sub.1] ([[beta].sub.1], T) is closer to F, then the equityholder invests in IA at a much higher level the second time. Since first-round IA was low precision, there is little replication of costs.

Figure VI, Panel B, shows optimal precision levels for [theta]= 1, (i.e., when first-time costs are not duplicated on second-time IA). In this case, the first round of IA is more intense than when [theta]= 0, but less so than the one-time case. Also, because it is not as costly to break up IA into two parts, the equityholder is expected to do more total IA (including more first-time IA) compared to the IA in the [theta]= 0 case. In fact, when the benefits to IA are highest (i.e., for m(T) near F), the cumulative IA levels for the two-time case are greater than the optimal level for the one-time case. We also note that the range over which the first-round IA is expected to exceed incremental second-round IA is wider in the [theta]= 1 case, because the low cost of the second-round IA feeds back to result in more intense first-round IA.

Figure VI, Panel C, shows the amount of cumulative precision conditional on various levels of first-time precision when [theta]= 1. In determining second-time optimal precision, the claimholder has already chosen [[beta].sub.1]. Thus, the only difference in the choice of [[[beta].sup.*].sub.2] for [theta]= 0 versus [theta]= 1 is in the cost function. Since [[beta].sub.1] is fixed at the time the claimholder chooses [[beta].sub.2], the optimal second-time precision levels are the same for [theta]= 1 and [theta]= 0, but the range over which a second round of IA is optimal is smaller when [theta]= 0.

We note that second-time IA occurs at high levels over a narrow range of conditional expected values centered near m(T) = F. This finding illustrates again the advantage of multiple-opportunity monitoring (i.e., using lower initial precision while retaining the option to pursue aggressive IA the second-time only when it is necessary). We also note that second-round IA takes place over increasingly narrower ranges and that the incremental level of IA (i.e., [[[beta].sup.*].sub.2] - [[[beta].sup.*].sub.1]) decreases as first-round IA is more intense. These effects occur because second-round IA is relatively more expensive at higher initial IA levels. The result is diminishing marginal returns when increasing levels of IA are realized.

IV. Exercise and IA Policy for Other Types of Options

In our analysis of the risky discount debt contract, the default option is effectively European, since no dividends are paid on the asset and no action (payment or default) is required until the debt payoff date. Therefore, it is optimal to wait until the contractual debt payoff date to acquire additional information on the Fl underlying asset value.

Many claims, such as coupon debt, are compound (as opposed to European), requiring that a periodic cash payment be made to keep the claim alive for another period. Compound options on noisy real assets require a dynamic IA policy. This policy will have characteristics similar to the multiple-opportunity IA analysis undertaken previously in which the value of delay reflects interim cash flows and the time value of money. We expect that the value ranges at which IA occurs prior to the option expiration date will be relatively narrow and depend on the debt's time to maturity.

Because there are often substantial irreversible costs incurred in fully developing new products, there are clear reasons for conducting low-cost experiments or undertaking related IA activities to resolve residual uncertainty as to the actual product value. In general, there are often opportunities to conduct a series of experiments prior to determining whether full product investment is warranted. We note that even for a one-time-only IA opportunity, the investor often acquires information at expected asset values below (but close to) the investment hurdle value. When the timing of option exercise is discretionary, acquiring information at expected asset values somewhat below the hurdle value helps reduce the probability of waiting too long to invest (i.e., noise hinders recognition that the investment option may be well into the money). It can also increase the accuracy of the asset value estimate when investment is delayed further. IA policy of this type is consistent with the findings of Roberts and Wei tzman (1981) from a search model approach to R&D.

Product market structure can affect private incentives to acquire and disseminate information and the social value of that information. For example, Admati and Pfleiderer (1998) find that, from a social standpoint, individual firms could choose to under-disclose private information when disclosure also results in more precise valuations by competitors. How product price information is correlated and communicated can have significant real effects for innovation and competitive new product market development. For example, in an emerging competitive market in a homogeneous good in which lead investment fully reveals the product demand curve, the ability to hinder communication or otherwise internalize the information externality from lead investment results in a first-mover advantage and avoids inefficient delay. Underinvestment incentives also suggest a role in the creation of preinvestment IA mechanisms that can be used to mitigate free-rider incentives and to result in more accurately timed investment decisi ons. [7]

V. Summary

Many real assets are traded infrequently in decentralized markets. Therefore, current asset values cannot be continuously and precisely observed. If a claimholder cannot know with certainty the value of an asset that underlies a contingent claim, then he must make both valuation and any exercise decisions with a noisy estimate of real asset value.

In this paper, we examine just such a setting. We use optimal filtering techniques to determine distributional parameters for three forms of noise propagation: 1) observation hindered by additional noise, 2) partial observation, and 3) mean-reverting noise. The level of noise changes dynamically after the initial, potentially noisy, observation. Because noise hinders the claimholder's ability to make fully informed exercise decisions, he has an incentive to more precisely determine asset value by acquiring costly information. To illustrate the economics of costly information acquisition, we determine optimal IA policy for a borrower who holds the default option embedded in a risky discount debt contract.

Our major findings are as follows. We show that the conditional asset value depends on the path of the observed value when outside noise accumulates over time or when noise is mean-reverting. However, the conditional asset value is path-independent when the FI asset value is partially observable. Claims on noisy real assets can be valued using the estimate of asset value and future dispersion of the estimate as inputs in standard option pricing models.

For the steadily increasing noise models without costly IA, the initial level of noise does not affect exercise policy or option value. However, accumulating noise affects the exercise policy and lowers (convex payoff) option values. In contrast, for the mean-reverting noise model or when information is purposefully acquired to increase precision in the asset value estimate, any noise that can be resolved is important for determining claim value and exercise policy.

We use a secured debt example to study information acquisition incentives. We show that value associated with the purposeful acquisition of information is analogous to the time premium found in the classical option pricing model. When the IA technology is sufficiently inexpensive and the conditional expected asset value is sufficiently close to the option exercise price, information is acquired to reduce potential errors in exercise policy.

We perform comparative statics for several variance measures. We show that for reasonable parameters, the intensity of information acquisition is most sensitive to the accumulating noise volatility, moderately sensitive to initial noise volatility, and least sensitive to FI asset value volatility.

When there are several opportunities to gather information, it is optimal for claimholders to acquire information in smaller increments, thereby reducing the potential of ex post over-and underinvestment in IA. When estimated asset value is close to the face value of debt, the intensity in the first round of IA is lower but similar to the single-opportunity case. However, the expected second round incremental IA is low. As the estimated asset value is further from the face value, it is best to perform low intensity IA in the first round, followed up by higher intensity IA.

Some of the effects of noise can be mitigated by purposefully acquiring information at debt payment dates. The borrower's ability to improve exercise decisions has a direct cost of gathering information and tempers the yield-lowering effect of noise.

In our model, efficient pricing of debt secured by noisy assets has lower yield-to-maturities than when noise is not present, since noise creates the potential for errors in default and payment exercise policies. For cases when the lender does not recognize that noise exists, debt financing becomes expensive and equity financing may be optimal. For companies that are capital constrained, equity financing of noisy assets through securitization may be a viable and less costly alternative to secured debt financing.

We are grateful to Carliss Baldwin, Michael Brennan, Jean Cooper Jerome Detemple, David Geltner, Steve Grenadier, David Mauer, Raymond Rishel, Jeremy Stein, Richard Stockbridge. Bruce Walcott, Bill Whearnon Joe Williams, seminar participants at the University of South Carolina, and participants at the 1999 International Conference on Real Options for helpful comments and discussions. We would also like to thank the Editors and an anonymous referee for helpful comments and suggestions.

(*.) Paul D. childs is an Associate Professor of Finance at the University of Kentucky. Steven H. Ott is an Associate Professor of Finance at the University of North Carolina at Charlotte. Timothy J. Riddiough is an Associate Professor of Real Estate Finance at the Massachusetts Institute of Technology.

(1.) For a review of the types of corporate real options and standard valuation principles in the absence of noise, see Trigeorgis (1993).

(2.) Pennings and Lint (1997) suggest that when the underlying asset value is not known with certainty, estimation errors do not affect the option value and, hence, can be ignored in their subsequent analysis. The Pennings and Lint assertion is only true when residual variance is constant. As we show, estimation errors have a negative (positive) impact on option value when residual variance increases (decreases).

(3.) Normally distributed values may be realistic real options scenarios where the underlying asset is an untraded project that may have a negative operating value (e.g., a R&D project).

(4.) When [delta] [less than] r, the critical threshold will not approach the traditional NPV threshold since the NPV approach does not capture any value to wait. Jorgenson (1963) shows that when there is no noise or uncertainty, the firm optimally invests when the project value has appreciated sufficiently so that the dollar cash flow from the project, [delta]X , exceeds the opportunity cost of the investment, rI (also see Dixit and Pindyck, 1994). Thus, when 0 [less than] [delta] [less than] r, the critical threshold decreases to rI/[delta] (rather than I) as noise increases.

(5.) For valuation of mines using real options see Schwartz (1998).

(6.) The formula for the total net benefits is provided in the Appendix.

(7.) Childs, Ott, and Riddiough (2001) examine an imperfectly competitive market for real estate development in which agents compete over the timing of lead investment. This application does not allow for the costly acquisition of information per se; information is revealed only through irreversible lead investment.


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Trigeorgis, L., 1993, "Real Options and Interactions with Financial Flexibility," Financial Management 22, 202-224.

Table I. Filtering Solutions

This table displays the dynamics, differential equations, conditional expected value, residual variance, and revealed variance for three special cases. The variables are normally distributed with Z(t) = X(t) + Y(t). Given the underlying dynamics, the table presents the solution for the conditional expected asset value, m, the residual variance, [gamma], and the revealed variance, v. The residual variance measures the uncertainty in the current asset value given the available information. Using m as the asset value and v as the variance rate, standard option pricing techniques can then be applied to value options on noisy assets.

Panel A. Additional Noise



Panel B. Partial Observation



Panel C. Mean Reverting Noise





The Liptser and Shiryayev (1978) Differential Equations

If the FI and observed value follow the dynamics in Equations 1 and 2, Liptser and Shiryayev (1978) show that the conditional expected value, m(t), and the residual variance, [gamma](t), are unique solutions of,

dm(t) = [[a.sub.0] + [a.sub.1]m(t)]dt + [b.sub.1][B.sub.1] + [b.sub.2][B.sub.2] + [gamma](t)[A.sub.1]/[[B.sup.2].sub.1] + [[B.sup.2].sub.2] [dZ(t) - ([A.sub.0] + [A.sub.1]m(t))dt] (A.1)

d[gamma] (t)/dt = 2[gamma](t)[a.sub.1] + [[b.sup.2].sub.1] + [[b.sup.2].sub.2] - [([b.sub.1][B.sub.1] + [b.sub.2][B.sub.2] + [gamma](t)[A.sub.1]).sup.2]/[[B.sup.2].sub.1] + [[B.sup.2].sub.2] (A.2)

with initial Conditions m(0) = E[X (0)\I(0)]= Z(0) and [gamma](0) = Var [X (0)\I(0)]. Even the initial observation might Contain uncertainty, that is, [gamma](0) = [[[sigma].sup.2].sub.y0] [greater than or equal to] 0. We note that [gamma](t) is the solution to an ordinary differential equation, but m(t) is the solution of a stochastic differential equation that depends on [gamma](t).

Proof of Equation 6

We define:

[[zeta].sup.2] = [b.sub.1][B.sub.1] + [b.sub.2][B.sub.2] + [gamma](t)[A.sub.1]/[[B.sup.2].sub.1] + [[B.sup.2].sub.2] (A.3)

When X is a full information value, the drift of X should not depend on m (i.e., [a.sub.1] = 0). Plugging Equation 2 into Equation A.1 provides:

dm =([a.sub.0] + [A.sub.1][[zeta].sup.2](X - m))dt + [[zeta].sup.2][square root][[B.sup.2].sub.1] + [[B.sup.2].sub.2]d[W.sub.3] (A.4)

The revealed variance rate is [[zeta].sup.4]([[B.sup.2].sub.1] + [[B.sup.2].sub.2]). This value is exactly the last term in Equation A.2. Rearranging yields:

[[zeta].sup.4]([[B.sup.2].sub.1] + [[B.sup.2].sub.2]) = [[b.sup.2].sub.1] + [[b.sup.2].sub.2] - d[gamma](t)/dt

Putting all of this together, the total revealed variance to time t is:

Total Revealed Variance = [[[integral].sup.t].sub.0][[zeta].sup.4] ([[B.sup.2].sub.1] + [[B.sup.2].sub.2])ds

= [[[integral].sup.t].sub.0](([[b.sup.2].sub.1] + [[b.sup.2].sub.2] - d[gamma](s)/ds)ds

= [[[integral].sup.t].sub.0](([[b.sup.2].sub.1] + [[b.sup.2].sub.2])ds - ([gamma](t) - [gamma](0)) (A.6

That is, the revealed variance is the variance of the FI value, [v.sup.FI] (0, t) = [[[integral].sup.t].sub.0]([[b.sup.2].sub.1] + [[b.sup.2].sub.2])ds, less the accumulated residual variance, [gamma](t) [gamma](0).

Merton's Debt Value Formula Adjusted for Noise

We state the time t [epsilon] [0,T] debt value when noise is present, D (m(t), F, r, T - t, [[rho].sup.2][[[sigma].sup.2].sub.x]), as,


[Fe.sup.-r(T-t)][phi]([d.sub.2](m,[Fe.sup.-r(T-t)],[[rho].sup.2][[[si gma].sup.2].sub.x](T-t)))+m[phi](-[d.sub.1] (m,[Fe.sup.-r(T-t)],[[rho].sup.2][[[sigma].sup.2].sub.x](T-t))) (A.7)


[d.sub.i](x,y,z)=ln(x/y)+(3/2-i)z/[square root]z, for i = 1,2 (A.8)

and where r is the risk-free rate of interest and [phi] is the normal distribution function.

The Expected Losses for Type 1 and Type 2 Errors

Without additional information as to the FI asset value, the expected loss due to a Type 1 error is,

[EL.sub.1](Z(T)) = [[[integral].sup.[infinity]].sub.F](X(T)- F)g(X(T)\I(T))dX (T)

= m(T)[phi]([e.sub.1](m(T),1,T))- F[phi]([e.sub.2](m(T),1,T)) (A.9)

where g is the lognormal conditional density function and:

[e.sub.i] (m, [beta], T) = [d.sub.i](m, F, [beta]([[rho].sup.2][[[sigma].sup.2].sub.y]T + [[[sigma].sup.2].sub.y0])), for i = 1,2 (A.10)

We note that the residual variance, ([[rho].sup.2][[[sigma].sup.2].sub.y]T + [[[sigma].sup.2].sub.y0]), is a measurement error that we use in the calculation of the expected loss. The expected loss from a Type 2 error is:

[EL.sub.2](Z(T)) = [[[integral].sup.F].sub.0] (F - X(T))g(X(T)\I(T))dx (T)

= F[phi](-[e.sub.2](m(T),1,T))- m(T)[phi](-[e.sub.1] (m(T),1,T)) (A.11)

Calculation of Partial Derivatives

It is convenient to suppress some of the notation. So, for example, [d.sub.i] = [d.sub.i](m/F, [square root][beta]s), where s = [square root][[rho].sup.2][[[sigma].sup.2].sub.y]T + [[[sigma].sup.2].sub.y0]. MB and MC are the marginal benefits and marginal cost functions with arguments suppressed. We use starred superscript to signify that we evaluate these functions at the optimal precision, [[beta].sup.*] (e.g., [d.sub.i], [MB.sup.*], and [MC.sup.*]). The partials for optimal precision are:

[partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.y] = [[rho].sup.4][[beta].sup.*]T(1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])/[s.sup.2](2[alpha] - (1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])) (A.12)

[partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.y0] = [[beta].sup.*](1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])/[s.sup.2](2[alpha] - (1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])) (A.13)

[partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.x] = [(1 - [[rho].sup.2]).sup.2] [[beta].sup.*]T(1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])/[s.sup.2](2[alpha] - (1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])) (A.14)

[partial][[beta].sup.*]/[partial]T = [[beta].sup.*] [[rho].sup.2][[[sigma].sup.2].sub.y](1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])/[s.sup.2](2[alpha] - (1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])) (A.15)

[partial][[beta].sup.*]/[partial]m = -2[square root][[beta].sup.*] [[d.sup.*].sub.2]/ms(2[alpha] - (1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])) (A.16)

[partial][[beta].sup.*]/[partial]F = 2[square root][[beta].sup.*] [[d.sup.*].sub.1]/Fs(2[alpha] - (1 + [[d.sup.*].sub.1][[d.sup.*].sub.2])) (A.17)

The pairwise relations between these partials are:

[partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.y] = [[rho].sup.4]T [partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.y0] (A.18)

[partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.x] = [(1 - [[rho].sup.2]).sup.2]T [partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.y0] (A.19)

[partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.y] = [[rho].sup.4]/[(1 - [[rho].sup.2]).sup.2] [partial][[beta].sup.*]/[partial][[[sigma].sup.2].sub.x] (A.20)

Below, we present the derivation for Equations A.12-A.14. Derivations for Equations A.15-A.17 are similar and are therefore not presented here.

The partials of total standard deviation are:

[partial]s/[partial][[[sigma].sup.2].sub.y0] = 1/2s (A.21)

[partial]s/[partial][[[sigma].sup.2].sub.y] = [[rho].sup.4]T/2s (A.22)

[partial]s/[partial][[[sigma].sup.2].sub.x] = [(1-[[rho].sup.2]).sup.2]T/2s (A.23)

The partial of [[d.sup.*].sub.2] with respect to variance is (where [[sigma].sup.2] is [[[sigma].sup.2].sub.y], [[[sigma].sup.2].sub.y0], or [[[sigma].sup.2].sub.x]):

[partial][[d.sup.*].sub.2]/[partial][[sigma].sup.2] = - [partial]s/[partial][[sigma].sup.2][ln m/F/[[beta].sup.* 1/2][s.sup.2] + 1/2 [[beta].sup.* 1/2]]-1/2 [partial][[beta].sup.*]/2[partial][[sigma].sup.2][ln m/F/[[beta].sup.* 3/2]s + s/2[[beta].sup.* 1/2]] (A.24)

= -[[d.sup.*].sub.1]/s [partial]s/[partial][[sigma].sup.2] - [[d.sup.*].sub.1]/2[[beta].sup.*] [partial][[beta].sup.*]/[partial][[sigma].sup.2] (A.25)

The optimal precision is defined implicitly by the point at which the marginal benefits equal marginal costs (for [[beta].sup.*] [epsilon] (0,1)):

SFn([[d.sup.2].sub.2])/2[[beta].sup.* 1/2] = [alpha][gamma][[beta].sup.*([alpha]-1)] (A.26)

We use implicit differentiation to find the partials of optimal precision. First, we take the partial of Equation A.26 with respect to variance:

Fn([[d.sup.*].sub.2])/2[[beta].sup.* 1/2] [partial]s/[partial][[sigma].sup.2] - sFN([[d.sup.*].sub.2])/4[[beta].sup.* 3/2] [partial][[beta].sup.*]/[partial][[sigma].sup.2] - s[[d.sup.*].sub.2]Fn([[d.sup.*].sub.2])/2[[beta].sup.* 1/2] [partial][[d.sup.*].sub.2]/[partial][[sigma].sup.2] = [alpha][gamma]([alpha] - 1)[[beta].sup.*([alpha]-2)] [partial][[beta].sup.*/[partial][[sigma].sup.2] (A.27)

Next, we substitute in [MB.sup.*], [MC.sup.*], and Equation A.25.

[MB.sup.*]/s [partial]s/[partial][[sigma].sup.2] - [MB.sup.*]/2[[beta].sup.*] [partial][[beta].sup.*]/[partial][[sigma].sup.2] + [[d.sup.*].sub.2][MB.sup.*] [[[d.sup.*].sub.1]/s [partial]s/[partial][[beta].sup.2] + [[d.sup.*].sub.1]/ 2[[beta].sup.*] [partial][[beta].sup.*]/[partial][[sigma].sup.2]] = [alpha]-1/ [[beta].sup.*] [MC.sup.*] [partial][[beta].sup.*]/[partial][[sigma].sup.2] (A.28)

Finally, we recognize that [MB.sup.*] = [MC.sup.*], multiply through by 2[[beta].sup.*]/[MB.sup.*] and collect terms:

((1 - [[d.sup.*].sub.1] [[d.sup.*].sub.2]) + 2([alpha] - 1))[partial][[beta].sup.*]/[partial][[sigma].sup.2] = 2[[beta].sup.*] (1 + [[d.sup.*].sub.1] [[d.sup.*].sub.2])/S [partial]S/[partial][[sigma].sup.2] (A.29)

This is solved for [partial][[beta].sup.*]/[partial][[sigma].sup.2]:

[partial][[beta].sup.*]/[partial][[sigma].sup.2] = [partial]S/[partial][[sigma].sup.2] 2[[beta].sup.*](1 + [[d.sup.*].sub.1] [[d.sup.*].sub.2])/S(2[alpha] - (1 - [[d.sup.*].sub.1] [[d.sup.*].sub.2])) (A.30)

Combining Equation A.30 with Equations A.21-A.23 provides the partial of optimal precision level with respect to the three variance rates, [[[sigma].sup.2].sub.y], [[[sigma].sup.2].sub.y0], and [[[sigma].sup.2].sub.x].

Equations A.18-A.20

Equations A.18-A.20 are a direct result of Equation A.30 and Equations A.21-A.23.

The Value of Total Net Benefits with Two Rounds of Information Acquisition

The expected incremental net benefits from the second round of information acquisition are,

E[[NB.sub.2] ([[beta].sub.1])\[I.sup.IA] (0, T)]= [[integral].sub.[R.sub.2]([[beta].sub.1])] [B(m, [[[beta].sup.*].sub.2] - [[beta].sub.1], T) - C([[beta].sub.1], [[[beta].sup.*].sub.2])]g(m\[I.sup.IA] ([[beta].sub.1], T)) dm (A.31)


[R.sub.2] ([[beta].sub.1]) = {mB (m, [[[beta].sup.*].sub.2] - [[beta].sub.1], T) - C ([[beta].sub.1], [[[beta].sup.*].sub.2])[greater than or equal to] 0} (A.32)

is the interval containing F for which a second round of IA is desirable.

The total net benefits from a first round of IA are the direct net benefits from the first round, plus the expected incremental net benefits from the second round:

B(m(T), [[beta].sub.1], T)- [C.sub.1] ([[beta].sub.1]) + E[[NB.sub.2] ([[beta].sub.1])[I.sup.IA] (0,T)] (A.33)

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Author:Childs, Paul D.; Ott, Steven H.; Riddiough, Timothy J.
Publication:Financial Management
Geographic Code:1USA
Date:Jun 22, 2001
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