How to design down-and-out barrier option contracts so that firms invest when it is socially efficient

Abstract

This paper investigates how to design down-and-out barrier options contracts so as firms invest when it is socially efficient. A government initially offers a firm a privileged right to exercise an investment opportunity that exhibits external benefits to society, but will eliminate this opportunity if its prospects are sufficiently bleak. The firm will invest at the date further away from that is socially efficient if the firm either is less uncertain about the return of the investment or incurs lower investment costs, or the government owns a more valuable knock-out option. Consequently, under these three scenarios the government can efficiently either offer the firm a higher investment tax credit or impose the firm a higher lease fee for holding the option to invest.

Keywords:

• American options
• down-and-out options
• investment tax credits
• irreversible investment
• socially efficient

JEL Classification:

• G13
• R52
• R58

Acknowledgements

The authors would like to thank Professor Chris J. Adcock (the editor), two anonymous referees, Toby Daglish, Min-Teh Yu, and participants of both the 20th Conference on the Theories and Practices of Securities and Financial Markets and the 17th New Zealand Finance Colloquium for their helpful comments on earlier versions of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Barriers options are path-dependent options whose payoffs depend on whether some barriers are reached within the lives of the options. Merton (1973) introduced the first pricing formula for down-and-out perpetual call and put barrier options. The pricing formula for eight types of standard European barrier options was later introduced by Reiner and Rubinstein (1991a, 1991b) who assume that the payoff of a barrier option is zero as soon as the barrier is touched. Zhang (1998) extended Reiner and Rubinstein (1991a, 1991b) by assuming that the payoff of a barrier option is a rebate (rather than zero) once the barrier is breached.

2. See also Jou and Lee (2001) and Saint-Paul (1992), both of which show that the government can use an investment tax subsidy to incorporate the externality benefits.

3. A government may implement regulatory policies that prohibit firms from investing at particular states of nature. For instance, if the government imposes a sufficiently stringent price ceiling control (e.g. Dixit 1991; Dixit and Pindyck 1994), then firms will be indirectly restricted from investing at a state of nature that results in an output price that exceeds the imposed price ceiling.

4. The VLSI Technology Development Project was the Phase III project launched by the ERSO. For a detailed discussion of this project, see Liu (1993) for details.

5. In the absence of any barrier option, our model reduces to the situation addressed in McDonald and Siegel (1986). McDonald and Siegel demonstrate that the interaction of uncertainty and investment irreversibility causes a firm to refrain from engaging in an investment project until the investment value exceeds the investment cost by a magnitude that is related to the (call) option value from delaying the investment.

6. Adkins and Paxson (2013) show that Tourinho abstracts from the convenience yield from holding the underlying asset of the option to invest, and thus Tourinho introduces the holding cost so as to depart from the trivial result indicating that uncertainty does not affect a firm's irreversible investment decision.

7. Note that the values of and in Equations (9) and (10) only apply to the region in which If , then we must impose the condition that ; in other words, the net option value to delay investment is worthless if the investment value is below its knock-out level.

8. We use to denote the efficient level of the project value that triggers the social planner to invest, that is, the choice of the investment timing that incorporates the positive external effect.

9. As shown in Figure2, the knock-out value (in the absolute value) is decreasing from 0.01628 to 0 when is increased from 0 to 0.0025. As a result, we must impose the lease fee to be lower than 0.0025 to ensure that the regulator has the obligation to knock out a firm's opportunity to undertake the investment.

10. As indicated by the signs of both Equations (B5) and (B13), both the investment tax credit and the lease fee are substitutable with each other. This suggests that if the government implements only the lease fee, it might take up ‘dead beat’ projects that private industry will not invest when the investment value is sufficiently low.

11. This type of policy contrasts with the policy that was discussed by Hasset and Metcalf (1999) in which a firm undertakes investment but faces policy uncertainty regarding the investment tax credit. They show that, if a government wishes to accelerate investment, then the government should enact a tax credit immediately, threaten to remove it in the near future, and commit to never restoring this credit (see Dixit and Pindyck 1994, 309).

12. See also the approximate analytical formulas that are provided by Carr (1995) and by Geske and Johnson (1984).

13. Our article focuses on how government policies and other macroeconomic variables may impact the investment behavior of a firm by affecting its real option value to delay the investment; which is analogous to a financial call option value. By contrast, a local project which may be ‘insured’ by super sovereign entities, is analogues to a financial ‘protective put’ option value, and has a payoff pattern similar to a financial call option. This research topic deserves further investigation as it involves greater uncertainty regarding if a local government can successfully obtain financial aid from super sovereign entities when necessary; that is, the knock-out level of the project value which triggers the investment project to be taken over, may also be stochastic.

Additional information

Funding

The financial support under grant 103R001-25 from the College of Social Sciences, National Taiwan University, is gratefully acknowledged.