Real options with a double continuation region

Abstract

If the average risk-adjusted growth rate of the project's present value V overcomes the discount rate but is dominated by the average risk-adjusted growth rate of the cost I of entering the project, a non-standard double continuation region can arise: The firm waits to invest in the project if V is insufficiently above I as well as if V is comfortably above I. Under a framework with diffusive uncertainty, we give exact characterization to the value of the option to invest, to the structure of the double continuation region, and to the subset of the primitives' values that support such a region.

Keywords:

• Asset pricing
• American options
• Capital investment theory
• Optimal stopping
• Free boundary

Acknowledgements

The authors acknowledge the valuable and detailed suggestions of an anonymous referee. The authors also thank Nicholas Barberis, Pauline Barrieu, Emilio Barucci, Andrea Berardi, Oleg Bondarenko, Nicole Branger, Andrea Buraschi, Francesco Corielli, Alessandra Cretarola, Enrico De Giorgi, Francesco Franzoni, Xavier Gabaix, Andrea Gamba, Jens Jackwerth, Christoph Kühn, Damien Lamberton, Abraham Lioui, Jan Krahnen, Piera Mazzoleni, Fausto Mignanego, Fulvio Ortu, Martijn Pistorius, Maurizio Pratelli, Frank Riedel, Christian Schlag, Claudio Tebaldi, Fabio Trojani, Stéphane Villeneuve, and the participants at the 5th World Congress of the Bachelier Finance Society (Imperial College, London), the 2008 Conference on Numerical Methods in Finance (University of Udine), the evening sessions of the 2008 European Summer Symposium in Financial Markets (Gerzensee), the 2007 Finance Seminar Series of the Goethe University Frankfurt am Main, the 30th AMASES Meetings (University of Trieste), the 8th Workshop on Quantitative Finance (University of Venice), and the 2007 Meeting of the Swiss Society of Economics and Statistics (University of St Gallen). Previous versions of the present work circulated under the titles ‘The double continuation region’ (http://ssrn.com/abstract=1018624) and ‘Closed-form optimal investment when present values and costs are jump-diffusions (http://ssrn.com/abstract=968172). Any remaining error is our sole responsibility. Alessandro Sbuelz is also associated with CAREFIN at Bocconi University.

Notes

†Although they do not explicitly appear, the correlations between V and I and their local volatilities contribute towards defining the drifts via the price(s) of risk and the associated risk adjustment.

†Conditioning the expectation upon ℱ₀ is intended to stress the dependence of the resulting quantities on the initial values for V and I.

‡Aase (2005) highlights the key role played by risk adjustment for an American perpetual option pricing problem, and works out exact solutions when jump sizes cannot be negative. He investigates when his solution is an approximation also for negative jumps.

§Broadie and Detemple (1997) and Detemple (2006) have shown that a single continuation region surrounded by multiple immediate-exercise regions arises with a variety of finite-maturity American derivatives written on two or more traded risky underlying assets. Battauz et al. (2009) investigate the double continuation region in a number of finite-maturity American problems.

†See, for example, Geman et al. (1995) and Battauz (2002) for a comprehensive application of change-of-numeraire techniques to American options.

†For an extensive review of valuation methods for American-style claims, see, for example, Broadie and Detemple (2004).

†Notice that a > 0 implies k _I  − k _V  > 0.

†The negative exponents ξ₁ and ξ₂ guarantee that the investment option value expressed in equation (11) remains increasing in V and decreasing in I also in the continuation region.

‡The project's predictable demise is considered in the real-options analysis of Magis and Sbuelz (2006).

†These properties are four of the five optimality properties Mordecki (1999) quotes in his lemma 1. It is worth remarking that Mordecki (1999) considers solely a zero ‘deflating rate’. We state the four properties in terms of the cost-to-value ratio X instead of lnX in order to save the convexity property of the put-payoff function. This allows us to apply the Meyer–Ito formula and minor modifications of its corollaries (Protter 2004). The missing property is a boundedness requirement, which is not satisfied in our case.