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Real options problem with nonsmooth obstacle. (English) Zbl 1480.91301

Summary: We consider a real options problem, which is posed as a stochastic optimal control problem. The investment strategy, which plays the role of control, involves a one-time option to expand (invest) and a one-time option to abandon (terminate) the project. The timing and amount of the investment and the termination time are parameters to be optimized in order to maximize the expected value of the profit. This stochastic optimization problem amounts to solving a deterministic variational inequality in dimension one, with the associated obstacle problem. Because we consider both cessation and expansion options and fixed and variable costs of expansion, the obstacle is nonsmooth. Due to the lack of smoothness, we use the concept of a weak solution. However, such solutions may not lead to a straightforward investment strategy. Therefore, we further consider strong (\(C^1\)) solutions based on thresholds. We propose sufficient conditions for the existence of such solutions to the variational inequality with a non-smooth obstacle in dimension one. When applied to the real options problem, these sufficient conditions yield a simple optimal investment strategy with the stopping times defined in terms of the thresholds.

MSC:

91G50 Corporate finance (dividends, real options, etc.)
93E20 Optimal stochastic control
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[1] L. H. R. Alvarez, On the properties of r-excessive mappings for a class of diffusions, Ann. Appl. Probab., 13 (2003), pp. 1517-1533. · Zbl 1072.60065
[2] L. H. R. Alvarez, A class of solvable impulse control problems, Appl. Math. Optim., 49 (2004), pp. 265-295. · Zbl 1060.93105
[3] L. H. R. Alvarez and J. Lempa, On the optimal stochastic impulse control of linear diffusions, SIAM J. Control Optim., 47 (2008), pp. 703-732, https://doi.org/10.1137/060659375. · Zbl 1157.49040
[4] A. Bensoussan and B. Chevalier-Roignant, Sequential capacity expansion options, Oper. Res., 67 (2018), pp. 33-57. · Zbl 1455.91268
[5] A. Bensoussan, B. Chevalier-Roignant, and A. Rivera, Interactions between Real Capacity Expansion and Shutdown Options: An Application to Renewable Energies, Tech. rep., SSRN, 2020.
[6] A. Bensoussan, B. Chevalier-Roignant, and A. Rivera, Does performance-sensitive debt mitigate debt overhang?, J. Econom. Dynam. Control, 131 (2021), 104203. · Zbl 1475.91394
[7] A. Bensoussan and J. Lions, Applications of Variational Inequalities in Stochastic Control, North-Holland, New York, 1982. · Zbl 0478.49002
[8] A. Bensoussan and J. Lions, Impulse Control and Quasi-Variational Inequalities, Gauthiers-Villars, Paris, France, 1984.
[9] A. Borodin and P. Salminen, Handbook on Brownian Motion: Facts and Formulae, 2nd ed., Birkhäuser, Basel, 2002. · Zbl 1012.60003
[10] A. Dixit and R. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1994.
[11] J. Duckworth and M. Zervos, An investment model with entry and exit decisions, J. Appl. Probab., 37 (2000), pp. 547-559. · Zbl 0959.93058
[12] K. Duckworth and M. Zervos, A model for investment decisions with switching costs, Ann. Appl. Probab., 11 (2001), pp. 239-260. · Zbl 1083.91055
[13] E. Dynkin, Markov Processes, Vol. 2, Springer-Verlag, Berlin, 1965. · Zbl 0132.37901
[14] J. Harrison, T. Sellke, and A. Taylor, Impulse control of Brownian motion, Math. Oper. Res., 8 (1983), pp. 454-466. · Zbl 0526.93066
[15] H. Leland, Corporate debt value, bond covenants, and optimal capital structure, J. Finance, 49 (1994), pp. 1213-1252.
[16] H. Morimoto, A singular control problem with discretionary stopping for geometric Brownian motions, SIAM J. Control Optim., 48 (2010), pp. 3781-3804, https://doi.org/10.1137/080734856. · Zbl 1205.93168
[17] A. Reppen, J.-C. Rochet, and H. Soner, Optimal dividend policies with random profitability, Math. Finance, 30 (2018), pp. 228-259. · Zbl 1508.91483
[18] L. Trigeorgis, Real options: Managerial Flexibility and Strategy in Resource Allocation, MIT Press, 1996.
[19] M. Zervos, A problem of sequential entry and exit decisions combined with discretionary stopping, SIAM J. Control Optim., 42 (2003), pp. 397-421, https://doi.org/10.1137/S036301290038111X. · Zbl 1037.93079
[20] M. Zervos, C. Oliveira, and K. Duckworth, An investment model with switching costs and the option to abandon, Math. Methods Oper. Res., 88 (2018), pp. 417-443. · Zbl 1411.91611
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