Abstract
We analyze the determination of a value maximizing dividend payout policy for a broad class of cash reserve processes modeled as spectrally negative jump diffusions. We extend previous results based on continuous diffusion models and characterize the value of the optimal dividend distribution strategy explicitly. We also characterize explicitly the values as well as the optimal dividend thresholds for a class of associated optimal liquidation and sequential lump sum dividend control problems. Our results indicate that both the value as well as the marginal value of the optimal policies are increasing functions of policy flexibility in the discontinuous setting as well.
Similar content being viewed by others
References
Alvarez LHR (2001) Reward functions, salvage values and optimal stopping. Math Methods Oper Res 54: 315–337
Alvarez LHR (2004) A class of solvable impulse control problems. Appl Math Optim 49: 265–295
Alvarez LHR, Rakkolainen T (2006) A class of solvable optimal stopping problems of spectrally negative jump diffusions, Aboa Centre for Economics, Discussion Paper No. 9
Alvarez LHR, Virtanen J (2006) A class of solvable stochastic dividend optimization problems: on the general impact of flexibility on valuation. Econ Theory 28: 373–398
Alili L, Kyprianou A (2005) Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann Appl Probab 15(3): 2062–2080
Avram F, Palmowski Z, Pistorius M (2007) On the optimal dividend problem for a spectrally negative Lévy process. Ann Appl Probab 17(1): 156–180
Azcue P, Muler N (2005) Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math Finance 15: 261–308
Bar-Ilan A, Perry D, Stadje W (2004) A generalized impulse control model of cash management. J Econ Dyn Control 28: 1013–1033
Bayraktar E, Egami M (2008) Optimizing venture capital investments in a jump diffusion model. Math Methods Oper Res 67(1): 21–42
Bernanke BS (1983) Irreversibility, uncertainty, and cyclical investment. Q J Econ 98(1): 85–103
Bertoin J (1996) Lévy processes. Cambridge University Press, Cambridge
Borodin A, Salminen P (2002) Handbook on Brownian motion–facts and formulae, 2nd edn. Birkhauser, Basel
Boyarchenko S (2004) Irreversible decisions and record-setting news principles. Am Econ Rev 94: 557–568
Boyarchenko S, Levendorskiĭ S (2000) Option pricing for truncated Lévy processes. Int J Theor Appl Finance 3(3): 549–552
Boyarchenko S, Levendorskiĭ S (2002) Perpetual American options under Lévy processes. SIAM J Control Optim 40(6): 1663–1696
Boyarchenko S, Levendorskiĭ S (2005) American options: the EPV pricing model. Ann Finance 1: 267–292
Boyarchenko S, Levendorskiĭ S (2006) General option exercise rules, with applications to embedded options and monopolistic expansion. Contrib Theor Econ 6:1 (article 2)
Boyarchenko S, Levendorskiĭ S (2007a) Optimal stopping made easy. Math Econ 43(2): 201–217
Boyarchenko S, Levendorskiĭ S (2007b) Practical guide to real options in discrete time. Int Econ Rev 48(1):275–306
Boyarchenko S, Levendorskiĭ S (2007) Irreversible decisions under uncertainty. Optimal stopping made easy. Springer, Berlin
Chan T, Kyprianou A (2006) Smoothness of scale functions for spectrally negative Lévy processes (preprint). http://www.maths.bath.ac.uk/~ak257/pubs.html
Dassios A, Embrechts P (1989) Martingales and insurance risk. Stoch Models 5: 181–217
Duffie D, Pan J, Singleton K (2000) Transform analysis and asset pricing for affine jump diffusions. Econometrica 68: 1343–1376
Gerber H, Landry B (1998) On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insur Math Econ 22: 263–276
Gerber H, Shiu E (1998) Pricing perpetual options for jump processes. North Am Actuar J 2: 101–112
Gerber H, Shiu E (2004) Optimal dividends analysis with Brownian motion. North Am Actuar J 8: 1–20
Jacob N, Schilling R et al (2001) Lévy type processes and pseudodifferential operators. In: Barndorff-Nielsen O(eds) Lévy processes: theory and applications. Birkhäuser, Basel, pp 139–167
Kyprianou A, Palmowski Z (2007) Distributional study of De Finetti’s dividend problem for a general Lévy insurance risk process. J Appl Probab 44: 428–443
Loeffen R (2008) On the optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann Appl Probab (to appear)
Miller M, Modigliani F (1961) Dividend policy, growth, and the valuation of shares. J Bus 34: 411–433
Mordecki E (2002a) Perpetual options for Lévy processes in the Bachelier model. Proc Steklov Math Inst 237: 256–264
Mordecki E (2002b) Optimal stopping and perpetual options for Lévy processes. Finance Stoch 6(4): 473–493
Mordecki E, Salminen P (2007) Optimal stopping of Hunt and Lévy processes. Stochastics 79: 233–251
Perry D, Stadje W (2000) Risk analysis for a stochastic cash management model with two types of customers. Insur Math Econ 26: 25–36
Peskir G, Shiryaev A (2006) Optimal stopping and free-boundary problems. Birkhäuser, Basel
Protter P (2004) Stochastic integration and differential equations, 2nd edn. Springer, Heidelberg
Rakkolainen T (2007) A class of solvable Dirichlet problems associated to spectrally negative jump diffusions (preprint)
Schmidli H (2006) Optimisation in non-life insurance. Stoch Models 22(4): 689–722
Schmidli H (2008) Stochastic control in insur‘ance. Springer, Heidelberg
Shreve SE, Lehoczky JP, Gaver DP (1984) Optimal consumption for general diffusions with absorbing and reflecting boundaries. SIAM J Control Optim 22: 55–75
Taksar M (2000) Optimal risk and dividend distribution control models for an insurance company. Math Methods Oper Res 51: 1–42
Øksendal B (2003) Stochastic differential equations. An introduction with applications 6th edn. Springer, Heidelberg
Øksendal B, Sulem A (2005) Applied stochastic control of jump diffusions. Springer, Heidelberg
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alvarez, L.H.R., Rakkolainen, T.A. Optimal payout policy in presence of downside risk. Math Meth Oper Res 69, 27–58 (2009). https://doi.org/10.1007/s00186-008-0228-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-008-0228-7
Keywords
- Dividend optimization
- Downside risk
- Impulse control
- Jump diffusion
- Optimal stopping
- Singular stochastic control