Optimal stopping in Levy models, for non-monotone discontinuous payoffs

Boyarchenko, Svetlana and Levendorskii, Sergei (2010): Optimal stopping in Levy models, for non-monotone discontinuous payoffs.

Abstract

We give short proofs of general theorems about optimal entry and exit problems in Levy models, when payoff streams may have discontinuities and be non-monotone. As applications, we consider exit and entry problems in the theory of real options, and an entry problem with an embedded option to exit.

• Item Type: MPRA Paper
• Language: English
• Keywords: optimal stopping, Levy processes, non-monotone discontinuous payoffs
• Subjects: D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis

References

[1] L. Alili, and A. Kyprianou "Some remarks on first passage of Levy process, the American put and pasting principles." Annals of Applied Probability 15 (2005), 2062-2080

[2] S. Assmusen, F. Avram, and M.R. Pistorius "Russian and American put options under exponential phase-type Levy models," Stochastic Processes and Applications, 109 (2004), 79-111.

[3] F. Avram, A.E. Kyprianou and M.R. Pistorius, "Exit problems for spectrally negative Levy processes and applications to (Canadized) Russian options", The Annals of Applied Probability, 14 (2004), 215-238

[4] Back, K. and D. Paulsen, "Open-Loop Equilibria and Perfect Competition in Option Exercise Games," Review of Financial Studies, 22:11 (2009), 4531-4552

[5] S. Boyarchenko, "Irreversible Decisions and Record-Setting News Principles", American Economic Review 94:3 (2004), pp. 557-568

[6] M. Boyarchenko and S.Z. Levendorskii, "Prices and sensitivities of barrier and first-touch digital options in Levy-driven models", International Journal of Theoretical and Applied Finance 12 (2009), pp. 1125-1170

[7] S.I. Boyarchenko and S.Z. Levendorskii, "Option pricing for truncated Levy processes", International Journal of Theoretical and Applied Finance, 3:3 (July 2000), pp. 549-552.

[8] S.I. Boyarchenko and S.Z. Levendorskii, "Perpetual American Options under Levy Processes," SIAM Journal on Control and Optimization, 40 (2002), 1663-1696.

[9] S.I. Boyarchenko and S.Z. Levendorskii, Non-Gaussian Merton-Black-Scholes theory. Singapore: World Scientific 2002.

[10] S.I. Boyarchenko and S.Z. Levendorskii, "American options: the EPV pricing model", Annals of Finance 1:3 (2005), 267-292.

[11] S.I. Boyarchenko and S.Z. Levendorskii, "General Option Exercise Rules, with Applications to Embedded Options and Monopolistic Expansion", Contributions to Theoretical Economics, 6:1 (2006), Article 2

[12] S.I. Boyarchenko and S.Z. Levendorskii, "Irreversible Decisions Under Uncertainty (Optimal Stopping Made Easy)", Springer, Berlin, 2007.

[13] S.I. Boyarchenko and S.Z. Levendorskii, "Practical guide to real options in discrete time", International Economic Review, 48:1 (2007), 275-306.

[14] S.I. Boyarchenko and S.Z. Levendorskii, \Exit Problems in Regime-Switching Models", Journ. of Mathematical Economics 44:2 (2008), 180-206

[15] S.I. Boyarchenko and S.Z. Levendorskii, "Pricing American Options in Regime-Switching Models", SIAM J Control and Optimization, 48:4 (2009), pp.1353-1375

[16] D.A. Darling, T. Ligget and H.M. Taylor, "Optimal Stopping for partial sums", Ann Math. Statistics, 43 (1972), 1363-1368.

[17] G. Deligiannidis, H. Le, and S. Utev, "Optimal Stopping for processes with independent increments, and applications", Journ. Appl. Probability, 46 (2009), 1130-1145.

[18] S. Grenadier, "Game Choices: The Intersection of Real Options and Game Theory," Risk Books, 2000

[19] S. Grenadier, "Option Exercise Games: An Application to the Equilibrium Investment Strategies of Firms," Review of Financial Studies 15:3 (2002), 691-721

[20] T.C. Johnson and M. Zervos, "The explicit solution to a sequential switching problem with non-smooth data," Stochastics: an International Journal of Probability and Stochastic Processes 82:1 (2010), 69-109

[21] A.E. Kyprianou, and B.A. Surya, "On the Novikov-Shiryaev optimal stopping problems in continuous time", Electronic Commun. Prob. 10 (2005), 146-154.

[22] A. Merhi and M. Zervos, "A model for reversible investment capacity expansion," SIAM Journal on Control and Optimization 46:3 (2007), 839-876

[23] E. Mordecki, "Optimal stopping and perpetual options for Levy processes", Finance Stoch. 6 (2002), 473-493.

[24] A.A. Novikov and A.N. Shiryaev, "On an effective case of the solution of the optimal stopping problem for random walks", Theory Prob. Appl. 49 (2005), 344-354.

[25] A.A. Novikov and A.N. Shiryaev, "On a solution of the optimal stopping problem for processes with independent increments", Stochastics 79 (2007), 393-406.

[26] G. Peskir and A. Shirayev, "Optimal Stopping and Free-Boundary Problems (Lectures in Mathematics. ETH Zurich)", Birkhauser, Basel Boston Berlin 2006

[27] F. Riedel, "Optimal stopping with multiple priors", Econometrica, 77:3 (2009), 857-908

[28] F. Riedel and Xia Su, "On Irreversible Investment," Finance and Stochastics, to appear

[29] L.C.G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Volume 1. Foundations, 2nd ed. John Wiley & Sons, Ltd., Chichester, 1994.

[30] M. Sirbu, and S.E. Shreve, "A two person game for pricing convertible bonds," SIAM Journal on Control and Optimization 45:4 (2006), 1508-1539

[31] Sato, Ken-Iti, Levy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press 1999.

[32] B.A. Surya, "An approach for solving perpetual optimal stopping problems driven by Levy processes." Stochastics, 79 (2007), 337-361.

[33] M.D. Whinston, "Exit with multiplant firms." RAND Journal of Economics, 19 (1988), 568-588.

[34] M. Zervos, "A problem of sequential entry and exit decisions combined with discretionary stopping", SIAM Journal on Control and Optimization 42:2 (2003), 397-421 2