Exit problems in regime-switching models.

*(English)*Zbl 1151.91485Summary: This paper provides a general framework for pricing of perpetual American and real options in regime-switching Lévy models. In each state of the Markov chain, which determines switches from one Lévy process to another, the payoff stream is a monotone function of the Lévy process labeled by the state. This allows for additional switching within each state of the Markov chain (payoffs can be different in different regions of the real line). The pricing procedure is efficient even if the number of states is large provided the transition rates are not very large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modeled as finite-state Markov chains. As an application, we solve exit problems for a price-taking firm, and study the dependence of the exit threshold on the interest rate uncertainty.

##### MSC:

91G20 | Derivative securities (option pricing, hedging, etc.) |

PDF
BibTeX
XML
Cite

\textit{S. Boyarchenko} and \textit{S. Levendorskiĭ}, J. Math. Econ. 44, No. 2, 180--206 (2008; Zbl 1151.91485)

Full Text:
DOI

##### References:

[1] | Abel, A.B.; Eberly, J.C., Optimal investment with costly reversibility, Review of economic studies, 63, 581-593, (1996) · Zbl 0864.90011 |

[2] | Alili, L.; Kyprianou, A.E., Some remarks on first passage of Lévy processes, the American put and pasting principles, Annals of applied probability, 15, 2062-2080, (2005) · Zbl 1083.60034 |

[3] | Alvarez, L.H.R.; Steinbacka, R., Adoption of uncertain multi-stage technology projects: a real options approach, Journal of mathematical economics, 35, 71-97, (2001) · Zbl 0970.91030 |

[4] | Asmussen, S.; Avram, F.; Pistorius, M.R., Russian and American put options under exponential phase-type Lévy models, Stochastic processes and their applications, 109, 79-111, (2004) · Zbl 1075.60037 |

[5] | Avram, F.; Kyprianou, A.E.; Pistorius, M.R., Exit problems for spectrally negative Lévy processes and applications to (canadized) Russian options, Annals of applied probability, 14, 215-238, (2004) · Zbl 1042.60023 |

[6] | Bakshi, G.; Cao, C.; Chen, Z., Empirical performance of alternative option pricing models, Journal of finance, 52, 2003-2049, (1997) |

[7] | Boyarchenko, S.I., Irreversible decisions and record setting news principles, American economic review, 94, 557-568, (2004) |

[8] | Boyarchenko, S.I.; Levendorskiĭ, S.Z., Perpetual American options under Lévy processes, SIAM journal on control and optimization, 40, 1663-1696, (2002) · Zbl 1025.60021 |

[9] | Boyarchenko, S.I.; Levendorskiĭ, S.Z., Non-Gaussian Merton-Black-Scholes theory, (2002), World Scientific · Zbl 0997.91031 |

[10] | Boyarchenko, S.I.; Levendorskiĭ, S.Z., American options: the EPV pricing model, Annals of finance, 1, 267-292, (2005) · Zbl 1233.91258 |

[11] | Boyarchenko, S.I., Levendorskiĭ, S.Z., 2006. General option exercise rules, with applications to embedded options and monopolistic expansion. Contributions to Theoretical Economics 6 (1) (Article 2). http://www.bepress.com/bejte/contributions/vol6/iss1/art2. |

[12] | Boyarchenko, S.I.; Levendorskiĭ, S.Z., Practical guide to real options in discrete time, International economic review, 48, 275-306, (2007) |

[13] | Buffington, J.; Elliott, R.J., American options with regime switching, International journal of theoretical and applied finance, 5, 497-514, (2002) · Zbl 1107.91325 |

[14] | Casassus, J., Collin-Dufresne, P., Routledge, B.R., 2005. Equilibrium Commodity Prices with Irreversible Investment and Non-linear Technology. National Bureau of Economic Research, Inc., NBER Working Papers: 11864. |

[15] | Chade, H.; Taub, B., Stable coalitions in a continuous-time model of risk sharing, Mathematical social sciences, 50, 24-38, (2005) · Zbl 1099.91073 |

[16] | Chernov, M.; Gallant, A.R.; Chysels, E.; Tauchen, G., Alternative models for stock price dynamics, Journal of econometrics, 116, 225-257, (2003) · Zbl 1043.62087 |

[17] | Dai, Q., Singleton, K.J., Yang, W., 2007. Regime shifts in a dynamic term structure model of U.S. treasury bond yields, Review of Financial Studies 20, 1669-1706. |

[18] | Davig, T., Regime switching debt and taxation, Journal of monetary economics, 51, 837-859, (2004) |

[19] | Dixit, A.K.; Pindyck, R.S., Investment under uncertainty, (1996), Princeton University Press |

[20] | Duffie, D.; Pan, J.; Singleton, K., Transform analysis and asset pricing for affine jump diffusions, Econometrica, 68, 1343-1376, (2000) · Zbl 1055.91524 |

[21] | Grenadier, S.R.; Weiss, M.S., Investment in technological innovations: an option pricing approach, Journal of financial economics, 44, 397-416, (1997) |

[22] | Grenadier, S.R., Game choices: the intersection of real options and games, (2000), Risk Books |

[23] | Grenadier, S.R., Option exercise games: an application to the equilibrium investment strategies of firms, Review of financial studies, 15, 691-721, (2002) |

[24] | Guo, X., An explicit solution to an optimal stopping problem with regime switching, Journal of applied probability, 38, 464-481, (2001) · Zbl 0988.60038 |

[25] | Guo, X.; Miao, J.; Morellec, E., Irreversible investment with regime shifts, Journal of economic theory, 122, 37-59, (2005) · Zbl 1118.91049 |

[26] | Guo, X.; Zhang, Q., Closed form solutions for perpetual American put options with regime switching, SIAM journal of applied mathematics, 64, 2034-2049, (2004) · Zbl 1061.90082 |

[27] | Guthrie, G., Regulating infrastructure: the impact on risk and investment, Journal of economic literature, 54, 925-972, (2006) |

[28] | Hamilton, J., A new approach to the economic analysis of non-stationary time series and the business cycle, Econometrica, 57, 357-384, (1989) · Zbl 0685.62092 |

[29] | Hsu, J.C., Schwartz, E.S., 2003. A model of R&D valuation and the design of research incentives. National Bureau of Economic Research, Inc., NBER Working Papers: 10041. |

[30] | Kou, S.G., A jump-diffusion model for option pricing, Management science, 48, 1086-1101, (2002) · Zbl 1216.91039 |

[31] | Kyprianou, A.E.; Pistorius, M.R., Perpetual options and canadization through fluctuation theory, Annals of applied probability, 13, 1077-1098, (2003) · Zbl 1039.60044 |

[32] | Levendorskiĭ, S.Z., Pricing of the American put under Lévy processes, International journal of theoretical and applied finance, 7, 303-336, (2004) · Zbl 1107.91050 |

[33] | Mauer, D.C.; Ott, S.H., The effect of growth options to expand, (), 151-182 |

[34] | Miltersen, K.R.; Schwartz, E.S., Pricing of options on commodity futures with stochastic term structures of convenience yields and interest rates, Journal of financial and quantitative analysis, 33, 33-59, (1998) |

[35] | Rogers, L.C.G.; Williams, D., Diffusions, Markov processes and martingales, (2000), Cambridge University Press · Zbl 0977.60005 |

[36] | Sarantis, N.; Piard, S., Credibility, macroeconomic fundamentals and Markov regime switching in the EMS, Scottish journal of political economy, 51, 453-476, (2004) |

[37] | Sato, K., Lévy processes and infinitely divisible distributions, (1999), Cambridge University Press · Zbl 0973.60001 |

[38] | Schulmerich, M., Real options valuation, (2005), Springer · Zbl 1085.91030 |

[39] | Smit, H.T.J.; Trigeorgis, L., Strategic investment, (2004), Princeton University Press |

[40] | Taub, B.; Chade, H., Segmented risk sharing in a continuous time setting, Economic theory, 20, 645-675, (2002) · Zbl 1037.91061 |

[41] | Trigeorgis, L., Real options: managerial flexibility and strategy in resource allocation, (1996), MIT Press |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.