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Pricing credit-risky bonds and spread options modelling credit-spread term structures with two-dimensional Markov-modulated jump-diffusion. (English) Zbl 1468.91163

Summary: The relationship between company hazard rates and the business cycle becomes more apparent after a financial crisis. To address this relationship, a regime-switching process with an intensity function is adopted in this paper. In addition, the dynamics of both interest rates and asset values are modelled with a Markov-modulated jump-diffusion model, and a 2-factor hazard rate model is also considered. Based on this more suitable model setting, a closed-form model of pricing risky bonds is derived. The difference in yield between a risky bond and risk-free zero coupon bond is used to model a term structure of credit spreads (CSs) from which a closed-form pricing model of a call option on CSs is obtained. In addition, the degree to which the explicit regime shift affects CSs and credit-risky bond prices is numerically examined using three forward-rate functions under various business-cycle patterns.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
91G40 Credit risk
60J74 Jump processes on discrete state spaces
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[1] Abudy, M. and Izhakian, Y., Pricing stock options with stochastic interest rate. Int. J. Portfol. Anal. Manage., 2013, 1, 250-277. · doi:10.1504/IJPAM.2013.054408
[2] A’Hearn, B. and Woitek, U., More international evidence on the historical properties of business cycles. J. Monet. Econ., 2001, 47, 321-346.10.1016/S0304-3932(01)00045-9 · doi:10.1016/S0304-3932(01)00045-9
[3] Bo, L., Wang, Y. and Yang, X., Markov-modulated jump-diffusions for currency option pricing. Insur: Math. Econ., 2010, 46, 461-469. · Zbl 1231.91425 · doi:10.1016/j.insmatheco.2010.01.003
[4] Bollen, N.P.B., Gray, S.F. and Whaley, R.E., Regime switching in foreign exchange rates: Evidence from currency option prices. J. Economet., 2000, 94, 239-276.10.1016/S0304-4076(99)00022-6 · Zbl 0970.62072 · doi:10.1016/S0304-4076(99)00022-6
[5] Chang, C., Fuh, C.D. and Lin, S.K., A tale of two regimes: Theory and empirical evidence for a Markov-modulated jump diffusion model of equity returns and derivative pricing implications. J. Bank. Finance, 2013, 37, 3204-3217.10.1016/j.jbankfin.2013.03.009 · doi:10.1016/j.jbankfin.2013.03.009
[6] Chiarella, C. and Sklibosios, C.N., A class of jump-diffusion bond pricing models within the HJM framework. Asia-Pacific Financ. Markets, 2003, 10, 87-127.10.1007/s10690-005-6006-0 · Zbl 1137.91438 · doi:10.1007/s10690-005-6006-0
[7] Chiarella, C., Sklibosios, C.N. and Schlogl, E., A Markovian defaultable term structure model with state dependent volatilities. Int. J. Theor. Appl. Finance, 2007, 10, 155-202.10.1142/S0219024907004147 · Zbl 1291.91220 · doi:10.1142/S0219024907004147
[8] Chiarella, C., Sklibosios, C. N. and Schlogl, E., Defaultable bond pricing within a hidden Markov HJM term structure class of models. Working Paper, School of Finance and Economics University of Technology, 2010.
[9] Choi, K. and Hammoudeh, S., Volatility behavior of oil, industrial commodity and stock markets in a regime-switching environment. Energy Policy, 2010, 38, 4388-4399.10.1016/j.enpol.2010.03.067 · doi:10.1016/j.enpol.2010.03.067
[10] Cox, J.C., Ingersoll, J.E. Jr. and Ross, S.A., A theory of the term structure of interest rates. Econometrica, 1985, 53, 385-407.10.2307/1911242 · Zbl 1274.91447 · doi:10.2307/1911242
[11] Das, S.R., The surprise element: Jumps in interest rates. J. Economet., 2002, 106, 27-65.10.1016/S0304-4076(01)00085-9 · Zbl 1051.62106 · doi:10.1016/S0304-4076(01)00085-9
[12] Duan, J.C., Ritchken, P.H. and Sun, Z., Approximating Garch-jump models, jump-diffusion processes, and option pricing. Math. Finance, 2006, 16, 21-52.10.1111/MAFI.2006.16.issue-1 · Zbl 1136.91427 · doi:10.1111/MAFI.2006.16.issue-1
[13] Duffee, G.R., On measuring credit risks of derivative instruments. J. Bank. Finance, 1996, 20, 805-833.10.1016/0378-4266(95)00030-5 · doi:10.1016/0378-4266(95)00030-5
[14] Duffie, D. and Singleton, K.J., Modeling term structures of defaultable bonds. Rev. Financ. Stud., 1999, 12, 687-720.10.1093/rfs/12.4.687 · doi:10.1093/rfs/12.4.687
[15] Elliott, R.J., Chan, L. and Siu, T.K., Option pricing and Esscher transform under regime switching. Ann. Finance, 2005, 1, 423-432.10.1007/s10436-005-0013-z · Zbl 1233.91270 · doi:10.1007/s10436-005-0013-z
[16] Elliott, R.J. and Siu, T.K., On Markov‐modulated exponential‐affine bond price formulae. Appl. Math. Finance, 2009, 16, 1-15.10.1080/13504860802015744 · Zbl 1169.91342
[17] Elliott, R.J., Siu, T.K., Chan, L. and Lau, J.W., Pricing options under a generalized Markov-modulated jump-diffusion model. Stoch. Anal. Appl., 2007, 25, 821-843.10.1080/07362990701420118 · Zbl 1155.91380
[18] Gerber, H.U. and Shiu, E.S.W., Option pricing by Esscher transform. Trans. Soc. Actuaries, 1994, 46, 99-191.
[19] Glasserman, P. and Kou, S.G., The term structure of simple forward rates with jump risk. Math. Finance, 2003, 13, 383-410.10.1111/mafi.2003.13.issue-3 · Zbl 1087.91024 · doi:10.1111/mafi.2003.13.issue-3
[20] Hackbarth, D., Miao, J. and Morellec, E., Capital structure, credit risk, and macroeconomic conditions. J. Financ. Econ., 2006, 82, 519-550.10.1016/j.jfineco.2005.10.003 · doi:10.1016/j.jfineco.2005.10.003
[21] Hamilton, J., A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 1989, 57, 357-384.10.2307/1912559 · Zbl 0685.62092 · doi:10.2307/1912559
[22] Heath, D., Jarrow, R. and Morton, A., Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica, 1992, 60, 77-105.10.2307/2951677 · Zbl 0751.90009 · doi:10.2307/2951677
[23] Hsu, P.P. and Chen, Y.H., Transforming one-factor mean reversion interest rate model into HJM. Int. Res. J. Appl. Finance, 2012, 3, 617-625.
[24] Hurd, T.R., Credit risk modeling using time-changed Brownian motion. Int. J. Theor. Appl. Finance, 2009, 12, 1213-1230.10.1142/S0219024909005646 · Zbl 1182.91188 · doi:10.1142/S0219024909005646
[25] Ishijima, H. and Kihara, T. Option pricing with hidden Markov models. Quantitative Methods in Finance 2005 Conference, Manly Pacific Sydney Hotel, December 14-17, 2005.
[26] Jarrow, R.A., Lando, D. and Turnbull, S.M., A Markov model for the term structure of credit risk spreads. Rev. Financ. Stud., 1997, 10, 481-523.10.1093/rfs/10.2.481 · doi:10.1093/rfs/10.2.481
[27] Johnson, G. and Schneeweis, T., Jump-diffusion processes in the foreign exchange markets and the release of macroeconomic news. Comput. Econ., 1994, 7, 309-329.10.1007/BF01299458 · Zbl 0824.90028 · doi:10.1007/BF01299458
[28] Kallsen, J. and Shiryaev, A.N., The cumulant process and Esscher’s change of measure. Finance Stoch., 2002, 6, 397-428.10.1007/s007800200069 · Zbl 1035.60042 · doi:10.1007/s007800200069
[29] Kijima, M. and Komoribayashi, K., A Markov chain model for valuing credit risk derivatives. J. Deriv., 1998, 6, 97-108.10.3905/jod.1998.408006 · doi:10.3905/jod.1998.408006
[30] Kodera, E., A Markov chain model with stochastic default rate for valuation of credit spreads. J. Deriv., 2001, 8, 8-18.10.3905/jod.2001.319159 · doi:10.3905/jod.2001.319159
[31] Lando, D., On Cox processes and credit risky securities. Rev. Deriv. Res., 1998, 2, 99-120. · Zbl 1274.91459 · doi:10.1007/BF01531332
[32] Liang, L.Z.J., Lemmens, D. and Tempere, J., Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model. Eur. Phys. J. B, 2010, 75, 335-342.10.1140/epjb/e2010-00109-3 · Zbl 1202.91321 · doi:10.1140/epjb/e2010-00109-3
[33] Liew, C.C. and Siu, T.K., A hidden Markov regime-switching model for option valuation. Insur.: Math. Econ., 2010, 47, 374-384. · Zbl 1231.91443 · doi:10.1016/j.insmatheco.2010.08.003
[34] Longstaff, F.A. and Schwartz, E.S., Valuing credit derivatives. J. Fixed Income, 1995, 5, 6-12.10.3905/jfi.1995.408138 · doi:10.3905/jfi.1995.408138
[35] Madan, D. and Unal, H., A two-factor hazard rate model for pricing risky debt and the term structure of credit spreads. J. Financ. Quant. Anal., 2000, 35, 43-65.10.2307/2676238 · doi:10.2307/2676238
[36] Merton, R.C., Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 1976, 3, 125-144.10.1016/0304-405X(76)90022-2 · Zbl 1131.91344 · doi:10.1016/0304-405X(76)90022-2
[37] Rabinovitch, R., Pricing stock and bond options when the default-free rate is Stochastic. J. Financ. Quant. Anal., 1989, 24, 447-457.10.2307/2330978 · doi:10.2307/2330978
[38] Raible, S., Lévy processes in finance: Theory, numerics, and empirical facts. Unpublished doctoral dissertation, University of Freiburg, 2000. · Zbl 0966.60044
[39] Schmid, B. and Kalemanova, A., Applying a three-factor defaultable term structure model to the pricing of credit default options. Int. Rev. Financ. Anal., 2002, 11, 139-158.10.1016/S1057-5219(02)00072-8 · doi:10.1016/S1057-5219(02)00072-8
[40] Schoutens, W., Lévy Processes in Finance, 2003 (Wiley Series in Probability and Statistics, John Wiley: New York).10.1002/0470870230 · doi:10.1002/0470870230
[41] Shirakawa, H., Interest rate option pricing with Poisson-Gaussian forward rate curve processes. Math. Finance, 1991, 1, 77-94.10.1111/mafi.1991.1.issue-4 · Zbl 0900.90107 · doi:10.1111/mafi.1991.1.issue-4
[42] Siu, T. K., Erlwein, C. and Mamon, R. S., The pricing of credit default swaps under a Markov-Modulated Merton’s structural model. N. Am. Actuar. J., 2008, 12, 18-46.10.1080/10920277.2008.10597498 · Zbl 1481.91211
[43] Siu, T.K., Tong, H. and Yang, H., On pricing derivatives under GARCH models: A dynamic Gerber-Shiu’s approach. N. Am. Actuar. J., 2004, 8(3), 17-31. · Zbl 1085.91531
[44] Siu, T. K., Yang, H. and Lau, J. W. Pricing currency options under two-factor Markov-modulated stochastic volatility models. Insur.: Math. Econ., 2008, 43, 295-302. · Zbl 1152.91550 · doi:10.1016/j.insmatheco.2008.05.002
[45] Tahani, N., Credit spread option valuation under GARCH. J. Deriv., 2006, 14, 27-39.10.3905/jod.2006.650197 · doi:10.3905/jod.2006.650197
[46] Tchuindjo, L., Pricing of multi‐defaultable bonds with a two‐correlated‐factor Hull-White model. Appl. Math. Finance, 2007, 14, 19-39.10.1080/13504860600658943 · Zbl 1281.91173
[47] Thomas, L.C., Allen, D.E. and Nigel, M.K., A hidden Markov chain model for the term structure of bond credit risk spreads. Int. Rev. Financ. Anal., 2002, 11, 311-329.10.1016/S1057-5219(02)00078-9 · doi:10.1016/S1057-5219(02)00078-9
[48] Valchev, S., Stochastic volatility Gaussian Heath-Jarrow-Morton models. Appl. Math. Finance, 2004, 11, 347-368.10.1080/1350486042000231902 · Zbl 1108.91327
[49] Wang, W. and Wang, W., Pricing vulnerable options under a Markov-modulated regime switching model. Commun. Stat. - Theor. Methods, 2010, 39, 3421-3433.10.1080/03610920903268873 · Zbl 1202.91328
[50] Zhou, C., The term structure of credit spreads with jump risk. J. Bank. Finance, 2001, 25, 2015-2040.10.1016/S0378-4266(00)00168-0 · doi:10.1016/S0378-4266(00)00168-0
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