×

The dynamic spread of the forward CDS with general random loss. (English) Zbl 1406.91453

Summary: We assume that the filtration \(\mathbb{F}\) is generated by a \(d\)-dimensional Brownian motion \(W = (W_1, \ldots, W_d)'\) as well as an integer-valued random measure \(\mu(d u, d y)\). The random variable \(\widetilde{\tau}\) is the default time and \(L\) is the default loss. Let \(\mathbb{G} = \{\mathcal{G}_t; t \geq 0 \}\) be the progressive enlargement of \(\mathbb{F}\) by \((\widetilde{\tau}, L)\); that is, \(\mathbb{G}\) is the smallest filtration including \(\mathbb{F}\) such that \(\widetilde{\tau}\) is a \(\mathbb{G}\)-stopping time and \(L\) is \(\mathcal{G}_{\widetilde{\tau}}\)-measurable. We mainly consider the forward CDS with loss in the framework of stochastic interest rates whose term structures are modeled by the Heath-Jarrow-Morton approach with jumps under the general conditional density hypothesis. We describe the dynamics of the defaultable bond in \(\mathbb{G}\) and the forward CDS with random loss explicitly by the BSDEs method.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
91G40 Credit risk
91G30 Interest rates, asset pricing, etc. (stochastic models)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brigo, D., Market models for CDS options and callable floaters, Risk Magazine
[2] Brigo, D., Constant maturity CDS valuation with market models, Risk Magazine (2006)
[3] Li, B.; Rutkowski, M., Market models of forward CDS spreads, Stochastic Analysis with Financial Applications. Stochastic Analysis with Financial Applications, Progress in Probability, 65, 361-411 (2011) · Zbl 1246.91101 · doi:10.1007/978-3-0348-0097-6_21
[4] Rutkowski, M.; Armstrong, A., Valuation of credit default swaptions and credit default index swaptions, International Journal of Theoretical and Applied Finance, 12, 7, 1027-1053 (2009) · Zbl 1198.91222 · doi:10.1142/S0219024909005579
[5] Xiong, D.; Kohlmann, M., Modeling the forward CDS spreads with jumps, Stochastic Analysis and Applications, 30, 3, 375-402 (2012) · Zbl 1242.91198 · doi:10.1080/07362994.2012.668435
[6] Björk, T.; Kabanov, Y.; Runggaldier, W., Bond market structure in the presence of marked point processes, Mathematical Finance, 7, 2, 211-239 (1997) · Zbl 0884.90014 · doi:10.1111/1467-9965.00031
[7] Jacod, J., Grossissement initial, Hypothèse (H) et théorème de Girsanov, Séminaire de Calcul Stochastique 1982/83. Séminaire de Calcul Stochastique 1982/83, Lecture Notes in Mathematics, 1118 (1987), Springer · Zbl 0568.60049
[8] Amendinger, J., Initial enlargement of filtrations and additional information in financial markets [Ph.D. thesis] (1999), Technischen Universität Berlin · Zbl 0936.91022
[9] Callegaro, G.; Jeanblanc, M.; Zargari, B., Carthaginian enlargement of filtrations, ESAIM: Probability and Statistics, 17, 550-566 (2013) · Zbl 1296.60106 · doi:10.1051/ps/2011162
[10] Dellacherie, C.; Meyer, P. A., A Propos des Réesultats de Yor Sur le Grossissement des Tribus Séminaire de Probabilités (1978) · Zbl 0378.60033
[11] El Karoui, N.; Jeanblanc, M.; Jiao, Y., What happens after a default: the conditional density approach, Stochastic Processes and Their Applications, 120, 7, 1011-1032 (2010) · Zbl 1194.91187 · doi:10.1016/j.spa.2010.02.003
[12] Jeanblanc, M.; Le Cam, Y., Progressive enlargement of filtrations with initial times, Stochastic Processes and Their Applications, 119, 8, 2523-2543 (2009) · Zbl 1175.60041 · doi:10.1016/j.spa.2008.12.009
[13] Jeanblanc, M.; Song, S., An explicit model of default time with given survival probability, Stochastic Processes and Their Applications, 121, 8, 1678-1704 (2011) · Zbl 1298.91176 · doi:10.1016/j.spa.2011.04.002
[14] Jeanblanc, M.; Song, S., Martingale representation property in progressively enlarged filtrations, Working Paper (2012) · Zbl 1328.60110
[15] Jacod, J.; Shiryaev, A. N., Limit Theorems for Stochastic Processes, 288 (1987), Berlin, Germany: Springer, Berlin, Germany · Zbl 0635.60021
[16] Kchia, Y.; Larsson, M.; Protter, P., On progressive filtration expansions with a process: applications to insider trading · Zbl 1303.91159 · doi:10.1007/s00780-010-0140-x
[17] Kchia, Y.; Larsson, M.; Protter, P., Linking progressive and initial filtration expansions, Malliavin Calculus and Stochastic Analysis. Malliavin Calculus and Stochastic Analysis, Springer Proceedings in Mathematics & Statistics, 34, 469-487 (2013), Springer · Zbl 1317.60047 · doi:10.1007/978-1-4614-5906-4_21
[18] Jeulin, T., Semi-Martingales et Grossissement d’une Filtration. Semi-Martingales et Grossissement d’une Filtration, Lecture Notes in Mathematics, 833 (1980), New York, NY, USA: Springer, New York, NY, USA · Zbl 0444.60002
[19] Jeanblanc, M.; Yor, M.; Chesney, M., Mathematical Methods for Financial Markets (2009), London, UK: Springer, London, UK · Zbl 1205.91003 · doi:10.1007/978-1-84628-737-4
[20] Pham, H., Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management, Stochastic Processes and their Applications, 120, 9, 1795-1820 (2010) · Zbl 1196.60141 · doi:10.1016/j.spa.2010.05.003
[21] Tian, K.; Xiong, D.; Ye, Z., The martingale representation in a progressive enlargement of a filtration with jumps
[22] Xiong, D.; Kohlmann, M., Defaultable bond markets with jumps, Stochastic Analysis and Applications, 30, 2, 285-321 (2012) · Zbl 1257.91055 · doi:10.1080/07362994.2012.649623
[23] Lando, D., On cox processes and credit risky securities, Review of Derivatives Research, 2, 2-3, 99-120 (1998) · Zbl 1274.91459
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.