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January  2017, 2: 12 doi: 10.1186/s41546-017-0023-6

Portfolio optimization of credit swap under funding costs

Lijun Bo 1,2,,

1 School of Mathematics and Statistics, Xidian University, Xian 710071, China;

2 School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui Province 230026, China

Received  January 27, 2017 Revised  October 17, 2017 Published  March 2017

Fund Project: The research was partially supported by NSF of China (No. 11471254), The Key Research Program of Frontier Sciences, CAS (No. QYZDB-SSWSYS009) and Fundamental Research Funds for the Central Universities (No. WK3470000008).

We develop a dynamic optimization framework to assess the impact of funding costs on credit swap investments. A defaultable investor can purchase CDS upfronts, borrow at a rate depending on her credit quality, and invest in the money market account. By viewing the concave drift of the wealth process as a continuous function of admissible strategies, we characterize the optimal strategy in terms of a relation between a critical borrowing threshold and two solutions of a suitably chosen system of first order conditions. Contagion effects between risky investor and reference entity make the optimal strategy coupled with the value function of the control problem. Using the dynamic programming principle, we show that the latter can be recovered as the solution of a nonlinear HJB equation whose coeffcients admit singular growth. By means of a truncation technique relying on the locally Lipschitzcontinuity of the optimal strategy, we establish existence and uniqueness of a global solution to the HJB equation.
Citation: Lijun Bo. Portfolio optimization of credit swap under funding costs. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 12-. doi: 10.1186/s41546-017-0023-6
References:
[1]

Azizpour, S, Giesecke, K, Schwenkler, G:Exploring the sources of default clustering. J. Finan. Econom.Forthcoming (2017)

[2]

Belanger, A, Shreve, S, Wong, D:A general framework for pricing credit risk. Math. Finan 14, 317-350(2004)

[3]

Bielecki, T, Jang, I:Portfolio optimization with a defaultable security. Asia-Pacific Finan. Markets 13, 113-127 (2006)

[4]

Bielecki, T, Jeanblanc, M, Rutkowski, M:Pricing and trading credit default swaps in a hazard process model. Ann. Appl. Probab 18, 2495-2529 (2008)

[5]

Bielecki, T, Rutkowski, M:Valuation and hedging of contracts with funding costs and collateralization.SIAM J. Finan. Math 6, 594-655 (2015)

[6]

Bo, L, Wang, Y, Yang, X:An optimal portfolio problem in a defaultable market. Adv. Appl. Probab 42, 689-705 (2010)

[7]

Bo, L, Capponi, A:Optimal investment in credit derivatives portfolio under contagion risk. Math. Finan 26, 785-834 (2016)

[8]

Bo, L, Capponi, A:Optimal credit investment with borrowing costs. Math Oper. Res 42, 546-575 (2017)

[9]

Capponi, A, Figueroa-López, JE:Dynamics portfolio optimization with a defaultable security and regime-switching. Math Finan 24, 207-249 (2014)

[10]

Chen, H:Macroeconomic conditions and the puzzles of credit spreads and capital structure. J. Finan 65, 2171-2212 (2010)

[11]

Crépey, S:Bilateral counterparty risk under funding constraints. Part I:pricing. Math Finan 25, 23-50(2015)

[12]

Cvitanić, J, Karatzas, I:Hedging contingent claims with constrained portfolios. Ann. Appl. Probab 3, 652-681 (1993)

[13]

Draouil, O, Oksendal, B:A donsker delta functional approach to optimal insider control and applications to finance. Commun. Math Stats 3, 365-421 (2015)

[14]

Duffie, D, Singleton, K:Credit Risk. Princeton University Press, Princeton (2003)

[15]

El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math Finan 7, 1-71 (1997)

[16]

Frey, R, Backhaus, J:Pricing and hedging of portfolio credit derivatives with interacting default intensities.Int. J. Theor. Appl. Finan 11, 611-634 (2008)

[17]

Giesecke, K, Kim, B, Kim, J, Tsoukalas, G:Optimal credit swap portfolios. Manage Sci 60, 2291-2307(2014)

[18]

Jarrow, R, Yu, F:Counterparty risk and the pricing of defaultable securities. J. Finan 56, 1765-1799 (2001)

[19]

Jiao, Y, Pham, H:Optimal investment with counterparty risk:a default density approach. Finan. Stoch 15, 725-753 (2011)

[20]

Korn, R:Contingent claim valuation in a market with different interest rates. Math. Meth. Oper. Res 42, 255-274 (1995)

[21]

Korn, R, Kraft, H:Optimal portfolios with defaultable securities-a firm value approach. Int. J. Theor. Appl.Finan 6, 793-819 (2003)

[22]

Kraft, H, Steffensen, M:Portfolio problems stopping at first hitting time with application to default risk.Math. Meth. Oper. Res 63, 123-150 (2005)

[23]

Kumagai, S:An implicit function theorem:comment. J. Optim. Theor. Appl 31, 285-288 (1980)

[24]

Mercurio, F:Bergman, Piterbarg, and Beyond:pricing derivatives under collateralization and differential rates. In:Londoño, J, Garrido, J, Hernández-Hernández, D (eds.) Actuarial Sciences and Quantitative

[25]

Finance. Springer Proceedings in Mathematics & Statistics, vol 135. Springer, Cham (2015)

show all references

References:
[1]

Azizpour, S, Giesecke, K, Schwenkler, G:Exploring the sources of default clustering. J. Finan. Econom.Forthcoming (2017)

[2]

Belanger, A, Shreve, S, Wong, D:A general framework for pricing credit risk. Math. Finan 14, 317-350(2004)

[3]

Bielecki, T, Jang, I:Portfolio optimization with a defaultable security. Asia-Pacific Finan. Markets 13, 113-127 (2006)

[4]

Bielecki, T, Jeanblanc, M, Rutkowski, M:Pricing and trading credit default swaps in a hazard process model. Ann. Appl. Probab 18, 2495-2529 (2008)

[5]

Bielecki, T, Rutkowski, M:Valuation and hedging of contracts with funding costs and collateralization.SIAM J. Finan. Math 6, 594-655 (2015)

[6]

Bo, L, Wang, Y, Yang, X:An optimal portfolio problem in a defaultable market. Adv. Appl. Probab 42, 689-705 (2010)

[7]

Bo, L, Capponi, A:Optimal investment in credit derivatives portfolio under contagion risk. Math. Finan 26, 785-834 (2016)

[8]

Bo, L, Capponi, A:Optimal credit investment with borrowing costs. Math Oper. Res 42, 546-575 (2017)

[9]

Capponi, A, Figueroa-López, JE:Dynamics portfolio optimization with a defaultable security and regime-switching. Math Finan 24, 207-249 (2014)

[10]

Chen, H:Macroeconomic conditions and the puzzles of credit spreads and capital structure. J. Finan 65, 2171-2212 (2010)

[11]

Crépey, S:Bilateral counterparty risk under funding constraints. Part I:pricing. Math Finan 25, 23-50(2015)

[12]

Cvitanić, J, Karatzas, I:Hedging contingent claims with constrained portfolios. Ann. Appl. Probab 3, 652-681 (1993)

[13]

Draouil, O, Oksendal, B:A donsker delta functional approach to optimal insider control and applications to finance. Commun. Math Stats 3, 365-421 (2015)

[14]

Duffie, D, Singleton, K:Credit Risk. Princeton University Press, Princeton (2003)

[15]

El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math Finan 7, 1-71 (1997)

[16]

Frey, R, Backhaus, J:Pricing and hedging of portfolio credit derivatives with interacting default intensities.Int. J. Theor. Appl. Finan 11, 611-634 (2008)

[17]

Giesecke, K, Kim, B, Kim, J, Tsoukalas, G:Optimal credit swap portfolios. Manage Sci 60, 2291-2307(2014)

[18]

Jarrow, R, Yu, F:Counterparty risk and the pricing of defaultable securities. J. Finan 56, 1765-1799 (2001)

[19]

Jiao, Y, Pham, H:Optimal investment with counterparty risk:a default density approach. Finan. Stoch 15, 725-753 (2011)

[20]

Korn, R:Contingent claim valuation in a market with different interest rates. Math. Meth. Oper. Res 42, 255-274 (1995)

[21]

Korn, R, Kraft, H:Optimal portfolios with defaultable securities-a firm value approach. Int. J. Theor. Appl.Finan 6, 793-819 (2003)

[22]

Kraft, H, Steffensen, M:Portfolio problems stopping at first hitting time with application to default risk.Math. Meth. Oper. Res 63, 123-150 (2005)

[23]

Kumagai, S:An implicit function theorem:comment. J. Optim. Theor. Appl 31, 285-288 (1980)

[24]

Mercurio, F:Bergman, Piterbarg, and Beyond:pricing derivatives under collateralization and differential rates. In:Londoño, J, Garrido, J, Hernández-Hernández, D (eds.) Actuarial Sciences and Quantitative

[25]

Finance. Springer Proceedings in Mathematics & Statistics, vol 135. Springer, Cham (2015)

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