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March  2021, 17(2): 889-908. doi: 10.3934/jimo.2020003

## On correlated defaults and incomplete information

 1 Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 2 Corresponding author. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China 3 Department of Mathematics, Imperial College, London, SW7 2AZ, UK

Received  June 2018 Revised  July 2019 Published  March 2021 Early access  January 2020

In this paper, we study a continuous time structural asset value model for two correlated firms using a two-dimensional Brownian motion. We consider the situation of incomplete information, where the information set available to the market participants includes the default time of each firm and the periodic asset value reports. In this situation, the default time of each firm becomes a totally inaccessible stopping time to the market participants. The original structural model is first transformed to a reduced-form model. Then the conditional distribution of the default time together with the asset value of each name are derived. We prove the existence of the intensity processes of default times and also give the explicit form of the intensity processes. Numerical studies on the intensities of the two correlated names are conducted for some special cases.

Citation: Wai-Ki Ching, Jia-Wen Gu, Harry Zheng. On correlated defaults and incomplete information. Journal of Industrial and Management Optimization, 2021, 17 (2) : 889-908. doi: 10.3934/jimo.2020003
##### References:
 [1] M. Abramowitz and I. Stegun (Eds.), Handbook of Mathematical Functions, US Department of Commerce, 1967. [2] T. Aven, A theorem for determining the compensator of a counting process, Scandinavian Journal of Statistics, 12 (1985), 69-72. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [4] C. Blanchet-Scalliet, A. Cousin and D. Dorobantu, Hitting time for correlated three-dimensional Brownian motion, working paper, 2013. [5] C. Blanchet-Scalliet and F. Patras, Counterparty risk valuation for CDS, Credit Risk Frontiers: Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings, and Liquidity, WILEY, 2011, 437–456. [6] D. Duffie and D. Lando, Term structures and credit spreads with incomplete accounting information, Econometrica, 69 (2001), 633-664. [7] R. J. Elliott, M. Jeanblanc and M. Yor, On models of default risk, Mathematical Finance, 10 (2000), 179-195. [8] M. Escobar, S. Ferrando and X. Wen, Three dimensional distribution of Brownian motion extrema, Stochastics an International Journal of Probability and Stochastic Processes, 85 (2013), 807-832.  doi: 10.1080/17442508.2012.660942. [9] K. Giesecke, Correlated default with incomplete information, Journal of Banking and Finance, 28 (2004), 1521-1545. [10] K. Giesecke and L. R. Goldberg, Sequential defaults and incomplete information, Journal of Risk, 7 (2004), 1-26. [11] J. Gu, W. Ching, T. Siu and H. Zheng, On pricing basket credit default swaps, Quantitative Finance, 13 (2013), 1845-1854.  doi: 10.1080/14697688.2013.783713. [12] X. Guo, R. Jarrow and Y. Zeng, Credit risk models with incomplete information, Mathematics of Operations Research, 34 (2009), 320-332.  doi: 10.1287/moor.1080.0361. [13] M. Harrison, Brownian Motion and Stochastic Flow Systems, John Wiley and Sons, Inc., New York, 1985. [14] S. Iyengar, Hitting lines with two-dimensional Brownian motion, SIAM Journal of Applied Mathematics, 45 (1985), 983-989.  doi: 10.1137/0145060. [15] R. Jarrow and S. Turnbull, Credit Risk: Drawing the Analogy, Risk Magazine, 5, 1992. [16] R. Jarrow and S. Turnbull, Pricing options of financial securities subject to default risk, Journal of Finance, 50 (1995), 53-86. [17] F. C. Klebaner, Introduction to Stochastic Calculus with Applications, Third edition, Imperial College Press, London, 2012.  doi: 10.1142/p821. [18] S. Kou and H. Zhong, First Passage Times of Two-dimensional Correlated Brownian Motion, presentation at "Nonlinear Expectation, Stochastic Calculus under Knightian Uncertainty, and Related Topics", Institute of Mathematical Sciences, National University of Singapore, Singapore, 3 June - 12 July, 2013. [19] R. C. Merton, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 29 (1974), 449-470.  doi: 10.1142/9789814759588_0003. [20] A. Metzler, On the first passage problem for correlated Brownian motion, Statistics and Probability Letters, 80 (2010), 277-284.  doi: 10.1016/j.spl.2009.11.001. [21] L. C. G. Rogers and L. Shepp, The correlation of the maxima of correlated Brownian motions, J. Applied Probability, 43 (2006), 880-883.  doi: 10.1239/jap/1158784954. [22] F. Yu, Correlated defaults in intensity-based models, Mathematical Finance, 17 (2007), 155-173. [23] H. Zheng and L. Jiang, Basket CDS pricing with interacting intensities, Finance and Stochastics, 13 (2009), 445-469.  doi: 10.1007/s00780-009-0091-2.

show all references

##### References:
 [1] M. Abramowitz and I. Stegun (Eds.), Handbook of Mathematical Functions, US Department of Commerce, 1967. [2] T. Aven, A theorem for determining the compensator of a counting process, Scandinavian Journal of Statistics, 12 (1985), 69-72. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [4] C. Blanchet-Scalliet, A. Cousin and D. Dorobantu, Hitting time for correlated three-dimensional Brownian motion, working paper, 2013. [5] C. Blanchet-Scalliet and F. Patras, Counterparty risk valuation for CDS, Credit Risk Frontiers: Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings, and Liquidity, WILEY, 2011, 437–456. [6] D. Duffie and D. Lando, Term structures and credit spreads with incomplete accounting information, Econometrica, 69 (2001), 633-664. [7] R. J. Elliott, M. Jeanblanc and M. Yor, On models of default risk, Mathematical Finance, 10 (2000), 179-195. [8] M. Escobar, S. Ferrando and X. Wen, Three dimensional distribution of Brownian motion extrema, Stochastics an International Journal of Probability and Stochastic Processes, 85 (2013), 807-832.  doi: 10.1080/17442508.2012.660942. [9] K. Giesecke, Correlated default with incomplete information, Journal of Banking and Finance, 28 (2004), 1521-1545. [10] K. Giesecke and L. R. Goldberg, Sequential defaults and incomplete information, Journal of Risk, 7 (2004), 1-26. [11] J. Gu, W. Ching, T. Siu and H. Zheng, On pricing basket credit default swaps, Quantitative Finance, 13 (2013), 1845-1854.  doi: 10.1080/14697688.2013.783713. [12] X. Guo, R. Jarrow and Y. Zeng, Credit risk models with incomplete information, Mathematics of Operations Research, 34 (2009), 320-332.  doi: 10.1287/moor.1080.0361. [13] M. Harrison, Brownian Motion and Stochastic Flow Systems, John Wiley and Sons, Inc., New York, 1985. [14] S. Iyengar, Hitting lines with two-dimensional Brownian motion, SIAM Journal of Applied Mathematics, 45 (1985), 983-989.  doi: 10.1137/0145060. [15] R. Jarrow and S. Turnbull, Credit Risk: Drawing the Analogy, Risk Magazine, 5, 1992. [16] R. Jarrow and S. Turnbull, Pricing options of financial securities subject to default risk, Journal of Finance, 50 (1995), 53-86. [17] F. C. Klebaner, Introduction to Stochastic Calculus with Applications, Third edition, Imperial College Press, London, 2012.  doi: 10.1142/p821. [18] S. Kou and H. Zhong, First Passage Times of Two-dimensional Correlated Brownian Motion, presentation at "Nonlinear Expectation, Stochastic Calculus under Knightian Uncertainty, and Related Topics", Institute of Mathematical Sciences, National University of Singapore, Singapore, 3 June - 12 July, 2013. [19] R. C. Merton, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 29 (1974), 449-470.  doi: 10.1142/9789814759588_0003. [20] A. Metzler, On the first passage problem for correlated Brownian motion, Statistics and Probability Letters, 80 (2010), 277-284.  doi: 10.1016/j.spl.2009.11.001. [21] L. C. G. Rogers and L. Shepp, The correlation of the maxima of correlated Brownian motions, J. Applied Probability, 43 (2006), 880-883.  doi: 10.1239/jap/1158784954. [22] F. Yu, Correlated defaults in intensity-based models, Mathematical Finance, 17 (2007), 155-173. [23] H. Zheng and L. Jiang, Basket CDS pricing with interacting intensities, Finance and Stochastics, 13 (2009), 445-469.  doi: 10.1007/s00780-009-0091-2.
Default Intensity process $\lambda_2$ when $\tau_1 = 2$
Default Intensities of firm B where $\rho = -0.5$
Default Intensity Process $\lambda_2$ when $\rho = 0$
Default Intensity Process $\lambda_2$ when $\tau_1 = 2$ and $\rho = 0.1$
Default Intensity Process $\lambda_2$ when $\tau_1 = 2$ and $\rho = -0.1$
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