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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/1906.05898 |
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| _version_ | 1866911532947865600 |
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| author | Hambly, Ben Kolliopoulos, Nikolaos |
| author_facet | Hambly, Ben Kolliopoulos, Nikolaos |
| contents | We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets. Both the asset value and the volatility processes are correlated through systemic Brownian motions, with default determined by the asset value reaching a lower boundary. We prove that if our volatility models are picked from a class of mean-reverting diffusions, the system converges as the portfolio becomes large and, when the vol-of-vol function satisfies certain regularity and boundedness conditions, the limit of the empirical measure process has a density given in terms of a solution to a stochastic initial-boundary value problem on a half-space. The problem is defined in a special weighted Sobolev space. Regularity results are established for solutions to this problem, and then we show that there exists a unique solution. In contrast to the CIR volatility setting covered by the existing literature, our results hold even when the systemic Brownian motions are taken to be correlated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_05898 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Stochastic PDEs for large portfolios with general mean-reverting volatility processes Hambly, Ben Kolliopoulos, Nikolaos Probability Analysis of PDEs Portfolio Management Pricing of Securities Risk Management 60H15, 60H07, 91G80 We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets. Both the asset value and the volatility processes are correlated through systemic Brownian motions, with default determined by the asset value reaching a lower boundary. We prove that if our volatility models are picked from a class of mean-reverting diffusions, the system converges as the portfolio becomes large and, when the vol-of-vol function satisfies certain regularity and boundedness conditions, the limit of the empirical measure process has a density given in terms of a solution to a stochastic initial-boundary value problem on a half-space. The problem is defined in a special weighted Sobolev space. Regularity results are established for solutions to this problem, and then we show that there exists a unique solution. In contrast to the CIR volatility setting covered by the existing literature, our results hold even when the systemic Brownian motions are taken to be correlated. |
| title | Stochastic PDEs for large portfolios with general mean-reverting volatility processes |
| topic | Probability Analysis of PDEs Portfolio Management Pricing of Securities Risk Management 60H15, 60H07, 91G80 |
| url | https://arxiv.org/abs/1906.05898 |