Stochastic PDEs for large portfolios with general mean-reverting volatility processes
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...
| Những tác giả chính: | , |
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| Định dạng: | Journal article |
| Ngôn ngữ: | English |
| Được phát hành: |
American Institute of Mathematical Sciences
2024
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| _version_ | 1826315388576595968 |
|---|---|
| author | Hambly, B Kolliopoulos, N |
| author_facet | Hambly, B Kolliopoulos, N |
| author_sort | Hambly, B |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-04-09T03:58:35Z |
| format | Journal article |
| id | oxford-uuid:e4c3504a-7a0b-48ab-bdb8-0aa0c46b18f6 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:24:59Z |
| publishDate | 2024 |
| publisher | American Institute of Mathematical Sciences |
| record_format | dspace |
| spelling | oxford-uuid:e4c3504a-7a0b-48ab-bdb8-0aa0c46b18f62024-11-27T09:31:43ZStochastic PDEs for large portfolios with general mean-reverting volatility processesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:e4c3504a-7a0b-48ab-bdb8-0aa0c46b18f6EnglishSymplectic ElementsAmerican Institute of Mathematical Sciences2024Hambly, BKolliopoulos, NWe 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. |
| spellingShingle | Hambly, B Kolliopoulos, N Stochastic PDEs for large portfolios with general mean-reverting volatility processes |
| title | Stochastic PDEs for large portfolios with general mean-reverting volatility processes |
| title_full | Stochastic PDEs for large portfolios with general mean-reverting volatility processes |
| title_fullStr | Stochastic PDEs for large portfolios with general mean-reverting volatility processes |
| title_full_unstemmed | Stochastic PDEs for large portfolios with general mean-reverting volatility processes |
| title_short | Stochastic PDEs for large portfolios with general mean-reverting volatility processes |
| title_sort | stochastic pdes for large portfolios with general mean reverting volatility processes |
| work_keys_str_mv | AT hamblyb stochasticpdesforlargeportfolioswithgeneralmeanrevertingvolatilityprocesses AT kolliopoulosn stochasticpdesforlargeportfolioswithgeneralmeanrevertingvolatilityprocesses |