2019/06/13 by Ben Hambly, Hambly, Ben, Nikolaos Kolliopoulos +1
Economics, Econometrics and Finance · Social Sciences · #60H07 #60H15 #91G80 #Analysis of PDEs (math.AP) #Credit Risk and Financial Regulations #FOS: Economics and business #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Portfolio Management (q-fin.PM) #Pricing of Securities (q-fin.PR) #Probability (math.PR) #Risk Management (q-fin.RM) #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.1906.05898
openalex publication_date 2019/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.