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Fast mean-reversion asymptotics for large portfolios of stochastic volatility models

Author

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  • Ben Hambly

    (University of Oxford)

  • Nikolaos Kolliopoulos

    (University of Oxford
    Beijing University)

Abstract

We consider an asymptotic SPDE description of a large portfolio model where the underlying asset prices evolve according to certain stochastic volatility models with default upon hitting a lower barrier. The asset prices and their volatilities are correlated through systemic Brownian motions, and the SPDE is obtained on the positive half-space along with a Dirichlet boundary condition. We study the convergence of the loss from the system, which is given in terms of the total mass of a solution to our stochastic initial-boundary value problem, under fast mean-reversion of the volatility. We consider two cases. In the first case, the volatilities are sped up towards a limiting distribution and the system converges only in a weak sense. On the other hand, when only the mean-reversion coefficients of the volatilities are allowed to grow large, we see a stronger form of convergence of the system to its limit. Our results show that in a fast mean-reverting volatility environment, we can accurately estimate the distribution of the loss from a large portfolio by using an approximate constant volatility model which is easier to handle.

Suggested Citation

  • Ben Hambly & Nikolaos Kolliopoulos, 2020. "Fast mean-reversion asymptotics for large portfolios of stochastic volatility models," Finance and Stochastics, Springer, vol. 24(3), pages 757-794, July.
    • Handle: RePEc:spr:finsto:v:24:y:2020:i:3:d:10.1007_s00780-020-00422-7
    DOI: 10.1007/s00780-020-00422-7
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    References listed on IDEAS

    as
    1. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers & Justin A. Sirignano, 2015. "Large Portfolio Asymptotics For Loss From Default," Mathematical Finance, Wiley Blackwell, vol. 25(1), pages 77-114, January.
    2. Ben Hambly & Nikolaos Kolliopoulos, 2019. "Stochastic PDEs for large portfolios with general mean-reverting volatility processes," Papers 1906.05898, arXiv.org, revised Mar 2024.
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    5. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers & Justin A. Sirignano, 2011. "Large Portfolio Asymptotics for Loss From Default," Papers 1109.1272, arXiv.org, revised Feb 2015.
    6. Spiliopoulos, Konstantinos & Sowers, Richard B., 2011. "Recovery rates in investment-grade pools of credit assets: A large deviations analysis," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2861-2898.
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    8. Justin Sirignano & Kay Giesecke, 2019. "Risk Analysis for Large Pools of Loans," Management Science, INFORMS, vol. 65(1), pages 107-121, January.
    9. Ben Hambly & Nikolaos Kolliopoulos, 2019. "ERRATUM: Stochastic evolution equations for large portfolios of stochastic volatility models," Papers 1905.04397, arXiv.org.
    10. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers, 2011. "Default clustering in large portfolios: Typical events," Papers 1104.1773, arXiv.org, revised Feb 2013.
    11. Michael B. Giles & Christoph Reisinger, 2012. "Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance," Papers 1204.1442, arXiv.org.
    12. Dai Pra, Paolo & Tolotti, Marco, 2009. "Heterogeneous credit portfolios and the dynamics of the aggregate losses," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2913-2944, September.
    13. Konstantinos Spiliopoulos & Richard B. Sowers, 2013. "Default Clustering in Large Pools: Large Deviations," Papers 1311.0498, arXiv.org, revised Feb 2015.
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    2. Tang, Qihe & Tong, Zhiwei & Yang, Yang, 2021. "Large portfolio losses in a turbulent market," European Journal of Operational Research, Elsevier, vol. 292(2), pages 755-769.

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    More about this item

    Keywords

    Large portfolio; Stochastic volatility; Distance to default; Systemic risk; Mean-field; SPDE; Fast mean-reversion; Large time-scale;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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