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Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models

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  • Richter, Anja

Abstract

Over the past few years quadratic Backward Stochastic Differential Equations (BSDEs) have been a popular field of research. However there are only very few examples where explicit solutions for these equations are known. In this paper we consider a class of quadratic BSDEs involving affine processes and show that their solution can be reduced to solving a system of generalized Riccati ordinary differential equations. In other words we introduce a rich and flexible class of quadratic BSDEs which are analytically tractable, i.e. explicit up to the solution of an ODE. Our results also provide analytically tractable solutions to the problem of utility maximization and indifference pricing in multivariate affine stochastic volatility models. This generalizes univariate results of Kallsen and Muhle-Karbe (2010) and some results in the multivariate setting of Leippold and Trojani (2010) by establishing the full picture in the multivariate affine jump-diffusion setting. In particular we calculate the interesting quantity of the power utility indifference value of change of numeraire. Explicit examples in the Heston, Barndorff-Nielsen–Shephard and multivariate Heston setting are calculated.

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  • Richter, Anja, 2014. "Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3578-3611.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:11:p:3578-3611
    DOI: 10.1016/j.spa.2014.05.004
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    1. JosE Da Fonseca & Martino Grasselli & Claudio Tebaldi, 2008. "A multifactor volatility Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 8(6), pages 591-604.
    2. Marcel Nutz, 2009. "The Opportunity Process for Optimal Consumption and Investment with Power Utility," Papers 0912.1879, arXiv.org, revised Jun 2010.
    3. Christa Cuchiero & Damir Filipovi'c & Eberhard Mayerhofer & Josef Teichmann, 2009. "Affine processes on positive semidefinite matrices," Papers 0910.0137, arXiv.org, revised Apr 2011.
    4. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    5. Mayerhofer, Eberhard & Muhle-Karbe, Johannes & Smirnov, Alexander G., 2011. "A characterization of the martingale property of exponentially affine processes," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 568-582, March.
    6. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    7. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    8. Andrea Buraschi & Paolo Porchia & Fabio Trojani, 2010. "Correlation Risk and Optimal Portfolio Choice," Journal of Finance, American Finance Association, vol. 65(1), pages 393-420, February.
    9. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    10. Marie-Amélie Morlais, 2009. "Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem," Finance and Stochastics, Springer, vol. 13(1), pages 121-150, January.
    11. Christian Gourieroux & Razvan Sufana, 2003. "Whishart Quadratic Term Structure Models," Working Papers 2003-50, Center for Research in Economics and Statistics.
    12. Briand, Philippe & Confortola, Fulvia, 2008. "BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 118(5), pages 818-838, May.
    13. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    14. José Fonseca & Martino Grasselli & Claudio Tebaldi, 2007. "Option pricing when correlations are stochastic: an analytical framework," Review of Derivatives Research, Springer, vol. 10(2), pages 151-180, May.
    15. N'Zi, Modeste & Ouknine, Youssef & Sulem, Agnès, 2006. "Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1319-1339, September.
    16. Qiang Dai & Kenneth J. Singleton, 2000. "Specification Analysis of Affine Term Structure Models," Journal of Finance, American Finance Association, vol. 55(5), pages 1943-1978, October.
    17. Morlais, Marie-Amelie, 2010. "A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1966-1995, September.
    18. Fred Espen Benth & Kenneth Hvistendahl Karlsen & Kristin Reikvam, 2003. "Merton's portfolio optimization problem in a Black and Scholes market with non‐Gaussian stochastic volatility of Ornstein‐Uhlenbeck type," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 215-244, April.
    19. Jan Kallsen & Johannes Muhle-Karbe, 2010. "Utility Maximization In Affine Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(03), pages 459-477.
    20. Duffie, Darrell, 2005. "Credit risk modeling with affine processes," Journal of Banking & Finance, Elsevier, vol. 29(11), pages 2751-2802, November.
    21. Jun Liu, 2007. "Portfolio Selection in Stochastic Environments," The Review of Financial Studies, Society for Financial Studies, vol. 20(1), pages 1-39, January.
    22. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    23. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    24. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," The Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-436.
    25. Imkeller, Peter & Dos Reis, Gonçalo, 2010. "Path regularity and explicit convergence rate for BSDE with truncated quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 348-379, March.
    26. Christian Gourieroux & Razvan Sufana, 2004. "Derivative Pricing with Multivariate Stochastic Volatility : Application to Credit Risk," Working Papers 2004-31, Center for Research in Economics and Statistics.
    27. Kallsen, Jan & Muhle-Karbe, Johannes, 2010. "Exponentially affine martingales, affine measure changes and exponential moments of affine processes," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 163-181, February.
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    Cited by:

    1. Da Fonseca, José, 2016. "On moment non-explosions for Wishart-based stochastic volatility models," European Journal of Operational Research, Elsevier, vol. 254(3), pages 889-894.
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    3. Jingtang Ma & Wenyuan Li & Harry Zheng, 2017. "Dual control Monte Carlo method for tight bounds of value function under Heston stochastic volatility model," Papers 1710.10487, arXiv.org.
    4. Carla Mereu & Robert Stelzer, 2015. "A BSDE arising in an exponential utility maximization problem in a pure jump market model," Papers 1508.07561, arXiv.org, revised Jan 2016.
    5. Scott Robertson & Hao Xing, 2014. "Long Term Optimal Investment in Matrix Valued Factor Models," Papers 1408.7010, arXiv.org.
    6. Cody B. Hyndman & Polynice Oyono Ngou, 2017. "A Convolution Method for Numerical Solution of Backward Stochastic Differential Equations," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 1-29, March.

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