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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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    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. 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.
    6. Scott Robertson & Hao Xing, 2014. "Long Term Optimal Investment in Matrix Valued Factor Models," Papers 1408.7010, arXiv.org.

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