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Over the past few years quadratic Backward Stochastic Differential Equations(BSDEs) have been a popular field of research. However there are only very fewexamples where explicit solutions for these equations are known. In this paperwe consider a class of quadratic BSDEs involving affine processes and show thattheir solution can be reduced to solving a system of generalized Riccatiordinary differential equations. In other words we introduce a rich andflexible class of quadratic BSDEs which are analytically tractable, i.e.explicit up to the solution of an ODE. Our results also provide analyticallytractable solutions to the problem of utility maximization and indifferencepricing in multivariate affine stochastic volatility models. This generalizesunivariate results of Kallsen and Muhle-Karbe and some results in themultivariate setting of Leippold and Trojani by establishing the full picturein the multivariate affine jump-diffusion setting. In particular we calculatethe interesting quantity of the power utility indifference value of change ofnumeraire. Explicit examples in the Heston, Barndorff-Nielsen-Shephard andmultivariate Heston setting are calculated.

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