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

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 and some results in the multivariate setting of Leippold and Trojani 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.

References in corpus (5)

• A regression-based Monte Carlo method to solve backward stochastic differential equations
• Pricing, Hedging and Optimally Designing Derivatives Via Minimization of Risk Measures
• BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces
• The Opportunity Process for Optimal Consumption and Investment with Power Utility
• An extended existence result for quadratic BSDEs with jumps with application to the utility maximization problem