Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models. (English) Zbl 1297.60040
Summary: 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 J. Kallsen and J. Muhle-Karbe [Stochastic Processes Appl. 120, No. 2, 163–181 (2010; Zbl 1185.60045)] and some results in the multivariate setting of M. Leippold and F. Trojani [“Asset pricing with matrix jump diffusions”, NCCR Finrisk Working Paper No. 681 (2008), http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1274482] 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.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
quadratic BSDEs; affine processes; Wishart processes; utility maximization; stochastic volatility; explicit solutionCitations:
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