Explicit solutions of quadratic FBSDEs arising from quadratic term structure models

TL;DR

This paper derives explicit solutions for quadratic FBSDEs linked to quadratic term structure models of interest rates, providing insights into bond pricing and asset derivatives through Riccati equations.

Contribution

It introduces explicit solutions for quadratic FBSDEs in interest rate models, connecting them to Riccati equations and quadratic control problems, extending affine model results.

Findings

Explicit solutions for quadratic FBSDEs in interest rate models.

Characterization of bond prices and asset derivatives via these solutions.

Connection between Riccati equations and solvability conditions.

Abstract

We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to similar results on affine term structure models of Hyndman (Math. Financ. Econ. 2(2):107-128, 2009) due to the relationship between quadratic functionals of Gaussian processes and linear functionals of affine processes. Similar to the affine case a sufficient condition for the explicit solutions to hold is the solvability in a fixed interval of Riccati-type ordinary differential equations. However, in contrast to the affine case, these Riccati equations are easily associated with those occurring in linear-quadratic control problems. We also consider quadratic models for a risky asset price and characterize the futures price and forward price of the asset in terms of similar FBSDEs. An example is considered, using an approach based on stochastic flows that is related to the FBSDE approach, to further emphasize the parallels between the affine and quadratic models. An appendix discusses solvability and explicit solutions of the Riccati equations.