2014/10/31 by Cody Hyndman, Xinghua Zhou
Economics, Econometrics and Finance · Mathematics · #Affine transformation #Bond valuation #Economic Policies and Impacts #Economic theories and models #Gaussian #Isotropic quadratic form #Ordinary differential equation #Quadratic equation #Riccati equation #Stochastic differential equation #Stochastic processes and financial applications #Term (time) #math.PR #msc:60G35 #msc:60H20 #msc:60H30 #msc:91B28 #msc:91B70 #q-fin.MF
paper · pdf · doi:10.1080/07362994.2015.1009548
23 pages, minor revisions (typos, restatement of main results to rely on additional lemmas)
arxiv created 2014/12/16 · openalex publication_date 2015/04/22 · arxiv updated 2015/05/14 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
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.