TY - JOUR

T1 - Utility maximization in a stochastic affine interest rate and CIR risk premium framework

T2 - a BSDE approach

AU - Zhang, Yumo

PY - 2023

Y1 - 2023

N2 - This paper investigates optimal investment problems in the presence of stochastic interest rates and stochastic volatility under the expected utility maximization criterion. The financial market consists of three assets: a risk-free asset, a risky asset, and zero-coupon bonds (rolling bonds). The short interest rate is assumed to follow an affine diffusion process, which includes the Vasicek and the Cox–Ingersoll–Ross (CIR) models, as special cases. The risk premium of the risky asset depends on a square-root diffusion (CIR) process, while the return rate and volatility coefficient are unspecified and possibly given by non-Markovian processes. This framework embraces the family of the state-of-the-art 4/2 stochastic volatility models and some non-Markovian models, as exceptional examples. The investor aims to maximize the expected utility of the terminal wealth for two types of utility functions, power utility, and logarithmic utility. By adopting a backward stochastic differential equation (BSDE) approach to overcome the potentially non-Markovian framework and solving two BSDEs explicitly, we derive, in closed form, the optimal investment strategies and optimal value functions. Furthermore, explicit solutions to some special cases of our model are provided. Finally, numerical examples illustrate our results under one specific case, the hybrid Vasicek-4/2 model.

AB - This paper investigates optimal investment problems in the presence of stochastic interest rates and stochastic volatility under the expected utility maximization criterion. The financial market consists of three assets: a risk-free asset, a risky asset, and zero-coupon bonds (rolling bonds). The short interest rate is assumed to follow an affine diffusion process, which includes the Vasicek and the Cox–Ingersoll–Ross (CIR) models, as special cases. The risk premium of the risky asset depends on a square-root diffusion (CIR) process, while the return rate and volatility coefficient are unspecified and possibly given by non-Markovian processes. This framework embraces the family of the state-of-the-art 4/2 stochastic volatility models and some non-Markovian models, as exceptional examples. The investor aims to maximize the expected utility of the terminal wealth for two types of utility functions, power utility, and logarithmic utility. By adopting a backward stochastic differential equation (BSDE) approach to overcome the potentially non-Markovian framework and solving two BSDEs explicitly, we derive, in closed form, the optimal investment strategies and optimal value functions. Furthermore, explicit solutions to some special cases of our model are provided. Finally, numerical examples illustrate our results under one specific case, the hybrid Vasicek-4/2 model.

KW - Affine diffusion process

KW - CIR risk premium

KW - Power utility

KW - Logarithmic utility

KW - Backward stochastic differential equation

U2 - 10.1007/s10203-022-00374-x

DO - 10.1007/s10203-022-00374-x

M3 - Journal article

VL - 46

SP - 97

EP - 128

JO - Rivista di Matematica per le Scienze Economiche e Sociali

JF - Rivista di Matematica per le Scienze Economiche e Sociali

SN - 1593-8883

ER -