An Investment and Consumption Problem with CIR Interest Rate and Stochastic Volatility

Abstract

We are concerned with an investment and consumption problem with stochastic interest rate and stochastic volatility, in which interest rate dynamic is described by the Cox-Ingersoll-Ross (CIR) model and the volatility of the stock is driven by Heston’s stochastic volatility model. We apply stochastic optimal control theory to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function and choose power utility and logarithm utility for our analysis. By using separate variable approach and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategy. A numerical example is given to illustrate our results and to analyze the effect of market parameters on the optimal investment and consumption strategies.

Optimal Investment for Insurers with the Extended CIR Interest Rate Model

“…Ferland and Watier [6] further analyzed the mean-variance efficiency of utility maximization problem under the extended CIR model. It is recently extended to incorporate stochastic volatility in [7].…”

Optimal Consumption and Portfolio Decision with Convertible Bond in Affine Interest Rate and Heston’s SV Framework

“…For example, Liu [22] and Noh and Kim [23] investigated the optimal consumption-investment problems with stochastic interest rate and stochastic volatility, but they did not obtain the explicit solutions to the optimal consumption-investment strategies. Chang and Rong [24] studied a special optimal investment-consumption problem with CIR interest rate and stochastic volatility under the assumption that interest rate dynamics are independent of stock price dynamics. However, stock market environment in practical is greatly affected by interest rate.…”

Legendre Transform-Dual Solution for a Class of Investment and Consumption Problems with HARA Utility

“…Theorem 3. Under HARA utility (6), the optimal investment and consumption strategy of the problem (5) is given by…”

“…In the following years, many scholars began to pay attention to the investment and consumption problems and obtained many research results. One can refer to the papers of Chacko and Viceira [3], Liu [4], Noh and Kim [5], Chang and Rong [6], and Chang et al [7]. But these results were generally achieved under power utility, which was taken as a special case of hyperbolic absolute risk aversion (HARA) utility function.…”