Optimal Investment Strategy under the CEV Model with Stochastic Interest Rate

Abstract

Interest rate is an important macrofactor that affects asset prices in the financial market. As the interest rate in the real market has the property of fluctuation, it might lead to a great bias in asset allocation if we only view the interest rate as a constant in portfolio management. In this paper, we mainly study an optimal investment strategy problem by employing a constant elasticity of variance (CEV) process and stochastic interest rate. The assets of investment for individuals are supposed to be composed of one risk-free asset and one risky asset. The interest rate for risk-free asset is assumed to follow the Cox–Ingersoll–Ross (CIR) process, and the price of risky asset follows the CEV process. The objective is to maximize the expected utility of terminal wealth. By applying the dual method, Legendre transformation, and asymptotic expansion approach, we successfully obtain an asymptotic solution for the optimal investment strategy under constant absolute risk aversion (CARA) utility function. In the end, some numerical examples are provided to support our theoretical results and to illustrate the effect of stochastic interest rates and some other model parameters on the optimal investment strategy.

An Investor’s Investment Plan with Stochastic Interest Rate under the CEV Model and the Ornstein-Uhlenbeck Process

“…Here, we consider an investor with utility function exhibiting constant relative risk aversion (CRRA) different from the one in [18] where the investor exhibited constant absolute risk aversion (CARA). Since our interest here is to determine the optimal investment plan for the investor with CRRA utility, we choose the logarithm utility function similar to the one in [20,21].…”

“…Next, we follow the approach in [18] by applying the asymptotic expansion method to solve the problem in (44). Assume that the volatility follows a slow fluctuating process, we attempt to find an asymptotic solution of (44) by a following slow-fluctuating process r α to replace (2), in which 0 < α 1 is a small positive parameter:…”

“…In [17], the optimal investment plan of an insurer with stochastic interest rate under the CEV model was studied using logarithm utility, they assumed that the risk free interest rate follows the Cox-Ingersoll-Ross (CIR) process and used the power transformation, change of variable and asymptotic approach to find the asymptotic solution of the optimal investment plan. In [18], the expected utility of an investor with exponential utility function was maximized using the Legendre transformation and asymptotic expansion method to find a closed form solution of the optimal investment plan. They pointed out that authors find it difficult to combine the CEV model and stochastic interest rate in determining investment strategies due to the complexity of the resultant optimization problem.…”

“…We also give some numerical simulations to explain our results. The main difference between our work and that of [18] is that we will consider an investor with logarithm utility where the investor's behaviour toward risks are not influenced by the risk aversion coefficient instead of the exponential utility which depend on the risk aversion coefficient. Also, investment is done with three assets instead of two assets and the risk free interest rate follows the O-U process instead of the CIR process.…”

Bond and Option Prices under Skew Vasicek Model with Transaction Cost

“…Classical models are popular financial models to forecast interest rate, where Brownian motion is used to display the random part of these models [2,4,5]. Since Brownian motion is a martingale, thus these models find a fair price or amount of the financial instruments.…”