Mean–variance efficiency with extended CIR interest rates

Abstract

We study a mean-variance investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. We construct a mean-variance efficient portfolio through the solutions of backward stochastic differential equations. We also give sufficient conditions under which an explicit analytic expression is available for the mean-variance optimal wealth of the investor. problem using the Hamilton-Jacobi-Bellman equation of dynamic programming. About 15 years later, the equivalent martingale measure concept developed by Harrison and Pliska [7] paved the way for solving several utility maximization problems in a market model with stochastic uniformly bounded parameters (see, for example, Karatzas [8]). Unfortunately one cannot adequately address the continuous-time mean-variance problem using one of the above techniques. This may explains in part why significant results were obtained only recently. Alternate methods had to be considered. For example, Duffie and Richardson [9] solved a deterministic market model mean-variance problem via orthogonal projection techniques. In the early 1990s Pardoux and Peng [10] introduced the notion of backward stochastic differential equations (BSDE) which provided the ideal tool for Lim and Zhou [11] to adequately solve the continuous-time mean-variance allocation problem in a Black-Scholes model with stochastic uniformly bounded market coefficients. It is worth mentioning that BSDE theory was also proved to be valuable in utility portfolio selection problems as shown in [12].Unfortunately, in these papers, the uniform boundedness hypothesis assumed for the interest rate process precludes the use of interest models such as Vasicek, Hull-White and Cox-Ingersoll-Ross (CIR) models which are highly valued by practitioners. In order to pass this limit a number of researchers drew on a more general market where in addition to the usual bank account and stocks an individual is allowed to invest part of his wealth in bonds or interest derivatives. Bajeux-Besnainou and Portait [13] incorporated a Vasicek interest rate model and zero-coupons bonds in their model and solved a mean-variance-type problem. But the Vasicek as well as the Hull-White model have a major drawback which is that there is a positive probability that one encounters negative values. Obviously, this is a highly unrealistic and undesirable feature in a real-world financial market. A few years later, Deelstra et al. [14] modified Bajeux-Besnainou and Portait's model by rather opting for a CIR interest rate process which is known to be almost surely positive under mild conditions. Applying essentially the martingale approach they obtained an optimal strategy for a power-utility maximization problem. Now the classical CIR process rely on three constant parameters, thus, from a practical point of view statistical fitting to real financial data may be greatly improved by considering a more flexible time-varying parameter model as shown in Maghsoodi's [15] study of 25 years of U.S. Treasury bill data....

Optimal Investment for Insurers with the Extended CIR Interest Rate Model

“…The term Σ ( )√ ( ) is the volatility factor of stock prices contributed by the interest rate while (Λ( )) and ( )√ ( ) are the market price of risk of the risky asset and the market price of interest rate risk, respectively. This financial market setting is also considered in [4][5][6].…”

“…However, Deelstra et al [4,5] are the pioneers who investigated the optimal investment problems with and without a minimum guarantee under the extended CIR 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].…”

Time-consistent strategy for a multi-period mean-variance asset-liability management problem with stochastic interest rate

“…For instance, Korn and Kraft (2002) [11] considered an optimal portfolio selection problem with the Vasicek and Ho-Lee stochastic interest rate models. Ferland and Watier (2010) [8] analyzed a portfolio selection problem with a extended CIR stochastic interest rate model. Yao et al (2016c) [37] investigated the dynamic mean-variance asset allocation with stochastic interest rate and inflation rate, where the interest rate follows the Vasicek model.…”

Portfolio Optimization Problem with Delay under Cox-Ingersoll-Ross Model

“…For instance, Chacko and Viceira [6], Fleming and Hernndez-Hernndez [7], Liu [8] and Zariphopoulou [9] consider an optimal investment and consumption problems under SV models, and they derive explicitly the optimal strategies and optimal value functions in some situations by applying Hamilton-JacobinBellman (HJB) technique; Kraft [10], Taksar and Zeng [11] investigate a portfolio optimization problem under SV models. Moreover, Ferland and Watier [12] consider a mean-variance portfolio optimization problem with the CIR interest rate in a continuous-time framework, and they derive the mean-variance efficient portfolio by solving backward stochastic differential equations. Li & Wu [13] and Noh & Kim [14] consider portfolio optimization problems with an SV asset price process and a stochastic interest rate to maximize the expected utility of the terminal wealth.…”