Logarithmutilityportf_省略_ochasticinteres

Logarithm utility portfolio for asset and liability management with stochastic interest rates
Hao CHANG1, Kai CHANG2
1.Department of Mathematics,Tianjin Polytechnic University,Tianjin 300387, China E-mail: ch8683897@ 2.Shenzhen Graduate School,Harbin Institute of Technology,Shenzhen 518055, China Email: changkai1898@ Abstract: This paper applies the maximum principle to obtain Hamilton-Jocabi-Bellman (HJB) equation for the asset and liability management problem under stochastic interest rate. And the optimal investment strategies under the Ho-Lee model and the Vasicek model are investigated respectively. Logarithm utility function is taken as the risky preference of investors and the closed-form solutions of the optimal investment strategy are derived via adopting Legendre transform approach. Keywords: Stochastic interest rate; asset anlity; portfolio; HJB equation; Legendre transform
~ (t ) , θ (t ),η ~ (t ) are assumed to be are constants and α deterministic and continuous function of the time t . The second asset is a stock with the price S 2 (t ) at ~ (t ) ,where λ > 0 and b > 0 with a (t ) = θ (t ) − λr (t ) + bη ~ ~
L2 (Ω, F , P ) of mean-square integrable random variables
constructs a natural Hilbert space structure. We consider a financial market where three assets are traded continuously over [0, T ] . One of the assets is a riskless asset with price S1 (t ) at time t , whose price process S1 (t ) satisfies dS1 (t ) = S1 (t )r (t )dt , S1 (0) = 1 , t ∈ [0, T ] (1) where r (t ) > 0 is the interest rate of the riskless asset at time t , and r (t ) is supposed to be stochastic process. In this paper interest rate is assumed to be driven by the stochastic differential equation dr (t ) = a (t )dt + bdW2 (t ) and as explicit examples we will consider the Ho-Lee ~ (t ) + bη ~ (t ) and a Vasicek model model given by a(t ) = α
978-1-4577-2074-1/12/$26.00 c 2012 IEEE
2035
time t , whose price process S 2 (t ) is given by the following stochastic differential equations(SDEs) dS 2 (t ) = S 2 (t )[(μ1 (t ) + r (t ) )dt + σ 1 (t )dW1 (t ) + σ 2 (t )dW2 (t )] (2) where μ1 (t ),σ 1 (t ),σ 2 (t ) are supposed to be deterministic and Borel-measurable bounded functions on [0, T ] . The third asset is a bond with maturity T1 > T , whose price dynamics P(t ) follows[5] dP(t ) = P(t )[(r (t ) + η (t )σ 3 (t ))dt + σ 3 (t )dW2 (t )] (3) where η (t ) is the deterministic function. As before, we have σ 3 (t ) = −b(T1 − t ) in the Ho-Lee model and
optimization problem. Section 3 applies Legendre transform-dual approach to the nonlinear HJB equation and obtains a linear HJB equation. Section 4 derives the optimal portfolio for logarithm utility under the Ho-Lee model and Vasicek model. Finally, the results are concluded in the section 5. 2. PROBLEM FORMULATION Throughout this paper we denote by E (⋅) the expectation of a random variable, [0, T ] the fixed and finite time horizon of the investment. Assumed that (W1 (t ),W2 (t )) is a 2-dimensional standard adapted and independent Brownian motion on a filtered complete probability space (Ω, F , P,{Ft }0≤t ≤T ) , in which {Ft }0≤t≤T is generated by (W1 (t ),W2 (t )) .The space
This research is supported by the Higher School Science and Technology Development Foundation of Tianjin (No: 20100821) and the Humanities and Social Science Research Youth Foundation of Ministry of Education(No:11YJC790006);
1. INTRODUCTION The Ho-Lee model[1]and the Vasicek model[2] are two important mathematical models in the term structure theory of interest rate. In recent years, portfolio selection theories with stochastic interest rates have achieved many research results[3-9]. Grasselli (2003) investigated portfolio selection via martingale approach and obtained a stability result for the HARA utility function, Korn and Kraft(2001) applied dynamic programming and HJB equation to the investment problem under the Ho-Lee model and the Vasicek model. Now, stochastic optimal control as a classical tool has been extended to pension funds management (Gao(2008), Josa-Fombellida and Rincón-Zapatero (2010)) and investment and consumption problem (Noh and Kim (2011)). The above mentioned authors did not incorporate liability into theirs models. However, in most of the real-world situations, liability is an important factor which almost all investors should cope with. Therefore, it is clear that the introduction of the liability in a portfolio selection model will make it more practical. This paper introduces liability into the portfolio selection model reported by Korn and Kraft(2001) and interest rate is supposed to be driven by Ho-Lee model and Vasicek model. As the same time, We consider that the financial market consists of three assets: a riskless asset, a stocks and a bond. The objective of an investor is to maximize the expected logarithm utility of the terminal wealth. By applying dynamic programming and HJB equation, we derive a nonlinear second-order partial different equation for the value function, which is hard to produce a closed-form solution. But we apply a Legendre transform to change the primary problem into a dual one and derive the optimal investment strategy by solving the dual equation. The rest of this paper is organized as follows. Section 2 presents the market model and investigated