On The Closed Form Strategies of an Investor under the CEV and CIR Processes

In this paper, the explicit solutions of the optimal investment plans of an investor with exponential utility function exhibiting constant absolute risk aversion (CARA) under constant elasticity of variance (CEV) and stochastic interest rate is studied. A portfolio comprising of a risk-free asset modelled by the Cox-Ingersoll-Ross (CIR) process and two risky assets modelled by the CEV process is considered, where the instantaneous volatilities of the two risky assets form a 2 x 2 matrix n = {np,q}2x2 such that nnT is positive definite. Using the power transformation and change of variable approach with asymptotic expansion technique, explicit solutions of the optimal investment plans are found. Moreover, numerical simulations are used to study the effects of the interest rate, elasticity parameter, correlation coefficient and the risk averse coefficient on the optimal investment plans.

A simple closed-form approximation for constant elasticity of variance spread options

By applying the Lie–Trotter operator splitting method and the idea of the WKB method, we have developed a simple, accurate and efficient analytical approximation for pricing the constant elasticity of variance (CEV) spread options. The derived option price formula bears a striking resemblance to Kirk’s formula of the Black–Scholes spread options. Illustrative numerical examples show that the proposed approximation is not only extremely fast and robust, but it is also remarkably accurate for typical volatilities and maturities of up to two years.

A Reinsurance and Investment Game between Two Insurers under the CEV Model

In this paper, the problem of nonzero-sum stochastic differential game between two competing insurance companies is considered, i.e., the relative performance concerns. A certain proportion of reinsurance can be taken out by each insurer to control his own risk. Moreover, each insurer can invest in a risk-free asset and risk asset with the price dramatically following the constant elasticity of variance (CEV) model. Based on the principle of dynamic programming, a general framework regarding Nash equilibrium for nonzero-sum games is established. For the typical case of exponential utilization, we, respectively, give the explicit solutions of the equilibrium strategy as well as the equilibrium function. Some numerical studies are provided at last which assist in obtaining some economic explanations.

Portfolio Strategy for an Investor with Logarithm Utility and Stochastic Interest Rate under Constant Elasticity of Variance Model

This paper is aim at maximizing the expected utility of an investor’s terminal wealth; to achieve this, we study the optimal portfolio strategy for an investor with logarithm utility function under constant elasticity of variance (CEV) model in the presence of stochastic interest rate. A portfolio comprising of one risk free asset and one risky asset is considered where the risk free interest rate follows the Cox- Ingersoll-Ross (CIR) model and the risky asset is modelled by CEV. Using power transformation, change of Variable and asymptotic expansion technique, an explicit solution of the optimal portfolio strategy and the Value function is obtained. Furthermore, numerical simulations are presented to study the effect of some parameters on the optimal portfolio strategy under stochastic interest rate.

Efficient exponential timestepping algorithm using control variate technique for simulating a functional of exit time of one-dimensional Brownian diffusion with applications in finance

The exponential timestepping Euler algorithm with a boundary test is adapted to simulate an expected of a function of exit time, such as the expected payoff of barrier options under the constant elasticity of variance (CEV) model. However, this method suffers from a high Monte Carlo (MC) statistical error due to its exponentially large exit times with unbounded samples. To reduce this kind of error efficiently and to speed up the MC simulation, we combine such an algorithm with an effective variance reduction technique called the control variate method. We call the resulting algorithm the improved Exp algorithm for abbreviation. In regard to the examples we consider in this paper for the restricted CEV process, we found that the variance of the improved Exp algorithm is about six times smaller than that of the Jansons and Lythe original method for the down-and-out call barrier option. It is also about eight times smaller for the up-and-out put barrier option, indicating that the gain in efficiency is significant without significant increase in simulation time.

Hedging with Liquidity Risk under CEV Diffusion

We study a discrete time hedging and pricing problem in a market with the liquidity risk. We consider a discrete version of the constant elasticity of variance (CEV) model by applying Leland’s discrete time replication scheme. The pricing equation becomes a nonlinear partial differential equation, and we solve it by a multi scale perturbation method. A numerical example is provided.

Optimal Investment Policy for Insurers under the Constant Elasticity of Variance Model with a Correlated Random Risk Process

This paper investigates the optimal portfolio choice problem for a large insurer with negative exponential utility over terminal wealth under the constant elasticity of variance (CEV) model. The surplus process is assumed to follow a diffusion approximation model with the Brownian motion in which is correlated with that driving the price of the risky asset. We first derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation and then obtain explicit solutions to the value function as well as the optimal control by applying a variable change technique and the Feynman–Kac formula. Finally, we discuss the economic implications of the optimal policy.