Optimal Investment Strategy under the CEV Model with Stochastic Interest Rate

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.

A Discontinuous Galerkin Method for Pricing American Options Under the Constant Elasticity of Variance Model

The pricing of option contracts is one of the classical problems in Mathematical Finance. While useful exact solution formulas exist for simple contracts, typically numerical simulations are mandated due to the fact that standard features, such as early-exercise, preclude the existence of such solutions. In this paper we consider derivatives which generalize the classical Black-Scholes setting by not only admitting the early-exercise feature, but also considering assets which evolve by the Constant Elasticity of Variance (CEV) process (which includes the Geometric Brownian Motion of Black-Scholes as a special case). In this paper we investigate a Discontinuous Galerkin method for valuing European and American options on assets evolving under the CEV process which has a number of advantages over existing approaches including adaptability, accuracy, and ease of parallelization.

Approximating Explicitly the Mean-Reverting CEV Process

We are interested in the numerical solution of mean-reverting CEV processes that appear in financial mathematics models and are described as nonnegative solutions of certain stochastic differential equations with sublinear diffusion coefficients of the form(xt)q,where1/2<q<1. Our goal is to construct explicit numerical schemes that preserve positivity. We prove convergence of the proposed SD scheme with rate depending on the parameterq. Furthermore, we verify our findings through numerical experiments and compare with other positivity preserving schemes. Finally, we show how to treat the two-dimensional stochastic volatility model with instantaneous variance process given by the above mean-reverting CEV process.

Continuous-Time Mean-Variance Portfolio Selection under the CEV Process

We consider a continuous-time mean-variance portfolio selection model when stock price follows the constant elasticity of variance (CEV) process. The aim of this paper is to derive an optimal portfolio strategy and the efficient frontier. The mean-variance portfolio selection problem is formulated as a linearly constrained convex program problem. By employing the Lagrange multiplier method and stochastic optimal control theory, we obtain the optimal portfolio strategy and mean-variance efficient frontier analytically. The results show that the mean-variance efficient frontier is still a parabola in the mean-variance plane, and the optimal strategies depend not only on the total wealth but also on the stock price. Moreover, some numerical examples are given to analyze the sensitivity of the efficient frontier with respect to the elasticity parameter and to illustrate the results presented in this paper. The numerical results show that the price of risk decreases as the elasticity coefficient increases.

VOLATILITY DERIVATIVES IN MARKET MODELS WITH JUMPS

It is well documented that a model for the underlying asset price process that seeks to capture the behaviour of the market prices of vanilla options needs to exhibit both diffusion and jump features. In this paper we assume that the asset price process S is Markov with càdlàg paths and propose a scheme for computing the law of the realized variance of the log returns accrued while the asset was trading in a prespecified corridor. We thus obtain an algorithm for pricing and hedging volatility derivatives and derivatives on the corridor-realized variance in such a market. The class of models under consideration is large, as it encompasses jump-diffusion and Lévy processes. We prove the weak convergence of the scheme and describe in detail the implementation of the algorithm in the characteristic cases where S is a CEV process (continuous trajectories), a variance gamma process (jumps with independent increments) or an infinite activity jump-diffusion (discontinuous trajectories with dependent increments).