Constant Elasticity Of Variance Research Articles

Numerical method for singular drift stochastic differential equation driven by fractional Brownian motion

In this paper, we study the stochastic differential equation with singular drift coefficient driven by fractional Brownian motion with Hurst parameter H∈(12,1). We obtain that the optimal convergence rate of the backward Euler method is 1.0. The constant elasticity of variance model and the Aït-Sahalia interest rate model are performed as numerical experiments to validate our theoretical results.

A new type of CEV model: properties, comparison, and application to portfolio optimization

This article proposes and studies a new type of constant elasticity of volatility (CEV) model, titled LVO-CEV, where the name indicates that the excess return is linear in the volatility. The model has either strong or weak solutions, depending on the elasticity parameter, thanks to a connection to radial Ornstein-Uhlenbeck processes. Attainability of lower bounds and the existence of pricing measures are provided. The model allows for closed-form solutions in the context of expected utility theory for investors with hyperbolic absolute risk aversion (HARA) utilities on terminal wealth and consumption. We estimate and implement our model as well as other popular models of indexes and stocks, providing a fair comparison in portfolio management.

Optimal portfolio strategy of wealth process: a Lévy process model-based method

This paper is concerned with the optimal portfolio problem for a company that can invest in two risky assets, where a novel Lévy-process-driven model is constructed to describe the dynamics of the wealth process by using a constant elasticity of variance model and a jump-diffusion process. A delicately designed value function is proposed under the mean–variance criterion to reflect the optimal portfolio for the stochastic volatility model. By using the verification theorem, the desired optimal portfolio strategy is proposed by the solution to certain Hamilton–Jacobi–Bellman equations. Furthermore, the corresponding expressions are achieved by using the stochastic analysis theory. Finally, a numerical simulation example is provided to verify the effectiveness of the proposed optimal portfolio strategy.

Pricing levered warrants under the CEV diffusion model

Much of the work on the valuation of levered (and unlevered) warrants assumes that the volatility of the underlying state variable is constant. This paper extends the literature on warrant pricing to a more general assumption for the state variable process, the so-called constant elasticity of variance (CEV) process. The CEV model is well-known for its ability to capture some empirical observations found in the financial economics literature, namely the asymmetry between equity returns and volatility and the implied volatility skew. Using the CEV process, we are able to reduce pricing bias as the volatility becomes a function of the underlying state variable. We price European-style call warrants without restrictions on the debt maturity. When warrants have the same maturity as debt, it is possible to obtain closed-form solutions for warrants prices. When the maturity of warrants is different from the maturity of debt, prices can be computed numerically through very efficient and simple to implement valuation methodologies.

A progressive approach to solving a generalized CEV-type model by applying symmetry-invariant surface conditions

<abstract><p>In this paper, we examine a type of constant elasticity of variance model that is subject to its terminal condition. We prove that certain transformations may be applied to obtain a simpler equation that has known solution processes. Four cases are obtained that play a role in specifying the many unknown parameters of the model. The corresponding terminal condition is transformed into an initial condition, and we then demonstrate how to solve this Cauchy problem by using Lie symmetries and Poisson's formula. Finally, we examine the behaviour of the obtained solutions.</p></abstract>

Optimal robust reinsurance contracts with investment strategy under variance premium principle

This paper investigates the optimal robust proportional reinsurance contracts with investment in a liquid financial market under variance premium principle in a principal-agent framework. The surplus process of the insurer (agent) is assumed to follow an approximating compound Poisson risk process. Both the insurer and reinsurer (principal) are allowed to invest in a risk-free asset and a risky asset whose price process is governed by the constant elasticity of variance model. The insurer and reinsurer aim to maximize the expected exponential utility of terminal wealth. The reinsurer is ambiguity-averse and has deterministic ambiguity aversion preferences against the diffusion risk caused by the financial market and the approximated diffusion risk which comes from the claim process. The reinsurance price is described by the reinsurer's safety loading which can be decided by the optimal strategies of the insurer and the reinsurer. By utilizing stochastic optimal control principle and HJB (or HJBI) equations, the optimal (robust) proportional reinsurance-investment strategies and the corresponding value functions are obtained explicitly. Numerical examples are provided.

The Finite-Horizon Reversible Investment Problem with the Constant Elasticity of Variance Model

The Finite-Horizon Reversible Investment Problem with the Constant Elasticity of Variance Model

Optimal reinsurance-investment problem with default risk for an insurer under the constant elasticity of variance model

Abstract Accepted by: Giorgio Consigli In this paper we study the optimal reinsurance-investment problem for an insurer with constant absolute risk aversion (CARA) utility. Because the insurer needs to avoid large claims and make profits in the actual operation process, he/she is allowed to buy proportional reinsurance and invest in a risk-free bond, a stock and a default bond. Moreover, the price process of the stock follows the constant elasticity of variance (CEV) model and the correlation between the risk process and the stock’s price is considered. For the objective of expected utility maximization, we establish the Hamilton–Jacobi–Bellman equation and derive the optimal reinsurance-investment strategy explicitly for the pre-default case and the post-default case via the dynamic programming. Finally, numerical examples and sensitivity analyses are provided to illustrate the effects of model parameters on the optimal strategy. We find that (i) the default risk has no impact on the optimal strategy of the stock and the parameters of the stock’s price have no impact on the optimal money amount invested in the defaultable bond. (ii) The optimal reinsurance strategy depends on the volatility of the stock’s price under the CEV model. (iii) The correlation between risk model and the stock’s price does have effects on the insurer’s optimal reinsurance-investment strategy. These findings may provide guidance to the management of insurers in practice.

SHORT-MATURITY ASYMPTOTICS FOR OPTION PRICES WITH INTEREST RATE EFFECTS

We derive the short-maturity asymptotics for option prices in the local volatility model in a new short-maturity limit [Formula: see text] at fixed [Formula: see text], where r is the interest rate and q is the dividend yield. In the case of practical relevance [Formula: see text] being small, however, our result holds for any fixed [Formula: see text]. The result is a generalization of the Berestycki–Busca–Florent formula ( Berestycki et al. ,  2002 ) [Asymptotics and calibration of local volatility models, Quantitative Finance 2, 61–69] for the short-maturity asymptotics of the implied volatility which includes interest rates and dividend yield effects of [Formula: see text] to all orders in n. We obtain the analytical results for the ATM volatility and skew in this asymptotic limit. Explicit results are derived for the CEV model. The asymptotic result is tested numerically against the exact evaluation in the square-root model [Formula: see text], which demonstrates that the new asymptotic result is in very good agreement with the exact evaluation in a wide range of model parameters relevant for practical applications.

Time-consistent strategies between two competitive DC pension plans with the return of premiums clauses and salary risk

. In practice, there is always competition among different pension managers. The competitive pension managers are concerned about their relative performance to make better decisions. In addition, to protect the rights of plan members who die before retirement, most of the defined contribution (DC) pension plans contain return of premiums clauses. In this article, we introduce the return of premiums clauses into two competitive DC pension plans with salary risk. Each pension manager cares about not only his own terminal wealth but also his relative wealth versus his competitor. Both the pension managers are allowed to invest the premiums in a financial market, which consists of one risk-free asset and one risky asset whose price process follows the constant elasticity of variance (CEV) model. We investigate the time-consistent investment strategies of the two pension managers under the mean-variance (MV) criterion. The aim of each pension manager is to maximize the mean and minimize the variance of his relative terminal wealth. The explicit expressions of the time-consistent strategies and the efficient frontiers are obtained via applying the extended stochastic optimal control theory. Some special cases are also discussed in detail. Finally, a numerical simulation demonstrates the results obtained.

A hybrid reinsurance-investment game with delay and asymmetric information

In this paper, we investigate the optimal reinsurance and investment problem in the framework of the hybrid stochastic differential game, which includes a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The stochastic Stackelberg differential subgame considers the leader–follower relationship between a reinsurer and two insurers. The reinsurer, as the leader, prices reinsurance premium and invests its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. On the other side, the two insurers, as the followers, purchase proportional reinsurance from the reinsurer and invest in the same financial market. Also, the competitive relationship between the two insurers is characterized by the non-zero-sum stochastic differential subgame in which each insurer considers the relative performance. By introducing the delay feature into the hybrid game, we are able to obtain the wealth processes depicted by the stochastic delay differential equations. Considering the effect of asymmetric information, we assume the reinsurer and insurers have different levels of information about the financial market. The objective of the reinsurer is to maximize the mean–variance functional of its terminal surplus with delay, while each insurer is to maximize the mean–variance functional of its terminal surplus with delay relative to that of its competitor. By applying techniques in stochastic control theory, we obtain the extended Hamilton–Jacobi–Bellman equations, and then by using the idea of backward induction, we establish the equilibrium strategies and the corresponding value functions. Finally, we include some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the equilibrium strategies.

A Cubic B-Spline Collocation Method for Barrier Options under the CEV Model

In this paper, we construct a new numerical algorithm for the partial differential equation of up-and-out put barrier options under the CEV model. In this method, we use the Crank-Nicolson scheme to discrete temporal variables and the cubic B-spline collocation method to discrete spatial variables. The method is stable and has second-order convergence for both time and space variables. The convergence analysis of the proposed method is discussed in detail. Finally, numerical examples verify the stability and accuracy of the method.

The investment and reinsurance game of insurers and reinsurers with default risk under CEV model

On the premise of considering the interests of insurance companies and reinsurance companies at the same time, this paper studies the investment and reinsurance game between them. Suppose that the compensation process faced by an insurance company is described by Brownian motion with drift. Insurance companies can purchase proportional reinsurance from reinsurance companies, and both companies can invest in a risk-free asset, a risky asset whose price process follows the constant elasticity of variance (CEV) model, and a defaultable bond. With the goal of maximizing the expected utility of weighted terminal wealth, the corresponding Hamilton–Jacobi–Bellman (HJB) equations are established and solved by using the principle of dynamic programming, and the analytical expressions of the equilibrium investment-reinsurance strategies of insurers and reinsurers are derived respectively. Finally, the influence of each model parameter on the equilibrium strategy is analyzed by numerical examples.

An existence result for two-dimensional parabolic integro-differential equations involving CEV model

Abstract In this paper, we present an existence result of weak solutions for some parabolic equations involving the so-called CEV model with jumps.

Robust optimal asset-liability management under square-root factor processes and model ambiguity: a BSDE approach

This article studies robust optimal asset-liability management problems for an ambiguity-averse manager in a possibly non-Markovian environment with stochastic investment opportunities. The manager has access to one risk-free asset and one risky asset in a financial market. The market price of risk relies on a stochastic factor process satisfying an affine-form, square-root, Markovian model, whereas the risky asset’s return rate and volatility are potentially given by general non-Markovian, unbounded stochastic processes. This financial framework includes, but is not limited to, the constant elasticity of variance (CEV) model, the family of 4/2 stochastic volatility models, and some path-dependent non-Markovian models, as exceptional cases. As opposed to most of the papers using the Hamilton-Jacobi-Bellman-Issacs (HJBI) equation to deal with model ambiguity in the Markovian cases, we address the non-Markovian case by proposing a backward stochastic differential equation (BSDE) approach. By solving the associated BSDEs explicitly, we derive, in closed form, the robust optimal controls and robust optimal value functions for power and exponential utility, respectively. In addition, analytical solutions to some particular cases of our model are provided. Finally, the effects of model ambiguity and market parameters on the robust optimal investment strategies are illustrated under the CEV model and 4/2 model with numerical examples.

Optimal reinsurance-investment game for two insurers with SAHARA utilities under correlated markets

In this paper, we study the optimal reinsurance-investment game between two insurers with the same insurance business but different wealth and risk preferences. Assume that the insurers who have the symmetric asymptotic hyperbolic absolute risk aversion (SAHARA) utilities and the price of risky asset obeys the constant elasticity of variance (CEV) model. It is impossible to obtain closed-form solution of the optimal reinsurance-investment strategy due to the non-homothetic property and the complicity of SAHARA utilities. According to establish a strong duality relationship of the value function, we successfully propose an efficient dual control Monte Carlo method for computing the Nash equilibrium strategies. Finally, numerical analysis is given to illustrate the impact of model parameters to Nash equilibrium strategies.

Analytical Formula for Conditional Moments of Extended Heston-CEV Hybrid Model with Time-dependent Parameters

This paper proposes an analytical formula for conditional moments of the extended Heston- CEV hybrid model, where the underlying asset dynamic is formed under the hybrid framework of a constant elasticity of variance (CEV) process that is the March–Rosenfeld process. The formula is received by successfully solving the partial differential equation corresponding to the infi nitesimal generator for two-dimensional process. The formula obtained is easy to use in practice and relatively more general compared with the currently existing results in the literature. In addition, numerical validations are conducted to illustrate the accuracy of our analytical formula by comparing the results gained from the Monte Carlo (MC) simulations.

Stochastic Differential Games on Optimal Investment and Reinsurance Strategy with Delay Under the CEV Model

Stochastic Differential Games on Optimal Investment and Reinsurance Strategy with Delay Under the CEV Model

A double obstacle problem in an optimal investment problem

In this paper, we study a double obstacle problem in partial differential equation that arises in an optimization problem in finance. Precisely, we consider the double obstacle problem which is related to the optimal investment problem with proportional transaction costs of an investor with the logarithmic utility in finite time under the constant elasticity of variance (CEV) model.First, we construct a solution of the double obstacle problem and prove the monotonicity of its free boundaries. From the solution to the double obstacle problem, we construct the solution of the optimization problem. Hence, our result regarding monotonicity indicates the optimal strategy for the optimization problem.

Finite difference scheme versus piecewise binomial lattice for interest rates under the skew CEV model

Interest rates frequently exhibit regulated or controlled characteristics, for example, the prevailing zero interest rate policy, or the leading role of central banks in short rate markets. In order to capture the regulated dynamics of interest rates, we introduce the skew constant-elasticity-of-variance (skew CEV) model. We then propose two numerical approaches: an improved finite difference scheme and a piecewise binomial lattice to evaluate bonds and European/American bond options. Numerical simulations show that both of these two approaches are efficient and satisfactory, with the finite difference scheme being more superior.