Constant Elasticity Of Variance Research Articles

Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models

The present paper studies time-consistent solutions to an investment-reinsurance problem under a mean-variance framework. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer jointly. The claim process of the insurer is governed by a Brownian motion with a drift. A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle. Both the insurer and the reinsurer are assumed to invest in a risky asset, which is distinct for each other and driven by a constant elasticity of variance model. The optimal decision is formulated on a weighted sum of the insurers and the reinsurers surplus processes. Upon a verification theorem, which is established with a formal proof for a more general problem, explicit solutions are obtained for the proposed investment-reinsurance model. Moreover, numerous mathematical analysis and numerical examples are provided to demonstrate those derived results as well as the economic implications behind.

A new statistic to capture the level dependence in stock price volatility

Abstract In this paper, we propose a new covariance estimator based on daily opening, high, low and closing prices. We prove theoretically that the new estimator is unbiased for a pure random walk and further validate it with simulation studies. However, upon examining empirically four indices namely: NIFTY, S&P500, FTSE100 and DAX over the sample period from January 1996 to March 2015, we find that the estimator is upward biased for all the indices under study. This overreaction in stock indices can be attributed to the level dependence in stock indices, something that is not captured by the random walk model. So we explore an alternative to random walk, namely: Constant Elasticity of Variance (CEV) specification. Simulation studies provide supporting evidence that the CEV specification can capture the level dependence that makes the estimator upward biased as seen in the data. Therefore, through this specification exercise, we can see that it is possible to isolate the effect of intraday level dependence in stock prices using our estimator.

Equilibrium investment strategy for DC pension plan with default risk and return of premiums clauses under CEV model

This paper considers an optimal investment problem for a defined contribution (DC) pension plan with default risk in a mean–variance framework. In the DC plan, contributions are supposed to be a predetermined amount of money as premiums and the pension funds are allowed to be invested in a financial market which consists of a risk-free asset, a defaultable bond and a risky asset satisfied a constant elasticity of variance (CEV) model. Notice that a part of pension members could die during the accumulation phase, and their premiums should be withdrawn. Thus, we consider the return of premiums clauses by an actuarial method and assume that the surviving members will share the difference between the return and the accumulation equally. Taking account of the pension fund size and the volatility of the accumulation, a mean–variance criterion as the investment objective for the DC plan can be formulated, and the original optimization problem can be decomposed into two sub-problems: a post-default case and a pre-default case. By applying a game theoretic framework, the equilibrium investment strategies and the corresponding equilibrium value functions can be obtained explicitly. Economic interpretations are given in the numerical simulation, which is presented to illustrate our results.

Asymptotic approach to the pricing of geometric asian options under the CEV model

This paper studies the pricing of Asian options whose payoffs depend on the average value of an underlying asset during the period to a maturity. Since the Asian option is not so sensitive to the value of underlying asset, the possibility of manipulation is relatively small than the other options such as European vanilla and barrier options. We derive the pricing formula of geometric Asian options under the constant elasticity of variance (CEV) model that is one of local volatility models, and investigate the implication of the CEV model for geometric Asian options.

Pricing maximum-minimum bidirectional options in trinomial CEV model

Maximum-minimum bidirectional options are a kind of exotic path dependent options. In the constant elasticity of variance (CEV) model, a combining trinomial tree was structured to approximate the non-constant volatility that is a function of the underlying asset. On this basis, a simple and efficient recursive algorithm was developed to compute the risk-neutral probability of each different node for the underlying asset reaching a maximum or minimum price and the total number of maxima (minima) in the trinomial tree. With help of it, the computational problems can be effectively solved arising from the inherent complexities of different types of maximum-minimum bidirectional options when the underlying asset evolves as the trinomial CEV model. Numerical results demonstrate the validity and the convergence of the approach mentioned above for the different parameter values set in the trinomial CEV model.

A delayed stochastic volatility correction to the constant elasticity of variance model

The Black-Scholes model does not account non-Markovian property and volatility smile or skew although asset price might depend on the past movement of the asset price and real market data can find a non-flat structure of the implied volatility surface. So, in this paper, we formulate an underlying asset model by adding a delayed structure to the constant elasticity of variance (CEV) model that is one of renowned alternative models resolving the geometric issue. However, it is still one factor volatility model which usually does not capture full dynamics of the volatility showing discrepancy between its predicted price and market price for certain range of options. Based on this observation we combine a stochastic volatility factor with the delayed CEV structure and develop a delayed hybrid model of stochastic and local volatilities. Using both a martingale approach and a singular perturbation method, we demonstrate the delayed CEV correction effects on the European vanilla option price under this hybrid volatility model as a direct extension of our previous work [12].

Optimal reinsurance and investment policies with the CEV stock market

In this paper, under the criterion of maximizing the expected exponential utility of terminal wealth, we study the optimal proportional reinsurance and investment policy for an insurer with the compound Poisson claim process. We model the price process of the risky asset to the constant elasticity of variance (for short, CEV) model, and consider net profit condition and variance reinsurance premium principle in our work. Using stochastic control theory, we derive explicit expressions for the optimal policy and value function. And some numerical examples are given.

Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

The optimal investment–consumption problem under the constant elasticity of variance (CEV) model is solved using the invariant approach. Firstly, the invariance criteria for scalar linear second‐order parabolic partial differential equations in two independent variables are reviewed. The criteria is then employed to reduce the CEV model to one of the four Lie canonical forms. It is found that the invariance criteria help in transforming the original equation to the second Lie canonical form and with a proper parameter selection; the required transformation converts the original equation to the first Lie canonical form that is the heat equation. As a consequence, we find some new classes of closed‐form solutions of the CEV model for the case of reduction into heat equation and also into second Lie canonical form. The closed‐form analytical solution of the Cauchy initial value problems for the CEV model under investigation is also obtained. Copyright © 2016 John Wiley & Sons, Ltd.

ARITHMETIC AVERAGE ASIAN OPTIONS WITH STOCHASTIC ELASTICITY OF VARIANCE

This article deals with the pricing of Asian options under a constant elasticity of variance (CEV) model as well as a stochastic elasticity of variance (SEV) model. The CEV and SEV models are underlying asset price models proposed to overcome shortcomings of the constant volatility model. In particular, the SEV model is attractive because it can characterize the feature of volatility in risky situation such as the global financial crisis both quantitatively and qualitatively. We use an asymptotic expansion method to approximate the no-arbitrage price of an arithmetic average Asian option under both CEV and SEV models. Subsequently, the zero and non-zero constant leverage effects as well as stochastic leverage effects are compared with each other. Lastly, we investigate the SEV correction effects to the CEV model for the price of Asian options.

Mean–variance asset–liability management under constant elasticity of variance process

This paper investigates a mean–variance asset–liability management (ALM) problem under the constant elasticity of variance (CEV) process. The company can invest in n+1 assets: one risk-free bond and n risky stocks. The uncontrollable liability process is modelled by a geometric Brownian motion. The feasibility is studied and potential optimal portfolio is proven to be admissible. We derive the efficient frontier and efficient feedback portfolio in terms of the solutions of two backward stochastic differential equations (BSDEs), which do not admit analytical solutions in general. The closed form solutions are obtained under some special cases. Applying the Monte Carlo simulation, we provide several numerical examples to demonstrate how the efficient frontier is influenced by the relevant parameters.

Optimal harvesting for a logistic growth model with predation and a constant elasticity of variance

In this paper, we address the problem of optimal management of renewable resources such as agricultural commodities and fishery production. For that purpose, we consider the population associated with such commodities and assume that its size evolves according to a logistic growth model with a predation term given by a Holling type-n functional response. Additionally, we assume that such population is subject to random fluctuations, modeled by a diffusive term driven by a one-dimensional Brownian motion and having a power-type coefficient, thus endowing the model under consideration with the property of having constant elasticity of variance. Since the stochastic differential equation associated with this model does not fit the standard assumptions in the stochastic optimal control literature, namely sublinear growth, we develop an appropriate version of the dynamic programming principle for the problem under consideration herein, proceeding also to provide a characterization of the optimal harvesting strategies and discuss some qualitative properties of the corresponding value function.

A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models

A novel time-stepping scheme, called transformed jump-adapted backward Euler method, is developed in this paper to simulate a class of jump-extended CIR and CEV models. The proposed scheme is able to preserve the positivity of the underlying problems. Furthermore, its strong convergence rate of order one is recovered for the considered models with non-Lipschitz diffusion coefficients. Numerical examples are finally reported to confirm our theoretical findings.

LEVERAGED ETF IMPLIED VOLATILITIES FROM ETF DYNAMICS

AbstractThe growth of the exchange‐traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local‐stochastic volatility models. A closed‐form approximation for prices is derived for European‐style options whose payoffs depend on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed‐form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three settings: Heston, limited constant elasticity of variance (CEV), and limited SABR; the last two are regularized versions of the well‐known CEV and SABR models.

The Binomial CEV Model and the Greeks

AbstractThis article compares alternative binomial approximation schemes for computing the option hedge ratios studied by Chung and Shackleton (2002), Chung, Hung, Lee, and Shih (2011), and Pelsser and Vorst (1994) under the lognormal assumption, but now considering the constant elasticity of variance (CEV) process proposed by Cox (1975) and using the continuous‐time analytical Greeks recently offered by Larguinho, Dias, and Braumann (2013) as the benchmarks. Among all the binomial models considered in this study, we conclude that an extended tree binomial CEV model with the smooth and monotonic convergence property is the most efficient method for computing Greeks under the CEV diffusion process because one can apply the two‐point extrapolation formula suggested by Chung et al. (2011). © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:90–104, 2017

Valuation of American Continuous-Installment Options Under the Constant Elasticity of Variance Model

AbstractThis article prices American-style continuous-installment options in the constant elasticity of variance (CEV) diffusion model where the volatility is a function of the stock price. We derive the semi-closed form formulas for the American continuous-installment options using Kim’s integral representation method and then obtain the closed-form solutions by approximating the optimal exercise and stopping boundaries as step functions. We demonstrate the speed-accuracy of our approach for different parameters of the CEV model. Furthermore, the effects on both option price and the optimal boundaries are discussed and the causes of underestimating or overestimating the option prices are analyzed under the classical Black-Scholes-Merton model, in particular, for the case of elasticity coefficient with numerical examples.

On convergence of Laplace inversion for the American put option under the CEV model

In this paper, we study the convergence of the inverse Laplace transform for valuing American put options when the dynamics of the risky asset is governed by the constant elasticity of variance (CEV) model. The CEV model is one popular alternative of the Black–Scholes model to describe well the real financial market. We calculate various coefficients explicitly and prove that the inverse Laplace transform converges absolutely using the properties of Whittaker functions.

On the multiplicity of option prices under CEV with positive elasticity of variance

The discounted stock price under the Constant Elasticity of Variance model is not a martingale when the elasticity of variance is positive. Two expressions for the European call price then arise, namely the price for which put-call parity holds and the price that represents the lowest cost of replicating the call option’s payoffs. The greeks of European put and call prices are derived and it is shown that the greeks of the risk-neutral call can substantially differ from standard results. For instance, the relation between the call price and variance may become non-monotonic. Such unfamiliar behavior then might yield option-based tests for the potential presence of a bubble in the underlying stock price.

Time-Consistent Investment Strategy for DC Pension Plan with Stochastic Salary Under CEV Model

This paper aims to derive the time-consistent investment strategy for the defined contribution (DC) pension plan under the mean-variance criterion. The financial market consists of a risk-free asset and a risky asset of which price process satisfies the constant elasticity of variance (CEV) model. Compared with the geometric Brownian motion model, the CEV model has the ability of capturing the implied volatility skew and explaining the volatility smile. The authors assume that the contribution to the pension fund is a constant proportion of the pension member’s salary. Meanwhile, the salary is stochastic and its volatility arises from the price process of the risky asset, which makes the proposed model different from most of existing researches and more realistic. In the proposed model, the optimization problem can be decomposed into two sub-problems: Before and after retirement cases. By applying a game theoretic framework and solving extended Hamilton-Jacobi-Bellman (HJB) systems, the authors derive the time-consistent strategies and the corresponding value functions explicitly. Finally, numerical simulations are presented to illustrate the effects of model parameters on the time-consistent strategies.

A Laplace Space Approach to American Options

In this paper, we extend the lower-upper bound approximation (LUBA) idea of Broadie and Detemple [Broadie, M., Detemple, J., (1996) American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies. 9(4): 1211-1250] to the Laplace space. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, for which there exist closed-form expressions for the Laplace transforms of the corresponding “capped (barrier) option prices”. The method is applied to a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures similar as Broadie and Detemple in the Laplace space. Finally we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option, and further utilize the early exercise premium (EEP) representation in the Laplace space to obtain the upper bound. We obtain explicit expressions in the case of the constant elasticity of variance (CEV) model (Wong and Zhao) and the double-exponential jump diffusion (DEJD) model (Leippold and Vasiljevic). Numerical examples show that our lower and upper bounds are accurate and efficient compared to results in the literature. To the best of authors’ knowledge, it is the first time that the LUBA idea of Broadie and Detemple is applied to a model with jumps, and this solves an open question stated on page 1181 of Kou and Wang.

Moderate deviations and central limit theorem for positive diffusions

In this paper, we establish a central limit theorem and a moderate deviation principle for the positive diffusions, including the CEV and CIR models. The proof is based on the exponential approximations theorem and Burkholder-Davis-Gundy’s inequality.