Constant Elasticity Of Variance Research Articles

VIX derivatives valuation and estimation based on closed-form series expansions

We propose a new methodology to evaluate VIX derivatives. The approach is based on a closed-form Hermite series expansion, and can be applied to general stochastic volatility models. We exemplify the proposed method using the Heston model, the mean-reverting CEV model and the 3/2 model. Numerical results show that the proposed method is accurate and efficient.

The CEV model and its application to financial markets with volatility uncertainty

We survey the financial markets whose risks are caused by uncertain volatilities. The financial markets focus on the assets which are effectively allocated in one risk-free asset and one risky asset, whose price process is governed by the constant elasticity of variance ( C E V for short) model which contains the G -Brownian motion rather than the classical Brownian motion. Such the C E V model which includes the G -Brownian motion utilized to financial markets is the extension of the classical C E V model. Applying the concept of arbitrage and the properties of G -expectation, we consider stock price dynamics which exclude arbitrage opportunities. Moreover, the interval of no-arbitrage price for the general European contingent claims is found in the Markovian case.

Pricing and hedging vulnerable option with funding costs and collateral

We explore the valuation and hedging strategies of a European vulnerable option with funding costs and collateralization for local volatility models. It is found that, in the absence of arbitrage opportunities, the option price must lie within a no-arbitrage band. The boundaries of no-arbitrage band are computed as solutions to backward stochastic differential equations (BSDEs in short) of replicating strategy and offsetting strategy. Under some conditions, we obtain the closed-form representations of the no-arbitrage band for local volatility models. In particular, the fully explicit expressions of the no-arbitrage band for Black–Scholes model and the constant elasticity of variance (CEV) model with time-dependent parameters are derived. Furthermore, we provide a strategy for the option holder by using the risky bond issued by the option writer to hedge the remaining potential losses. By virtue of numerical simulation, the impact of the default risk, funding costs and collateral can be observed visually.

A multiquadric quasi-interpolations method for CEV option pricing model

The pricing of option contracts when the underlying process follows the constant elasticity of variance (CEV) model is considered. For CEV European options, the closed-form solutions involve the non-central chi-square distribution, whose computations by the current literatures are rather unstable and extremely expensive. Based on multiquadric quasi-interpolation methods, this study suggests a stable and fast numerical algorithm for CEV option pricing model. The method is confirmed to be a multinomial tree, in which the underlying variable moves from its initial value to an infinity of possible values of the next time step. The probabilities in the associated tree are ensured to be positive, which is a sufficient condition for stability and convergence. The method is flexible, since it is simple to implement with the nonuniform knots. Moreover, the method is easy to value the Greek letters which are important parameters in financial engineering, as the multiquadric function is infinitely continuously differentiable. Besides, the method does not require solving a resultant full matrix, the ill-conditioning problem arising when using the radial basis functions as a global interpolant can be avoided. Numerical experiments imply that the method is highly effective to calculate the stock options and its Greeks under the CEV model.

Finite-maturity stock loans under the constant elasticity of variance model

ABSTRACTThis article expresses the constant elasticity of variance stock loans pricing formula in terms of the noncentral chi-square distribution. By using the numerical solutions of the integral equation for the early exit boundary, we calculated the valuation of the stock loans. Numerical analysis is carried out to establish explicitly the value of such a loan, as well as the loan amount and the fee for providing such a service.

Optimal excess-of-loss reinsurance and investment problem with delay and jump–diffusion risk process under the CEV model

In this paper, we investigate an optimal investment and excess-of-loss reinsurance problem with delay and jump–diffusion risk process for an insurer. Specifically, the insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market, where the surplus of insurer is represented by a jump–diffusion model and the financial market consists of one risk-free asset and one risky asset whose price process is governed by a constant elasticity of variance model. In addition, the performance-related capital inflow/outflow is introduced, the wealth process of insurer is modeled by a stochastic differential delay equation. The insurer aims to seek the optimal excess-of-loss reinsurance and investment strategy to maximize the expected exponential utility of the combination of terminal wealth and average performance wealth. By solving a Hamilton–Jacobi–Bellman equation, the closed-form expressions for the optimal strategy and the optimal value function are derived. Finally, some special cases of our model and results are presented, and some numerical examples for our results are provided.

The Early Exercise Boundary Under the Jump to Default Extended CEV Model

This paper proves the existence, uniqueness, monotonicity and continuity of the early exercise boundary attached to American-style standard options under the jump to default extended constant elasticity of variance model of Carr and Linetsky (Financ Stoch 10(3):303–330, 2006).

Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

AbstractThe optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws.

Approximation of Non-Lipschitz SDEs by Picard Iterations

ABSTRACTIn this article, we propose an approximation method based on Picard iterations deduced from the Doléans–Dade exponential formula. Our method allows to approximate trajectories of Markov processes in a large class, e.g., solutions to non-Lipchitz stochastic differential equation. An application to the pricing of Asian-style contingent claims in the constant elasticity of variance model is presented and compared to other methods of the literature.

Ethical culture as predictor of work motivation: An application of the CEV model

This study examines the relationship between ethical culture and work motivation, whereby specific dimensions of ethical culture are measured by employing the Corporate Ethical Virtues Model (CEV) developed by Muel Kaptein and his associates. The analysis is based on data collected by means of an online survey conducted in Croatia on a sample of 400 employees of private and public sector organizations. The results presented in the paper show that clarity is rated the highest, while supportability is the lowest-rated ethical virtue in the aforementioned organizations. Ethical culture as a whole is a good predictor of work motivation, with congruence of the management being the most important predictor. An interesting fi nding of this study is also a negative association between sanctionability and work motivation when all other predictors are being held constant. It is thus concluded that the CEV model represents a good measurement tool with a proven construct validity outside its original social and cultural context. The paper closes with suggestions for future research, especially for a possible broadening of the application of the CEV model.

Optimal investment strategies for general utilities under dynamic elasticity of variance models

This paper studies the optimal investment strategies under the dynamic elasticity of variance (DEV) model which maximize the expected utility of terminal wealth. The DEV model is an extension of the constant elasticity of variance model, in which the volatility term is a power function of stock prices with the power being a nonparametric time function. It is not possible to find the explicit solution to the utility maximization problem under the DEV model. In this paper, a dual-control Monte-Carlo method is developed to compute the optimal investment strategies for a variety of utility functions, including power, non-hyperbolic absolute risk aversion and symmetric asymptotic hyperbolic absolute risk aversion utilities. Numerical examples show that this dual-control Monte-Carlo method is quite efficient.

Time-consistent mean–variance proportional reinsurance and investment problem in a defaultable market

In this paper, we consider an optimal time-consistent reinsurance-investment problem incorporating a defaultable security for a mean–variance insurer under a constant elasticity of variance (CEV) model. In our model, the insurer’s surplus process is described by a jump-diffusion risk model, the insurer can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset, a defaultable bond and a risky asset whose price process is assumed to follow a CEV model. Using a game theoretic approach, we establish the extended Hamilton–Jacobi–Bellman system for the post-default case and the pre-default case, respectively. Furthermore, we obtain the closed-from expressions for the time-consistent reinsurance-investment strategy and the corresponding value function in both cases. Finally, we provide numerical examples to illustrate the impacts of model parameters on the optimal time-consistent strategy.

Testing CEV stochastic volatility models using implied volatility index data

We test the goodness-of-fit of stochastic volatility (SV) models using the implied volatility index of the KOSPI200 options (VKOSPI). The likelihood ratio tests reject the Heston and Hull–White SV models, whether or not they include jumps. Our estimation results advocate the unconstrained constant elasticity of variance (CEV) model with return jumps for describing the physical-measure dynamics of the spot index. The sub-period analysis shows that there was a significant increase in the size and frequency of jumps during the crisis period, when compared to those in the normal periods.

Quantization Under the Real-world Measure: Fast and Accurate Valuation of Long-dated Contracts

This paper provides a methodology for fast and accurate pricing of the long-dated contracts that arise as the building blocks of insurance and pension fund agreements. It applies the recursive marginal quantization (RMQ) and joint recursive marginal quantization (JRMQ) algorithms outside the framework of traditional risk-neutral methods by pricing options under the real-world probability measure, using the benchmark approach. The benchmark approach is reviewed, and the real-world pricing theorem is presented and applied to various long-dated claims to obtain less expensive prices than suggested by traditional risk-neutral valuation. The growth-optimal portfolio (GOP), the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model (TCEV). Analytic European option prices are derived and the RMQ algorithm is used to efficiently and accurately price Bermudan options on the GOP. The TCEV model is then combined with a $3/2$ stochastic short-rate model and RMQ is used to price zero-coupon bonds and zero-coupon bond options, highlighting the departure from risk-neutral pricing.

Recursive marginal quantization of higher-order schemes

Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive marginal quantization (RMQ) of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform RMQ for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. We further extend the applicability of RMQ by showing how absorption and reflection at the zero boundary may be incorporated, when necessary. To illustrate the improved accuracy of the higher-order schemes, various computations are performed using geometric Brownian motion and the constant elasticity of variance model. For both models, we provide evidence of improved weak order convergence and computational efficiency. By pricing European, Bermudan and barrier options, further evidence of improved accuracy of the higher-order schemes is demonstrated.

Spectral study of options based on CEV model with multidimensional volatility

This article studies the derivatives pricing using a method of spectral analysis, a theory of singular and regular perturbations. Using a risk-neutral assessment, the authors obtain the Cauchy problem, which allows to calculate the approximate price of derivative assets and their volatility based on the diffusion equation with fast and slow variables of nonlocal volatility, and they obtain a model with multidimensional stochastic volatility. Applying a spectral theory of self-adjoint operators in Hilbert space and a theory of singular and regular perturbations, an analytic formula for approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on l-fast variables and r-slowly variables, l ≥ 1, r ≥ 1, l ∈ N, r ∈ N and a local variable. Applying the Sturm-Liouville theory, Fredholm’s alternatives, as well as the analysis of singular and regular perturbations at different time scales, the authors obtained explicit formulas for derivatives price approximations. To obtain explicit formulae, it is necessary to solve 2l Poisson equations.

Correlation of Brownian Motions and Its Impact on a Reinsurer’s Optimal Investment Strategy and Reinsured Proportion under Exponential Utility Maximization and Constant Elasticity of Variance Model

This work investigated a reinsurer’s optimal investment strategy and the pro-portion he accepted for reinsurance under proportional reinsurance and expo-nential utility preference in the cases where the Brownian motions were corre-lated and where they did not correlate. The reinsurer invested in a market in which the price process of the risky asset is governed by constant elasticity of variance (CEV) model. The required Hamilton-Jacobi-Bellman Equations (HJB) were derived using the Ito’s lemma from which the optimal investment strategy and reinsured proportion were calculated. Also investigated were the implications of the correlation coefficient.

Optimal Investment Policy for Insurers under the Constant Elasticity of Variance Model

Optimal Investment Policy for Insurers under the Constant Elasticity of Variance Model

Optimal Portfolio Management in a Modified Constant Elasticity of Variance Model

Optimal Portfolio Management in a Modified Constant Elasticity of Variance Model

Time-Consistent Mean-Variance Reinsurance-Investment Problems Under Unbounded Random Parameters: BSDE and Uniqueness

To strike the best balance between insurance risk and profit, insurers transfer insurable risk through reinsurance and enhance yield by participating into the financial market. The long-term commitment of insurance contracts makes insurers necessary to consider time-consistent (TC) reinsurance-investment policies. Using the open-loop TC mean-variance (MV) reinsurance-investment framework, we investigate the equilibrium reinsurance-investment problems for the financial market with unbounded random coefficients or, specifically, an unbounded risk premium. We characterize the problem via a backward stochastic differential equation (BSDE) framework. An explicit solution to the equilibrium strategies is derived for a constant risk aversion under a general class of stochastic models, embracing the constant elasticity of variance (CEV) and Ornstein-Uhlenbeck (OU) processes as special cases. For state-dependent risk aversions, the problem is related to the existence of a solution to a quadratic BSDE with unbounded parameters. A semi-closed form solution is derived, up to the solution to a nonlinear partial differential equation. By examining properties of the equilibrium strategies numerically, we find that the reinsurance decision is greatly affected by the market situation under the state-dependent risk aversion case. We prove the uniqueness of equilibrium strategies for both cases.