Constant Elasticity Of Variance Research Articles

CEV模型下鲁棒最优投资和超额损失再保险问题研究

CEV模型下鲁棒最优投资和超额损失再保险问题研究

Appendix: Recursive Marginal Quantization of Higher-Order Schemes

This document serves as the online appendix to McWalter et al. [2017]. It provides efficient matrix implementations for both vector quantization and the generalized recursive marginal quantization, which allows for the implementation of the higher-order discretization schemes presented in the main paper. It includes graphical examples of how the higher-order discretization schemes reduce the error in the implied marginal distributions for both the geometric Brownian motion process (GBM) and its generalization, the constant elasticity of variance (CEV) process. It concludes with an example of the effect of modelling the CEV process with an absorbing or reflecting boundary at zero. The full-text version of this paper can be found at http://ssrn.com/abstract=2894753.

RBF-FD schemes for option valuation under models with price-dependent and stochastic volatility

Radial basis functions based finite difference schemes for the solution of partial differential equations have the advantage that an optimal choice of the shape parameter can yield better accuracies than standard finite difference discretisations based on the same number of nodal points. Such schemes known as local radial basis functions methods are considered for the pricing of options under the constant elasticity of variance and the Heston stochastic volatility model. A general methodology for approximating first and second order derivative terms in the finance pdes is presented and the resulting schemes are applied for option valuation. For one-dimensional problems, we derive a compact-RBF scheme which achieves a higher order accuracy when combined with a local mesh refinement strategy. Numerical results and comparisons made for European, American and barrier options illustrate the good performances of the localized radial basis functions methods.

Leverage Effect and Stochastic Volatility in the Agricultural Commodity Market under the CEV Model

The problem of price fluctuation is crucial to the concept of financial engineering nowadays. The aim of this paper is twofold; first to investigate the leverage effect of the main agricultural commodities – wheat and corn, i. e. the relationship between monetary returns and the volatility of commodity prices and, secondly to capture their stochastic volatility by forming an appropriate model. The data are considered as ‘post‑crisis’ data. That means the period after the biggest shock to the world economy. Thus, the Constant Elasticity of Variance (CEV) model is used calibrated to the Generalized Method of Moments (GMM). The paper is briefly based on the research of Geman and Shih (2009), who propose an extension in capturing the leverege effect in the commodity market. Their results show a positive relationship between commodity price returns and the volatility in both the corn and wheat derivative market. According to these results, corn futures prices are characterized significantly under the CEV model. On the other side in the wheat futures market exists a driftless condition by using stochastic volatility models.

Approximate option pricing and hedging in the CEV model via path-wise comparison of stochastic processes

This paper presents a methodology of finding explicit boundaries for some financial quantities via comparison of stochastic processes. The path-wise comparison theorem is used to establish domination of the stock price process by a process with a known distribution that is relatively simple. We demonstrate how the comparison theorem can be applied in the constant elasticity of variance model to derive closed-form expressions for option price bounds, an approximate hedging strategy and a conditional value-at-risk estimate. We also provide numerical examples and compare precision of our method with the distribution-free approach.

Liquidity Risk and Volatility Risk in Credit Spread Models: A Unified Approach

AbstractWe present an integrated framework incorporating both exogenous liquidity risk in the secondary corporate bond market and volatility risk in the dynamics of asset value in debt rollover models. Using an innovative theoretical approach we derive general expressions for the debt and equity values in all cases. Taking advantage of the analytical expressions for the asset value with the constant elasticity of variance (CEV) process, we show numerically using realistic parameter values from empirical studies that volatility risk, together with deteriorating bond market liquidity, decreases both debt and equity values and significantly increases the credit spreads.

On the Theoretical Valuation of V-KOSPI 200 Futures

Although the V-KOSPI 200 Futures markets opened in November 2014, trading has not been active until recently. One of the reasons for the illiquidity is due to the lack of a market consensus on the stochastic process model for the underlying volatility index (V-KOSPI 200). Given this fact, there is no theoretical pricing model that can be used for the determination of the benchmark price for the V-KOSPI 200 Futures. In this paper, we use the generalized method of moments method to search for a model that fits well with the time series of V-KOSPI 200 under the historical measure. In addition, we compare the performance of each model for the pricing of the V-KOSPI 200 Futures under the risk neutral measure. In the empirical analysis, we find that the CEV (constant elasticity of variance) parameter with the value about 1.5 is needed to price both the underlying V-KOSPI 200 process (under the physical measure) and the V-KOSPI 200 Futures (under the risk neutral measure). We also find that the mean reversion property is necessary to explain the dynamics of V-KOSPI 200.

Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model

In this paper, we consider the optimal investment problem for both an insurer and a reinsurer. The insurer’s wealth process is described by a jump diffusion risk model and the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset whose price process follows the constant elasticity of variance (CEV) model. Moreover, the correlation between risk model and the risky asset’s price is considered. The objective is maximizing the expected utility of the insurer’s and the reinsurer’s terminal wealth. Applying stochastic control theory, we establish the corresponding Hamilton–Jacobi–Bellman (HJB) equations and derive optimal investment–reinsurance strategies for exponential utility function. Finally, numerical examples are provided to analyze the effects of parameters on the optimal strategies.

Option Price Decomposition in Spot-Dependent Volatility Models and Some Applications

We obtain a Hull and White type option price decomposition for a general local volatility model. We apply the obtained formula to CEV model. As an application we give an approximated closed formula for the call option price under a CEV model and an approximated short term implied volatility surface. These approximated formulas are used to estimate model parameters. Numerical comparison is performed for our new method with exact and approximated formulas existing in the literature.

Hybrid Laplace transform and finite difference methods for pricing American options under complex models

In this paper, we propose a hybrid Laplace transform and finite difference method to price (finite-maturity) American options, which is applicable to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump–diffusion (HEJD), Markov regime switching models, and the finite moment log stable (FMLS) models. We first apply Laplace transforms to free boundary partial differential equations (PDEs) or fractional partial differential equations (FPDEs) governing the American option prices with respect to time, and obtain second order ordinary differential equations (ODEs) or fractional differential equations (FDEs) with free boundary, which is named as the early exercise boundary in the American option pricing. Then, we develop an iterative algorithm based on finite difference methods to solve the ODEs or FDEs together with the unknown free boundary values in the Laplace space. Both the early exercise boundary and the prices of American options are recovered through inverse Laplace transforms. Numerical examples demonstrate the accuracy and efficiency of the method in CEV, HEJD, Markov regime switching models and the FMLS models.

Errata for the article “Pricing and static hedging of American-style options under the jump to default extended CEV model”

Abstract This errata corrects an error in Ruas et al. (2013, Equation 27) and updates the numerical results contained in Ruas et al. (2013, Tables 4 and 5). The material provided here is meant to be read strictly in conjunction with Ruas et al. (2013).

On the Performance of the Comonotonicity Approach for Pricing Asian Options in Some Benchmark Models from Equities and Commodities

In this paper, we investigate the applicability of the comonotonicity approach in the context of various benchmark models for equities and commodities. Instead of classical Lévy models as in Albrecher et al. we focus on the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and Schwartz’ 1997 stochastic convenience yield model. We show how the technical difficulties of inverting the distribution function of the sum of the comonotonic random vector can be overcome and that the method delivers rather tight upper bounds for the prices of Asian Options in these models, at least for strikes which are not too large. As a by-product the method delivers super-hedging strategies which can be easily implemented.

Estimating the Constant Elasticity of Variance Model with Data-Driven Markov Chain Monte Carlo Methods

The constant elasticity of variance (CEV) model is widely studied and applied for volatility forecasting and optimal decision making in both areas of financial engineering and operational management, especially in option pricing, due to its good fitting effect for the volatility process of various assets such as stocks and commodities. However, it is extremely difficult to conduct parameter estimation for the CEV model in practice since the precise likelihood function cannot be derived. Motivated by the gap between theory and practice, this paper initiatively applies the Markov Chain-Monte Carlo (MCMC) method into parameter estimation for the CEV model. We first construct a theoretical structure on how to implement the MCMC method into the CEV model, and then execute an empirical analysis with big data of CSI 300 index collected from the Chinese stock market. The final empirical results reveal insights on two aspects: On one aspect, the simulated results of the convergence test are convergent, which demonstrates that the MCMC estimation method for the CEV model is effective; On the other aspect, by a comparison with other two most frequently used estimation methods, the maximum likelihood estimation (MLE) and the generalized moment estimation (GMM), our method is proved to be of high accuracy and has a simpler implementation and wider application.

Rollover Risk and Volatility Risk in Credit Spread Models: A Unified Approach

We present a new approach for the derivation of debt and equity values in structural models of the firm with stationary debt structure for general asset dynamics that include state-dependent and stochastic volatility in the presence of liquidity shocks. The volatility risk affects the credit spread and the optimal capital structure through the cumulative first passage default probability, thus altering the endogenous default boundary and inducing rollover costs. We derive general expressions for all cases and apply them to the constant elasticity of variance (CEV) process. Taking advantage of the analytical expression for a digital option in CEV, we show numerically that volatility risk with negative elasticity decreases both debt and equity values and increases significantly the credit spreads.

Investor’s Power Utility Optimization with Consumption, Tax, Dividend and Transaction Cost under Constant Elasticity of Variance Model

Investor’s Power Utility Optimization with Consumption, Tax, Dividend and Transaction Cost under Constant Elasticity of Variance Model

Hybrid Laplace Transform and Finite Difference Methods for Pricing American Options under Complex Models

In this paper, we propose a hybrid Laplace transform and finite difference method to price (finite-maturity) American options, which is applicable to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump–diffusion (HEJD), Markov regime switching models, and the finite moment log stable (FMLS) models. We first apply Laplace transforms to free boundary partial differential equations (PDEs) or fractional partial differential equations (FPDEs) governing the American option prices with respect to time, and obtain second order ordinary differential equations (ODEs) or fractional differential equations (FDEs) with free boundary, which is named as the early exercise boundary in the American option pricing. Then, we develop an iterative algorithm based on finite difference methods to solve the ODEs or FDEs together with the unknown free boundary values in the Laplace space. Both the early exercise boundary and the prices of American options are recovered through inverse Laplace transforms. Numerical examples demonstrate the accuracy and efficiency of the method in CEV, HEJD, Markov regime switching models and the FMLS models.

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 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 recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. As part of this generalization a simple matrix formulation is presented, allowing efficient implementation. We further extend the applicability of recursive marginal quantization by showing how absorption and reflection at the zero boundary may be incorporated, when this is necessary. To illustrate the improved accuracy of the higher order schemes, various computations are performed using geometric Brownian motion and its generalization, the constant elasticity of variance model. For both processes, we show numerical evidence of improved weak order convergence and we compare the marginal distributions implied by the three schemes to the known analytical distributions. By pricing European, Bermudan and Barrier options, further evidence of improved accuracy of the higher order schemes is demonstrated.

On a Generalized Squared Gaussian Diffusion Model for Option Valuation

In financial mathematics, option pricing models are vital tools whose usefulness cannot be overemphasized. Modern approaches and modelling of financial derivatives are therefore required in option pricing and valuation settings. In this paper, we derive via the application of Ito lemma, a pricing model referred to as Generalized Squared Gaussian Diffusion Model (GSGDM) for option pricing and valuation. Same approach can be considered via Stratonovich stochastic dynamics. We also show that the classical Black-Scholes, and the square root constant elasticity of variance models are special cases of the GSGDM. In addition, general solution of the GSGDM is obtained using modified variational iterative method (MVIM).

Shapes of Implied Volatility with Positive Mass at Zero

We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyze the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behavior at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can---theoretically---distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically test our model-free results in stochastic models with absorption at the boundary, such as the constant elasticity of variance (CEV) process, and in jump-to-default models. Note that while Lee's moment formula [R. W. Lee, Math. Finance, 14 (2004), pp. 469--480] tells us that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as...

Asset price bubbles and the quantification of credit risk capital with sensitivity analysis, empirical implementation and an application to stress testing

This paper presents an analysis of the impact of asset price bubbles on standard credit risk measures, extending research by Jacobs published in 2015 in which the author constructed a model, provided evidence that asset price bubbles understate economic credit capital and proposed a new credit risk measure that is robust to this bias (the expected holding period credit loss, or EHPCL). We perform a sensitivity analysis of the model parameters on the resulting credit risk measures as well as the changes in their relationship to the constant elasticity of variance (CEV) parameter; this controls the degree of market departure from fair value, illustrating an application of an important model validation procedure. We also perform an exercise in which we calibrate the model to historical equity prices and project credit losses on both baseline and stressed conditions for bubble and nonbubble parameter estimate settings. Through the estimation of the CEV model parameters from a long time series, we find statistically significant evidence that the historical Standard & Poor’s index exhibits only mild bubble behavior, but this translates into underestimation of potential extreme credit losses according to standard measures by an order of magnitude. However, while there is still some underestimation of unexpected credit losses under the EHPCL measure, it is of a much lower severity than in the case of traditional measures: it is on the order of 1.5% in the former, compared with tenfold in the latter. However, the degree of relative underestimation of risk due to asset price bubbles is significantly attenuated under stressed parameter settings in our model. The implication of these findings is that risk managers should be wary of measuring tail credit losses according to standard credit risk measures. Alternative measures, such as the EHPCL, should be considered, including benchmarking to stress testing generated credit losses.