Multiscale analysis of a perpetual American option with the stochastic elasticity of variance

A perpetual American option is considered under a generalized model of the constant elasticity of variance model where the constant elasticity is perturbed by a small fast mean-reverting Ornstein–Uhlenbeck process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on option prices as well as optimal exercise prices. Our results improve the existing option price structure in view of flexibility and applicability through the market price of risk. The revealed results may provide useful information on real option problems.     

A Mellin transform approach to pricing barrier options under stochastic elasticity of variance

This article considers a problem of evaluating barrier option prices when the underlying dynamics are driven by stochastic elasticity of variance (SEV). We employ asymptotic expansions and Mellin transform to evaluate the option prices. The approach is able to efficiently handle barrier options in a SEV framework and produce explicitly a semi-closed form formula for the approximate barrier option prices. The formula is an expansion of the option price in powers of the characteristic amplitude scale and variation time of the elasticity and it can be calculated easily by taking the derivatives of the Black–Scholes price for a barrier option with respect to the underlying price and computing the one-dimensional integrals of some linear combinations of the Greeks with respect to time. We confirm the accuracy of our formula via Monte-Carlo simulation and find the SEV effect on the Black–Scholes barrier option prices.     

Pricing turbo warrants under stochastic elasticity of variance

We consider an extended constant elasticity of variance (CEV) model in which the elasticity follows a stochastic process driven by a fast mean-reverting Ornstein–Uhlenbeck process. Then, we use the proposed model to examine a turbo warrant option, which is a type of exotic option. Based on an asymptotic analysis, we derive the partial differential equation of the leading and the corrected terms, which we use to determine the analytic formula for the turbo warrant call option. The parameter analysis using the extended CEV model provides us with a better understanding of the price structure of a turbo warrant call. Moreover, by comparing the turbo warrant call with a European vanilla call, we can examine the sensitivity of options with respect to the model parameters.     

A new fourth-order numerical scheme for option pricing under the CEV model

The empirically observed negative relationship between a stock price and its return volatility can be captured by the constant elasticity of variance option pricing model. For European options, closed form expressions involve the non-central chi-square distribution whose computation can be slow when the elasticity factor is close to one, volatility is low or time to maturity is small. We present a fast numerical scheme based on a high-order compact discretisation which accurately computes the option price. Various numerical examples indicate that for comparable computational times, the option price computed with the scheme has higher accuracy than the Crank–Nicolson numerical solution. The scheme accurately computes the hedging parameters and is stable for strongly negative values of the elasticity factor.     

New approach and analysis of the generalized constant elasticity of variance model

Generally, it is well known that the constant elasticity of variance (CEV) model fails to capture the empirical results verifying that the implied volatility of equity options displays smile and skew curves at the same time. In this study, to overcome the limitation of the CEV model, we introduce a new model, which is a generalization of the CEV model, and show that it can capture the smile and skew effects of implied volatility. Using an asymptotic analysis for two small parameters that determine the volatility shape, we obtain approximated solutions for option prices in the extended model. In addition, we demonstrate the stability of the solution for the expansion of the option price. Furthermore, we show the convergence rate of the solutions in Monte-Carlo simulation and compare our model with the CEV, Heston, and other extended stochastic volatility models to verify its flexibility and efficiency compared with these other models when fitting option data from the S&P 500 index.     

Lie-algebraic approach for pricing moving barrier options with time-dependent parameters

In this paper we apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters. The value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process. By exploiting the dynamical symmetry of the pricing partial differential equations, the new approach enables us to derive the analytical kernels of the pricing formulae straightforwardly, and thus provides an efficient way for computing the prices of the moving barrier options. The method is also able to provide tight upper and lower bounds for the exact prices of CEV barrier options with fixed barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, our new approach could facilitate more efficient comparative pricing and precise risk management in equity derivatives with barriers by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.     

A numerical method to estimate the parameters of the CEV model implied by American option prices: Evidence from NYSE

We develop a highly efficient procedure to forecast the parameters of the constant elasticity of variance (CEV) model implied by American options. In particular, first of all, the American option prices predicted by the CEV model are calculated using an accurate and fast finite difference scheme. Then, the parameters of the CEV model are obtained by minimizing the distance between theoretical and empirical option prices, which yields an optimization problem that is solved using an ad-hoc numerical procedure. The proposed approach, which turns out to be very efficient from the computational standpoint, is used to test the goodness-of-fit of the CEV model in predicting the prices of American options traded on the NYSE. The results obtained reveal that the CEV model does not provide a very good agreement with real market data and yields only a marginal improvement over the more popular Black–Scholes model.     

A closed-form analytic correction to the Black–Scholes–Merton price for perpetual American options

This is a complementary study of a recent work by Yoon et al. (2013) [1] [J.-H. Yoon, J.-H. Kim, S.-Y. Choi, Multiscale analysis of a perpetual American option with the stochastic elasticity of variance, Appl. Math. Lett. 26 (7) (2013)] which excludes a certain level of the elasticity of variance. A second-order correction to the Black–Scholes option price and optimal exercise boundary for a perpetual American put option is made under the stochastic elasticity of variance of a risky asset. Contrary to the case of Yoon et al. (2013) [1], it is given by an explicit closed-form analytic expression so that one can access clearly the sensitivity of the option price and the optimal exercise boundary to changes in model parameters as well as the impact of the presence of a stochastic elasticity term on the option price and the optimal time to exercise.     

Valuing American options under the CEV model by Laplace-Carson transforms

This paper investigates American option pricing under the constant elasticity of variance (CEV) model. Taking the Laplace-Carson transform (LCT) to the corresponding free-boundary problem enables the determination of the optimal early exercise boundary to be separated from the valuation procedure. Specifically, a functional equation for the LCT of the early exercise boundary is obtained. By applying Gaussian quadrature formulas, an efficient method is developed to compute the early exercise boundary, American option price and Greeks under the CEV model.     

Asymptotic option pricing under the CEV diffusion

In finance, many option pricing models generalizing the Black-Scholes model do not have closed form, analytic solutions so that it is hard to compute the solutions or at least it requires much time to compute the solutions. Therefore, asymptotic representation of options prices of various type has important practical implications in finance. This paper presents asymptotic expansions of option prices in the constant elasticity of variance model as the parameter appearing in the exponent of the diffusion coefficient tends to 2 which corresponds to the well-known Black-Scholes model. We use perturbation theory for partial differential equations to obtain the relevant results for European vanilla, barrier, and lookback options. We make our application of perturbation theory mathematically rigorous by supplying error bounds.     

动态随机弹性在期权定价中的应用

给出动态随机弹性的概念及运算性质,讨论了动态随机弹性在期权定价模型中的应用.主要结果有:(1)在波动率为常数时,期权价格对的弹性,得到了动态随机弹性服从运动,并给出了相应的经济解释;(2)由于波动率一般不是常数,也是随机过程,因此本文进一步研究了期权价格对波动率的弹性,就股票价格的波动情况给出了数学描述和金融意义上的解释.     

A recursive pricing formula for a path-dependent option under the constant elasticity of variance diffusion

In this paper, we consider a path-dependent option in finance under the constant elasticity of variance diffusion. We use a perturbation argument and the probabilistic representation (the Feynman–Kac theorem) of a partial differential equation to obtain a complete asymptotic expansion of the option price in a recursive manner based on the Black–Scholes formula and prove rigorously the existence of the expansion with a convergence error.     

Optimal reinsurance–investment problem in a constant elasticity of variance stock market for jump‐diffusion risk model

In this paper, we consider the jump‐diffusion risk model with proportional reinsurance and stock price process following the constant elasticity of variance model. Compared with the geometric Brownian motion model, the advantage of the constant elasticity of variance model is that the volatility has correlation with the risky asset price, and thus, it can explain the empirical bias exhibited by the Black and Scholes model, such as volatility smile. Here, we study the optimal investment–reinsurance problem of maximizing the expected exponential utility of terminal wealth. By using techniques of stochastic control theory, we are able to derive the explicit expressions for the optimal strategy and value function. Numerical examples are presented to show the impact of model parameters on the optimal strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

基于EGARCH模型的分类商品零售价格波动的信息效应研究

利用EGARCH模型对我国部分具有代表性的分类商品零售价格波动的信息效应进行了实证分析.分析结果显示,我国分类商品零售价格波动特征是不同的.在六个具有代表性的分类商品零售价格指数中,有四个指数的方差具有时变性特征.在四个当中,有三个指数有非对称信息效应,即非期望的价格上涨或下降信息对价格波动的影响是非对称的.另外的两个价格指数的方差为常数,价格波动稳定.     

基于CEV过程带交易费的脆弱期权定价的数值算法

主要研究基于CEV过程且支付交易费的脆弱期权定价的数值计算问题.首先通过构造无风险投资组合,导出了基于CEV过程且支付交易费用的脆弱期权定价的偏微分方程模型;其次应用有限差分方法将定价模型离散化,并设计数值算法;最后以看跌期权为例进行数值试验,分析各定价参数对看跌期权价值的影响.     

The American put is log-concave in the log-price

We show that the American put option price is log-concave as a function of the log-price of the underlying asset. Thus the elasticity of the price decreases with increasing stock value. We also consider related contracts of American type, and we provide an example showing that not all American option prices are log-concave in the stock log-price.     

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].     

风险敏感性控制在CEV模型的应用研究

假设股票的价格遵循CEV过程,经济因子满足两个相互独立的布朗运动,运用风险敏感性随机最优控制理论得到新的结论,最后对于简化的模型,得到最优长期增长率的解析解.     

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.     

具有复合泊松损失和随机利率的巨灾期权定价

分析了带有复合泊松损失过程和随机利率的巨灾看跌期权的定价问题.资产价格通过跳扩散过程刻画,该过程与损失过程相关.当利率过程服从CIR模型时,获得了期权定价的显式解,并给出相关证明.通过一个实例,讨论了资产价格与期权价格的关系.