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].     

中国股市期权波动率微笑特征的实证分析

波动率微笑现象显示了期权隐含波动率和执行价格之间的关系.在理想的完全符合Black-Scholes期权定价模型假设的情况下,期权隐含波动率关于执行价格应该是一条水平线.然而,在实证分析中,对隐含波动率和执行价格进行拟合并绘制曲线,会产生一个倾斜或微笑形状的曲线,证明Black-Scholes期权定价模型存在一定的缺陷....     

快速均值回归随机波动率模型参数估计及应用

基于快速均值回归随机波动率模型, 研究双限期权的定价问题, 同时推导了考虑均值回归随机波动率的双限期权的定价公式。 根据金融市场中SPDR S&P 500 ETF期权的隐含波动率数据和标的资产的历史收益数据, 对快速均值回归随机波动率模型中的两个重要参数进行估计。 利用估计得到的参数以及定价公式, 对双限期权价格做了数值模拟。 数值模拟结果发现, 考虑了随机波动率之后双限期权的价格在标的资产价格偏高的时候会小于基于常数波动率模型的期权价格。     

The simulation of option prices with application to LIFFE options on futures

We build a framework for modelling the deviation of observed option prices from the Black & Scholes prices. We use a flexible model for a density, a two sided switching Weibull, to capture the implied volatility. The model can be used to generate prices, it can take into account no-arbitrage bounds for option prices and is capable of generating such stylised facts as the smile effect. We apply this methodology to LIFFE options on German government bond futures.     

The Heston model with stochastic elasticity of variance

Empirical evidence suggests that single factor models would not capture the full dynamics of stochastic volatility such that a marked discrepancy between their predicted prices and market prices exists for certain ranges (deep in‐the‐money and out‐of‐the‐money) of time‐to‐maturities of options. On the other hand, there is an empirical reason to believe that volatility skew fluctuates randomly. Based upon the idea of combining stochastic volatility and stochastic skew, this paper incorporates stochastic elasticity of variance running on a fast timescale into the Heston stochastic volatility model. This multiscale and multifactor hybrid model keeps analytic tractability of the Heston model as much as possible, while it enhances capturing the complex nature of volatility and skew dynamics. Asymptotic analysis based on ergodic theory yields a closed form analytic formula for the approximate price of European vanilla options. Subsequently, the effect of adding the stochastic elasticity factor on top of the Heston model is demonstrated in terms of implied volatility surface. Copyright © 2016 John Wiley & Sons, Ltd.     

Pricing VIX options with stochastic skew and asymmetric jumps

This paper performs several empirical exercises to provide evidence that the stochas-tic skew behavior and asymmetric jumps exist in VIX markets.In order to adequately capture all of the features,we develop a general valuation model and obtain quasi-analytical solutions for pricing VIX options.In addition,we make comparative studies of alternative models to illustrate the e ects after taking into account these features on the valuation of VIX options and investigate the relative value of an additional volatility factor and jump components.The empirical results indicate that the multi-factor volatility structure is vital to VIX option pricing due to providing more exibility in the modeling of VIX dynamics,and the need for asymmetric jumps cannot be eliminated by an additional volatility factor.     

Option pricing with mean reversion and stochastic volatility

Many underlying assets of option contracts, such as currencies, commodities, energy, temperature and even some stocks, exhibit both mean reversion and stochastic volatility. This paper investigates the valuation of options when the underlying asset follows a mean-reverting lognormal process with stochastic volatility. A closed-form solution is derived for European options by means of Fourier transform. The proposed model allows the option pricing formula to capture both the term structure of futures prices and the market implied volatility smile within a unified framework. A bivariate trinomial lattice approach is introduced to value path-dependent options with the proposed model. Numerical examples using European options, American options and barrier options demonstrate the use of the model and the quality of the numerical scheme.     

Pricing Exotic Discrete Variance Swaps under the 3/2-Stochastic Volatility Models

Abstract

We consider pricing of various types of exotic discrete variance swaps, like the gamma swaps and corridor variance swaps, under the 3/2-stochastic volatility models (SVMs) with jumps in asset price. The class of SVMs that use a constant-elasticity-of-variance (CEV) process for the instantaneous variance exhibits good analytical tractability only when the CEV parameter takes just a few special values (namely 0, 1/2, 1 and 3/2). The popular Heston model corresponds to the choice of the CEV parameter to be 1/2. However, the stochastic volatility dynamics implied by the Heston model fails to capture some important empirical features of the market data. The choice of 3/2 for the CEV parameter in the SVM shows better agreement with empirical studies while it maintains a good level of analytical tractability. Using the partial integro-differential equation (PIDE) formulation, we manage to derive quasi-closed-form pricing formulas for the fair strike prices of various types of exotic discrete variance swaps with various weight processes and different return specifications under the 3/2-model. Pricing properties of these exotic discrete variance swaps with respect to various model parameters are explored.     

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes’ transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.     

Pricing VIX options in a stochastic vol‐of‐vol model

This paper proposes and makes a study of a new model for volatility index option pricing. Factors such as mean‐reversion, jumps, and stochastic volatility are taken into consideration. In particular, the positive volatility skew is addressed by the jump and the stochastic volatility of volatility. Daily calibration is used to check whether the model fits market prices and generates positive volatility skews. Overall, the results show that the mean‐reverting logarithmic jump and stochastic volatility model (called MRLRJSV in the paper) serves as the best model in all the required aspects. Copyright © 2015 John Wiley & Sons, Ltd.     

Pricing VIX options in a 3/2 plus jumps model

This paper proposes and makes a study of a new model (called the 3/2 plus jumps model) for VIX option pricing. The model allows the mean-reversion speed and volatility of volatility to be highly sensitive to the actual level of VIX. In particular, the positive volatility skew is addressed by the 3/2 plus jumps model. Daily calibration is used to prove that the proposed model preserves its validity and reliability for both in-sample and out-of-sample tests. The results show that the models are capable of fitting the market price while generating positive volatility skew.     

基于GARCH-GH模型的上证50ETF期权定价研究

上证50ETF期权是中国推出的首支股票期权.为描述上证50ETF收益率偏态、尖峰、时变波动率等特征,结合GARCH模型和广义双曲(Generalized Hyperbolic,GH)分布两方面的优势,建立GARCH-GH模型为上证50ETF期权定价.在等价鞅测度下,利用蒙特卡罗方法估计上证50ETF欧式认购期权价格.实证表明,相比较Black-Scholes模型和GARCH-Gaussian模型,GARCH-GH模型得到的结果更接近于上证50ETF期权的实际价格,其定价误差最小.     

基于外汇汇率的期权定价及其实证分析

在外汇汇率服从连续扩散过程模型下,研究了外汇汇率的几何平均亚式期权和附有汇率范围的示性函数的新型幂期权定价问题。在实证分析中,通过美元/人民币汇率的真实数据来计算以上所研究期权的价格,并和Black-Scholes模型下的期权定价进行比较,同时对相关期权的隐含波动率进行了分析。     

对数正态随机波动率期权局部风险最小定价与风险对冲

随机波动率模型是著名的Black-Scholes模型的推广,该模型描述的市场是不完备的,相应期权的定价与保值和投资者的风险态度有关．本文假设标的资产波动率为对数正态过程,根据局部风险最小准则,运用梯度算子方法,得到了欧式看涨期权的局部风险最小定价及套期保值策略的显式解．     

Pricing VXX option with default risk and positive volatility skew

This paper proposes and makes a comparative study of alternative models for VXX option pricing. Factors such as mean-reversion, jumps, default risk and positive volatility skew are taken into consideration. In particular, default risk is characterized by jump-to-default framework and the “positive volatility skew” issue is addressed by stochastic volatility of volatility and jumps. Daily calibration is conducted and comparative study of the models is performed to check whether they properly fit market prices and generate reasonable positive volatility skews and deltas. Overall, jump-to-default extended LRJ model with positive correlated stochastic volatility (called JDLRJSV in the paper) serves as the best model in all the required aspects.     

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.     

Small-Maturity Asymptotics for the At-The-Money Implied Volatility Slope in Lévy Models

We consider the at-the-money (ATM) strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the behaviour of the slope for infinite activity exponential Lévy models including a Brownian component. As auxiliary results, we obtain asymptotic expansions of short maturity ATM digital call options, using Mellin transform asymptotics. Finally, we discuss when the ATM slope is consistent with the steepness of the smile wings, as given by Lee’s moment formula.     

An inverse problem of determining the implied volatility in option pricing

In the Black-Scholes world there is the important quantity of volatility which cannot be observed directly but has a major impact on the option value. In practice, traders usually work with what is known as implied volatility which is implied by option prices observed in the market. In this paper, we use an optimal control framework to discuss an inverse problem of determining the implied volatility when the average option premium, namely the average value of option premium corresponding with a fixed strike price and all possible maturities from the current time to a chosen future time, is known. The issue is converted into a terminal control problem by Green function method. The existence and uniqueness of the minimum of the control functional are addressed by the optimal control method, and the necessary condition which must be satisfied by the minimum is also given. The results obtained in the paper may be useful for those who engage in risk management or volatility trading.     

Trading volume in models of financial derivatives

This paper develops a subordinated stochastic process model for an asset price, where the directing process is identified as information. Motivated by recent empirical and theoretical work, the paper makes use of the under-used market statistic of transaction count as a suitable proxy for the information flow. An option pricing formula is derived, and comparisons with stochastic volatility models are drawn. Both the asset price and the number of trades are used in parameter estimation. The underlying process is found to be fast mean reverting, and this is exploited to perform an asymptotic expansion. The implied volatility skew is then used to calibrate the model.     

Short Positions,Rally Fears and Option Markets

Abstract

Index option pricing on world market indices are investigated using Lévy processes with no positive jumps. Economically this is motivated by the possible absence of longer horizon short positions while mathematically we are able to evaluate for such processes the probability of a rally before a crash. Three models are used to effectively calibrate index options at an annual maturity, and it is observed that positive jumps may be needed for FTSE, N225 and HSI. Rally before a crash probabilities are shown to have fallen by 10 points after July 2007. Typical implied volatility curves for such models are also described and illustrated. They have smirks and never smile.