均值回复跳扩散随机波动率模型下的欧式期权定价研究

考虑到金融市场数据波动的不确定性,本文提出了一个新的对数均值回复跳扩散4/2随机波动率(LMRJ-4/2-SV)模型.首先,构建了LMRJ-4/2-SV模型,并利用FFT等方法获得了基于LMRJ-4/2-SV模型的欧式期权定价公式.其次,对实际市场数据进行描述性统计分析,探讨标的资产价格变化特征及LMRJ-4/2-SV模型的适用性,并通过粒子群优化算法估计模型参数.最后,基于LMRJ-4/2-SV模型下的期权定价公式及模型参数估计值对欧式期权进行定价,并将其定价结果与4/2、3/2、Heston模型估计值及市场价格进行对比.结果表明:基于LMRJ-4/2-SV模型的欧式期权定价误差最小,定价结果较其它随机波动率模型而言具有明显优势.     

高维欧式期权定价模型的奇摄动解

讨论了一类欧式期权定价问题的随机波动率模型,其随机波动率采用快速均值回归的随机波动率模型.通过采用奇摄动方法,得到了多风险资产欧式期权价格的形式渐近展开式,得到该合成展开式的一致有效误差估计.     

基于带跳随机波动率模型的双币种重置期权定价研究

研究了外国标的资产价格,汇率及其波动率过程满足仿射跳扩散模型的双币种重置期权定价问题,其中波动率过程与标的资产,汇率相关,且具有共同跳跃风险成分.利用多维Feynman-Kac定理,Fourier逆变换等方法,获得了双币种重置期权价格的表达式.应用数值计算分析了波动率过程主要参数对期权价格的影响.数值结果表明,波动率因素以及跳跃风险参数对期权价格的影响是显著的.     

随机波动率跳跃扩散模型下复合期权定价

复合期权是一类以期权作为标的物的奇异型合约,它已广泛应用于许多金融实践。本文在股价满足一类随机波动率及跳跃均存在于股价和波动率的仿射跳跃扩散模型下(也称随机波动率混合跳跃扩散模型)考察了复合期权的定价。应用二维特征函数和Fourier反变换方法获到了标的为欧式标准看涨期权的欧式复合看涨期权的定价半封闭公式,并将其应用于推导扩展期权的定价。最后,借助于离散快速Fourier变换法(FFT)数值计算定价公式,并用数值实例分析了期权价格对波动率的敏感性。数值结果表明扩散波动和跳跃波动对期权价格都有正的影响,而且跳跃波动的冲击非常显著。     

非仿射随机波动率跳扩散模型的利差期权定价

在两标的资产价格满足一类随机利率、随机波动率及跳跃均存在于资产价格和波动率的非仿射跳扩散模型下考察了利差期权的定价.首先,利用泰勒公式将非线性微分方程线性化,得到了两标的资产对数价格的近似联合密度特征函数;然后,使用Fourier逆变换等方法,获得了利差期权定价理论的半封闭公式,并将其推广到价差期权的定价.最后,通过数值实验,表明非仿射随机波动率跳扩散的利差期权定价模型比仿射随机波动率模型具有更高的精确性,并且扩散波动和跳跃波动对期权价格影响显著.     

基于随机波动率模型的上证50ETF期权定价研究

传统上,期权定价主要基于Black-Scholes (B-S)模型。但B-S模型不能描述时变波动率以及解释"波动率微笑"现象,导致期权定价存在较大的误差。随机波动率模型克服了B-S模型的这些缺陷,能够合理地刻画波动率动态性和波动率微笑。基于此,本文考虑随机波动率模型下的期权定价问题,并针对我国上证50ETF期权进行实证分析。为了解决定价模型的参数估计问题,采用上证50ETF及其期权价格数据,建立两步法对定价模型的参数进行估计。该估计方法保证了定价模型在客观与风险中性测度下的一致性。采用2016年1月到2017年10月的上证50ETF期权价格数据为研究样本,对随机波动率模型进行了实证检验。结果表明,无论是在样本内还是样本外,随机波动率模型相比传统的常数波动率B-S模型都能够获得明显更为精确和稳定的定价结果,B-S模型的定价误差总体偏大且呈现较高波动,凸显了随机波动率对于期权定价的重要性。另外,随机波动率模型对于短期实值期权的定价相比对于其它期权的定价要更精确。     

随机波动模型下算术亚式期权的Monte Carlo模拟定价

在随机波动模型下,研究亚式期权的定价问题.推导出了标的资产及其随机波动模型的路径,利用对偶变量法对亚式期权进行数值模拟计算,并对随机波动模型下与B-S模型下的欧式期权和亚式期权定价结果进行比较,最后给出了具有固定敲定价格和浮动敲定价格的算术亚式期权的数值计算结果.     

基于加权可能性均值的亚式期权模糊定价

主要探讨不确定环境下用模糊集理论处理亚式期权的定价问题.运用梯形模糊数来表示标的资产价格、无风险利率、红利率和波动率,建立了亚式期权的加权可能性均值模糊定价模型,得到连续几何和算术亚式期权的模糊价格公式.最后通过数值例子表明:亚式期权的加权可能性均值模糊定价模型具有很大的灵活性,更符合现实的不确定情况,具有较强的实用价值.     

随机波动率与跳扩散组合模型的双币种期权定价

在股价和汇率满足随机波动率与跳扩散组合模型下应用半鞅Ito公式、多维随机变量的联合特征函数、Girsanov测度变换以及Fourier反变换等随机分析技巧给出了双币种欧式期权价格的封闭式解,并利用数值实例分析了波动参数对期权价格的影响,结果表明:波动率参数对期权价格有显著的影响作用.     

期权标的资产波动率的重构方法

在B lack-Scho les公式中,波动率σ是一个非常重要的参数.并且在诸如股票、利率、股指期货等标的资产的交易市场中,人们往往希望知道标的资产未来价格的波动率,从而知道该资产的未来风险结构.但一般来说,由于事件还没有发生,人们对σ的未来走向很难预测.但可以运用B lack-Scho les的理论框架,从期权市场获取的信息去重构标的资产价格的波动率.论文使用的是基于T ikhonov正则化的数值微分方法,利用Dup ire公式去重构标的资产的未来预期波动率.相对于其他方法,该算法更加快速有效,并且能识别标的资产的预期风险突变.     

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.     

Market calibration under a long memory stochastic volatility model

In this article, we study a long memory stochastic volatility model (LSV), under which stock prices follow a jump-diffusion stochastic process and its stochastic volatility is driven by a continuous-time fractional process that attains a long memory. LSV model should take into account most of the observed market aspects and unlike many other approaches, the volatility clustering phenomenon is captured explicitly by the long memory parameter. Moreover, this property has been reported in realized volatility time-series across different asset classes and time periods. In the first part of the article, we derive an alternative formula for pricing European securities. The formula enables us to effectively price European options and to calibrate the model to a given option market. In the second part of the article, we provide an empirical review of the model calibration. For this purpose, a set of traded FTSE 100 index call options is used and the long memory volatility model is compared to a popular pricing approach – the Heston model. To test stability of calibrated parameters and to verify calibration results from previous data set, we utilize multiple data sets from NYSE option market on Apple Inc. stock.     

Pricing vulnerable options under a stochastic volatility model

In this paper, we consider the pricing of vulnerable options when the underlying asset follows a stochastic volatility model. We use multiscale asymptotic analysis to derive an analytic approximation formula for the price of the vulnerable options and study the stochastic volatility effect on the option price. A numerical experiment result is presented to demonstrate our findings graphically.     

Pricing perpetual American options under multiscale stochastic elasticity of variance

This paper studies pricing the perpetual American options under a constant elasticity of variance type of underlying asset price model where the constant elasticity is replaced by a fast mean-reverting Ornstein–Ulenbeck process and a slowly varying diffusion process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on the option prices and the optimal exercise prices with respect to model parameters. Our results enhance the existing option price structures in view of flexibility and applicability through the market prices of elasticity risk.     

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.     

Stochastic volatility asymptotics of stock loans: Valuation and optimal stopping

A stock loan, or equity security lending service, is a loan which uses stocks as collateral. The borrower has the right to repay the principal with interest and regain the stock, or make no repayment and surrender the stock. Therefore, the valuation of stock loan is an optimal stopping problem related to a perpetual American option with a negative effective interest rate. The negative effective interest rate makes standard techniques for perpetual American option pricing failure. Using a fast mean-reverting stochastic volatility model, we applied a perturbation technique to the free-boundary value problem for the stock loan price. An analytical pricing formula and optimal exercise boundary are derived by means of asymptotic expansion.     

基于跳跃波动率的未定权益定价模型

讨论了具有随机波动率的未定权益定价问题,建立了两状态波动率的股票价格行为模型,在股票价格过程是连续过程、跳风险不可定价的假设下,推导出未定权益的定价公式.     

Unifying pricing formula for several stochastic volatility models with jumps

In this paper, we introduce a unifying approach to option pricing under continuous‐time stochastic volatility models with jumps. For European style options, a new semi‐closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro‐differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way. In particular, we focus on a log‐normal and a log‐uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out‐of‐the money contracts. Copyright © 2017 John Wiley & Sons, Ltd.     

Heston模型下的方差互换定价研究

通过实证分析论证了波动率具有均值回复性质的合理性.在Heston模型下,利用Ito积分推导出了方差互换在其存续期内任意时刻的价格与公平执行价格的定价公式.得到公平执行价格是波动率的平方的初始水平与长期均值水平的线性组合的性质,并利用该性质对Heston模型参数的敏感性进行了分析.     

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.