混合分数布朗运动环境下的欧式期权定价

混合双分数布朗运动下期权的定价研究摘要：本文研究了混合双分数布朗运动下期权的定价。

首先，我们介绍了双分数布朗运动和混合双分数布朗运动的定义，分析了混合双分数布朗运动的性质，并给出了其表示式。

接着，我们介绍了Black-Scholes期权定价模型及其在标准布朗运动下的应用，然后将其扩展到混合双分数布朗运动下，并给出了相应的定价公式。

最后，通过实际数据的计算和模拟，验证了所得定价公式的正确性和可行性。

关键词：混合双分数布朗运动、双分数布朗运动、期权定价模型、Black-Scholes模型、定价公式Abstract：In this paper, we studied the pricing of options under the mixed fractional Brownian motion. Firstly, we introduce the definition of the fractional Brownian motion and the mixed fractional Brownian motion, analyze the properties of mixed fractional Brownian motion, and give its expression. Then, we introduce the Black-Scholes option pricing model and its application in the standard Brownian motion.Furthermore, we extend it to the mixed fractional Brownian motion and give the corresponding pricing formula. Finally, the correctness and feasibility of the obtained pricing formula are verified by calculating and simulating actual data.Keywords: mixed fractional Brownian motion, fractional Brownian motion, option pricing model, Black-Scholes model, pricing formulaOption pricing has become a critical issue in the financial market, as there is an increasing demand for financial instruments that can help manage and hedge financial risks. The Black-Scholes option pricing model is widely used to price financial derivatives such as options. The model assumes that the underlying asset follows a standard Brownian motion, which is characterized by its constant volatility and drift. However, in reality, the volatility and drift of financial assets may vary over time, and their behavior may not be accurately represented by standard Brownian motion.The fractional Brownian motion (fBm) offers a more flexible framework for modeling the behavior of financial assets. Compared to the standard Brownian motion, fBm allows for varying volatility and drift,and exhibits long-range dependence. The mixed fractional Brownian motion (m-fBm) is an extension of fBm that incorporates both long- and short-range dependence, and has been used to model the behavior of stock prices and other financial assets.In this paper, we present a pricing formula for options based on the Black-Scholes option pricing model, but with the underlying asset modeled by m-fBm. We derive the formula using Ito's lemma and the risk-neutral pricing approach, and show that it reduces to the standard Black-Scholes formula when the underlying asset is modeled by standard Brownian motion.To test the validity of the pricing formula, we apply it to actual data on stock prices and compare the results with those obtained using the standard Black-Scholes formula. We find that the pricing formula based on m-fBm provides a better fit to the observed prices, particularly in cases where the underlying asset exhibits long-range dependence.In conclusion, we have shown that the m-fBm provides a more flexible and accurate framework for modeling the behavior of financial assets, and can be used to develop pricing models for financial derivatives such as options. The pricing formula presented in thispaper demonstrates the feasibility and effectiveness of using m-fBm in option pricingIn addition to its applications in modeling financial assets, m-fBm has also been used in other fields such as image processing, speech recognition, and geology. Its ability to capture long-range dependence and multifractal properties makes it a valuable tool in studying complex systems.One potential future direction for m-fBm research isin the development of more complex and realistic models that incorporate additional factors such as jumps, stochastic volatility, and other forms of nonlinearity. These factors are often present in real-world financial markets and can have a significant impact on asset prices. Developing models that can accurately capture these dynamics could lead to better pricing and risk management strategies for financial instruments.Another potential area for future research is in the application of m-fBm to other types of financial instruments such as futures, swaps, and credit derivatives. While options are a popular focus for financial modeling research, there are many other types of financial instruments that can benefit fromaccurate pricing models.Overall, the use of m-fBm in financial modeling represents an important development in the field of quantitative finance. Its ability to capture long-range dependence and multifractal properties makes it a valuable tool for understanding and predicting the behavior of financial assets. While there are still many challenges to overcome in developing more accurate and realistic models, the potential benefits of using m-fBm in financial modeling make it a promising area for future researchOne area where the use of m-fBm in financial modeling could be particularly useful is in risk management. By accurately modeling the multifractal properties of financial assets, it would be possible to better understand the risk associated with different types of investments. This could help investors make more informed decisions and avoid potential losses.Another potential application of m-fBm in finance is in the development of trading strategies. By analyzing the long-range dependence of financial assets, it may be possible to identify patterns that can be exploited for profit. This could lead to the development of more effective trading algorithms and better investmentstrategies.However, there are also several challenges that needto be overcome in order to fully realize the potential of m-fBm in financial modeling. One major challenge is the lack of high-quality data. Multifractal analysis requires long and accurate time series data, which may be difficult to obtain in the financial markets. Additionally, there is a need for more sophisticated modeling techniques that can accurately capture the complex dynamics of financial markets.Despite these challenges, the use of m-fBm infinancial modeling has already shown promising results in several areas. Its ability to capture long-range dependence and multifractal properties make it a valuable tool for understanding and predicting the behavior of financial assets. As research in this area continues, it is likely that we will see further advancements in our understanding of financial markets and their underlying dynamicsIn conclusion, multifractional Brownian motions (m-fBm) have become an increasingly popular tool for modeling financial markets due to their ability to capturelong-range dependence and multifractal properties. While the use of m-fBm in financial modeling presentsseveral challenges, there have been promising results in predicting the behavior of financial assets. Further advancements in research are likely to provide a deeper understanding of financial markets and their underlying dynamics。

混合双分数布朗运动下欧式期权的定价徐峰【摘要】提出一种新的不具有平稳增量的随机过程—混合双分数布朗运动，用来刻画标的资产的价格，进行欧式期权定价的研究。

假设标的资产由混合双分数布朗运动驱动，运用对冲原理建立混合双分数布朗运动环境下的欧式期权价值所满足的偏微分方程，并采用边界条件和变量代换的方法得到该偏微分方程的解，即欧式期权的定价公式，其结果可看作是混合分数布朗运动和双分数布朗运动驱动下的一种推广。

%Assuming that the underlying asset is driven by the mixed bi-fractional Brownian motion,this paper proposes a partial differential equation formulation for valuing European option by hedge principle. Moreover, using the boundary condition and the method of variable substitution,we obtain the solution to this partial differential equation-the pricing formula for European option.【期刊名称】《苏州市职业大学学报》【年(卷),期】2015(000)001【总页数】4页(P50-53)【关键词】混合双分数布朗运动;欧式期权;定价;长记忆性【作者】徐峰【作者单位】苏州市职业大学商学院，江苏苏州 215104【正文语种】中文【中图分类】F830.9;O211.6传统的期权定价都是在假设标的资产服从几何布朗运动的基础上进行研究的，然而近年来大量的实证研究表明，金融资产的对数收益率并非服从正态分布，而是服从一种“尖峰厚尾”的分布，而且其价格之间也并非是随机游走的，存在着长记忆性和自相似性等分形特征，这导致了大量由布朗运动驱动的定价模型不符合真实的市场.分数布朗运动[1]已成为弥补上述模型缺陷最为简单的方法.但是，文献[2]指出分数布朗运动不是半鞅，许多研究者用不同的方法给出了分数布朗运动的离散逼近，并指出直接将分数布朗运动应用于金融环境将会产生套利机会[3-4]，这使得分数布朗运动似乎不适合用于刻画金融资产价格变化的行为模式.从而，部分学者开始研究修正的分数布朗运动，如混合分数布朗运动、双分数布朗运动等[5-6]，由于双分数布朗运动不仅具有自相似性和长记忆性的特征，而且在一定的限制条件下是半鞅，因此可以应用于期权定价领域.本文提出一种新的不具有平稳增量的随机过程—混合双分数布朗运动，用来刻画标的资产(如股票)的价格，进行欧式期权定价的研究.本文的结果可作为混合分数布朗运动和双分数布朗运动驱动下的一种推广.1.1 混合双分数布朗运动的定义与性质定义1 如果满足均值为0，协方差为则中心高斯过程称为混合双分数布朗运动，其中，σ，ε为两个常数，过程是双分数布朗运动，{Bt}t≥0是标准布朗运动，与独立，当K=1时，混合双分数布朗运动就退化成混合分数布朗运动；当时，混合双分数布朗运动就退化成双分数布朗运动.由定义易知，混合双分数布朗运动具有以下性质.性质1是HK-自相似的，即对任意α＞0，过程具有相同的分布；性质2 当具有长记忆性；性质3 当不是半鞅.这些性质的证明可见参考文献[6].1.2 模型假设对金融市场做如下假设：市场无摩擦，即交易费用为零，无税收，不存在无风险套利机会；没有对交易头寸方向的限制，允许买空卖空证券；无风险利率r为常数；标的资产(如股票)的价格变化过程St服从过程式中：μ表示标的资产的收益率.在以下研究中假设根据文献[7]易得到下面的引理.引理1 随机微分方程(1)的解为定理1 设Ct=C(t，St)是欧式看涨期权在t时刻的价格，股票价格满足方程(1)，则Ct满足偏微分方程证明构建一个买入一份期权C和卖空Δ份股票S的资产组合Π，即Π=C-ΔS，则选取适当的Δ使得资产组合Π在(t，t+dt)上是无风险的，即dΠ=rΠdt.令，则有即有定理2 假设到期日为T，履约价格为K，则混合双分数布朗运动下欧式看涨期权在任意时刻t∈ [0，T]的价格Ct为式中为标准正态函数.证明由定理1得Ct满足偏微分方程(3)，且边界条件为C(T，S)=(S-K)+.令S=ex，C=V(t，x)，则易得将上式代入式(3)，则有同时边界条件变为令则有将上式代入式(4)，则有式中边界条件为根据热传导方程经典解理论[8]，式(5)有唯一强解将边界条件代入可得11对式(6)做逆变换易得定理2成立.推论1 当K=1时，可得到混合分数布朗运动驱动下的欧式看涨期权在t时刻的价格为其中注1 该结论与文献[9](当n=1时)中得出的结果一致.注2 当ε=0时，推论1的结果即为双分数布朗运动环境下欧式期权的定价公式，与文献[10]的结果一致.采用类似的方法同样可以推导出欧式看跌期权的定价公式，不加证明地给出下面的定理.定理3 假设到期日为T，履约价格为K，则混合双分数布朗运动下欧式看跌期权在任意时刻t∈ [0，T]的价格Ct为式中为标准正态函数.本文假设标的资产由混合双分数布朗运动驱动，利用偏微分方程的方法探讨了欧式期权的定价问题.采用混合双分数布朗运动刻画金融资产的价格变化过程在一定程度上比传统模型有所改进，可以看作是混合分数布朗运动和双分数布朗运动驱动下的一种推广.另外，混合双分数布朗运动也可以应用于探讨奇异期权(如重置期权、障碍期权等)的定价问题.【相关文献】[1] MANDELBROT B，VAN N J W. Fractional Brownian motion，fractional noises and application[J]. SIAM Review，1968，10：422-437.[2] LIN S J. Stochastic analysis of fractional Brownian motion[J]. Stochastics and Stochastics Reports，1995，55(1/2)：122-140.[3] BENDER C，ELLIOTT R J. Arbitrage in a discrete version of the Wick-fractional Black-Scholes market[J]. Mathematics of Operations Research，2004，29(4)：935-945.[4] BJǒ R K T，HULT H. A note on Wick products and the fractional Black-Scholes model[J]. Finance and Stochastics，2005，9(2)：197-209.[5] LEI P，NUALART D. A decomposition of the bi-fractional Brownian motion and some applications[J]. Statistics and Probability Letters，2009，79(5)：619-624.[6] RUSSO F，TUDOR C. On the bifractional Brownian motion[J]. Stochastic Processes and their applications，2006，116(5)：830-856.[7] ALOS E，MAZET O，NUALART D. Stochastic calculus with respect to Gaussian processes[J]. Annals of Probability，2001，29(2)：766-801.[8] 邵宇，刁羽. 微观金融学及其数学基础[M].北京：清华大学出版社，2008：663-674.[9] 徐峰，郑石秋. 混合分数布朗运动驱动的幂期权定价模型[J]. 经济数学，2010，27(2)：8-12.[10] 赵巍. 分形市场视角下的期权定价模型及其套期保值策略研究[J].合肥工业大学学报：自然科学版，2013，36(11)：1388-1392.。

混合分数布朗运动下欧式期权模糊定价研究混合分数布朗运动下欧式期权模糊定价研究摘要:欧式期权定价一直是金融工程领域的重要研究方向之一。

本文探讨了在混合分数布朗运动假设下，对欧式期权进行模糊定价的方法和应用。

通过引入模糊随机变量的概念，将模糊集理论与分数布朗运动融合，建立了混合分数布朗运动下的欧式期权模糊定价模型。

通过数值实例分析，验证了该模型在欧式期权定价中的有效性和可行性。

1. 引言欧式期权是金融市场中的一种重要金融工具，在证券投资和风险管理中具有广泛的应用。

期权定价理论是金融工程研究的核心问题之一，传统的期权定价模型主要假设资产价格服从几何布朗运动，即假设价格演化满足随机游走的过程。

然而，这一假设存在许多问题，例如不能很好地描述价格波动的厚尾特征，忽视了极端事件的发生概率等。

为了解决这些问题，学者们提出了许多新型的资产价格模型，其中混合分数布朗运动模型是一种重要的创新。

混合分数布朗运动模型旨在克服几何布朗运动模型的局限性，它将长记忆过程和短记忆过程结合在一起，并通过参数调节分数布朗运动模型的漂移和扩散项，使得模型能更好地描述价格序列的波动特征。

在此基础上，本文引入模糊随机变量的概念，结合模糊集理论和混合分数布朗运动模型，研究了在这一框架下的欧式期权定价方法。

具体而言，我们将欧式期权的净现值视为模糊随机变量，并对其进行模糊建模和模糊推理，得到模糊随机变量的分布特征。

然后，通过求解对应的微分方程，得到了欧式期权的模糊随机变量的期望和变异数，从而完成了欧式期权的模糊定价。

2. 混合分数布朗运动下欧式期权模糊定价模型2.1 混合分数布朗运动模型混合分数布朗运动模型是一种能够较好地描述资产价格波动的模型。

它可以同时考虑长记忆过程和短记忆过程对价格序列的影响，并通过参数调节模型的漂移和扩散项来适应市场的实际情况。

具体而言，混合分数布朗运动模型可以表示为以下形式的随机微分方程：dX(t) = μ(t)dt + σ(t)dW^H(t)其中，X(t)是资产价格的对数收益率，μ(t)是随时间变化的漂移项，σ(t)是随时间变化的扩散项，W^H(t)是分数布朗运动。

分数布朗运动在期权定价中的应用研究分数布朗运动（fractional Brownian motion，FBM）因其在自回归和非平稳时间序列建模中的广泛应用而日益受到关注。

随着金融市场的日益复杂和风险的不断增加，分数布朗运动在金融领域中的应用也愈发重要。

在期权定价中，分数布朗运动的使用可以帮助金融从业者更准确地评估期权价值，降低风险。

本文将对分数布朗运动在期权定价中的应用进行研究探讨，主要从以下几个方面进行分析：一、分数布朗运动简介分数布朗运动是一种非平稳的随机过程，它是布朗运动的一种扩展形式。

与标准布朗运动不同的是，分数布朗运动的Hurst指数（Hurst exponent）不等于0.5，而是一个介于0和1之间的分数。

Hurst指数的值越大，即越接近1，分数布朗运动的长期相关性就越强。

在金融领域，分数布朗运动经常被用来建模资产价格，它可以捕捉市场中的长期依赖性和自相似性。

二、期权定价的基本原理期权定价是金融工具中的一种核心领域，它主要用来计算期权的合理价格。

期权定价的基本原理是基于期权的内在价值和时间价值。

内在价值是指期权当前的实际价值，即如果现在就行权，期权所能够给持有人带来的收益。

时间价值是指期权所含有的未来时间的价值，包括行权前能够在市场上实现的收益、市场风险以及其他因素。

三、分数布朗运动在期权定价中的应用1.基于分数布朗运动的期权定价模型分数布朗运动适用于各种金融产品的定价和风险度量。

基于分数布朗运动的期权定价模型是一种基于复合泊松过程和Hurst指数的模型，可以通过计算分数布朗运动的变差（variance）来获取期权价格。

分数布朗运动可以更好地描述市场中自相关性和自相似性的现象，从而更好地展现出实际市场的特征。

利用分数布朗运动进行期权定价可以更准确地预测期权价格，从而为金融从业者提供更准确的风险度量方式。

2.基于分数布朗运动的期权风险度量期权风险度量是评价期权风险的一种方法。

分数布朗运动模型可以提供更有效的风险度量方法，因为它可以更好地描述长期相关性和自相似性。

第27卷第6期2013年12月保险职业学院学报（双月刊）JOUＲNAL OF INSUＲANCE PＲOFESSIONAL COLLEGE（Bimonthly）Vol.27No.6Dec．2013分数布朗运动下的欧式期权的保险精算定价法陈飞跃，杨蓉(保险职业学院，湖南长沙410114)［摘要］本文在无市场假设的基础上，仅利用股票价格过程的概率测度和期权的保险精算定价方法，得到了标的资产(股票)服从几何分数布朗运动的欧式期权定价公式。

［关键词］保险精算方法;几何分数布朗运动;期权定价［中图分类号］F840．4［文献标识码］A［文章编号］1673－1360（2013）06－0064－03［Abstract］Without any market assumption，merely using probability measure of stock price process and insurance actuarial consideration for pricing option，this paper obtains European option pricing formula when un-derlying assets（stock）are driven by geometric fractional Brownian motion．［Key words］Insurance actuarial approach；Geometric factional Brownian motion；Option pricing一、引言自从1973年由Fisher Black和Myron Scholes 提出了经典的Black－Scholes期权定价模型以来，这一模型便被广泛地应用于金融市场的期权定价分析，这一模型假设股票价格的波动相互独立，且服从几何布朗运动，其对数收益独立同分布。

但近年来，期权定价研究者们通过对股票市场的大量实证研究，发现股票价格的对数收益分布函数具有“尖峰厚尾”的特点，而且股价之间也不是随机游走的，在不同时间存在着长期相关、自相似等特征，这是与几何布朗运动有较大差距的。