【作者】 李峥；

【导师】 韩月才；

【作者基本信息】 吉林大学 ， 概率论与数理统计， 2020， 博士

【摘要】 分数维布朗运动是一类连续的高斯过程,它既不是Markov过程也不是半鞅,当Hurst指数大于1/2时具有长时记忆性且增量具有正相关性,当Hurst指数小于H<1/2时增量具有负相关性.也正是由于这些性质分数维布朗运动比标准布朗运动能够更好的解释自然现象,所以围绕分数维布朗运动相关的随机分析理论被广泛研究,并且该理论被应用到了许多领域,例如气候学,水文学,金融数学等.本文主要研究由分数维布朗运动描述的随机波动率下的期权定价问题及分数维噪声驱动的离散随机控制系统的最优控制问题.上个世纪七十年代著名的欧式期权定价的Black-Scholes公式被提出后,对于金融市场的研究有了一个质的飞跃.尽管Black-Scholes公式被广泛应用,其得出的定价结果却与实际有一定的偏差,即现实市场存在波动率微笑现象,因此恒定为常数的波动率并不能解释现有的价格波动.随后许多学者开始投入到对随机波动率模型的研究中,其中比较经典被广泛应用的有Heston模型,Hull-White模型和Stein-Stein模型等.然而大量市场数据表明,与标准正态分布曲线相比真实的收益率分布曲线是倾斜的呈尖峰状并且具有长期的记忆性以及增量具有相关性.为了更加贴切的描述实际金融市场的价格规律,在波动率方程中分数维布朗运动替代了标准布朗运动.结合上述实际情况,在本文中,我们考虑两种分数维随机波动率模型下的欧式看涨期权定价问题.为了更好的刻画实际的市场价格走势,波动率方程是由分数维布朗运动和标准布朗运动同时驱动的.而引入分数维布朗运动同时也增加了问题求解的难度,需要考虑关于分数维布朗运动的Ito公式以及Malliavin计算.本文第一个模型中考虑一般情形的Hull-White模型,波动率方程非均值回复过程,此时波动率呈指数型增长,为了增加产品的灵活性规避由于波动率过大而对投资者造成的风险我们对波动率加入了障碍.在我们又考虑了另一随机波动率模型种,假设波动率满足分数维Ornstein-Uhlenbeck过程,由于波动率过程满足均值回复性,使得价格波动更加稳定同时也有效减少了投资者的风险.在这两个模型中我们运用期权复制策略结合分数维布朗运动的Ito公式和Malliavin计算得到了欧式看涨期权价格满足的偏微分方程,通过引入格林函数并运用待定系数法得到了欧式看涨期权价格的闭型解.最后对于得出的两个结果我们分别做了相关的数值模拟来进一步说明结果的合理性.下面我们简单介绍上述两种分数维随机波动率下的期权定价的主要结果.在第三章中我们考虑分数维Hull-White随机波动率模型其中St为风险资产价格过程,vt为瞬时波动率.模型中μ是风险资产定价过程的漂移率并且α是波动率过程的漂移率,γ1和γ2为波动率的波动率.Bts和Btv为(Ω,F,P)上的相关系数为ρ的两个标准布朗运动.BtH是Hurst参数H>1/2的(Ω,F,P)上的分数维布朗运动并且独立于BtS和Btv.由于波动率具有显性解且该解呈指数型增长,因此我们考虑波动率上障碍和下障碍分别为B和A运用Malliavin计算以及Ito公式通过复制投资组合的方法得到欧式看涨期权价格满足的偏微分方程为其中τ=T-t,T为欧式看涨期权的到期日.通过引入格林函数并且运用待定系数法对偏微分方程进行求解,得到定价公式的闭型解为在第四章中我们考虑分数维Ornstein-Uhlenbeck过程描述的随机波动率,假设波动率方程是一个均值回复过程并且不呈指数型增长.该模型如下其中St为标的资产的价格,是严格正的,vt为随机的瞬时波动率,rt为随机的瞬时利率,μ为标的资产的预期收益,β为波动率的均值回复速度,α为波动率的长期均值,γ1和γ2为波动率的波动率.我们运用复制投资组合方法得到欧式看涨期权价格满足的偏微分方程为这里τ=T-t,k=β+λ,θ=βα/β+λ并且运用待定系数法对偏微分方程进行求解,得到定价公式的闭型解为其中A(τ,k),B(τ,k)和 C(τ,k)如下这里庞特里亚金最大值原理作为解决最优控制问题的必要条件,由庞特里亚金及其团队于上世纪五十年代提出.对于实际的应用中,往往得到的是一些离散数据,因此离散最优控制问题比连续情形更具有实际意义.然而连续情形的方法并不能直接应用到离散情形中,最初一些学者考虑加入“凸性”条件来得到离散最优控制系统的最大值原理.随着随机问题的出现,随机最优控制问题被广泛研究.值得一提的是Peng[98]研究了随机系统情形的最大值原理,该结果中控制域可以非凸且扩散项中包含控制变量.然而相对于标准布朗运动更加一般情形的分数维布朗运动因其具有长时记忆性和自相似性因此能够更好的解释实际的现象和规律.近些年有许多学者在白噪声驱动的离散随机控制系统上给出了一些结果.综上我们考虑由分数维噪声驱动的离散随机控制系统的最大值原理问题.令{BtkH}k=0,1,2,…,N-1为一个d维分数维布朗运动序列,其中1/2<H<1,满足下列条件:(ⅰ){BtkH}k=0,1,2,…,N是Fk-1可测的.(ⅱ)BtkH的增量是平稳的但不是相互独立的.(ⅲ)对任意的BtkH=(BtkH,1,BtkH,2,…,BtkH,d)T,BtkH,1,BtkH,2,…,BtkH,d是相互独立的实值高斯随机变量.(ⅳ)E[BtkH]=0,E[BtiH,lBtjH,l]=1/2(ti2H+tj2H-|ti-tj|2H),其中 l=1,2,…,d.我们用(?)gtk=Btk+1H-BtkH来代表分数维噪声D代表一个有界域,则容许控制空间为Uad={Uk}k=0N-1(?){Utk∈Uk∈Rn:Ω→D\vtk是Fk-1可测的,E[vtTvtk]<+∞,k=0,1,…N-1.本文中离散随机控制系统为其性能指标为最优控制问题为在Uad范围内最小化性能指标J(v).也就是找到一个最优的u*∈Uad 使得令u*(·)为最优控制且(x*(·),u*(·))为最优对.我们通过经典变分法以及Malliavin计算得出了一个最优控制满足的必要条件:其中DH(·)和D(·)分别为分数维布朗运动和标准布朗运动的Malliavin导数,φtk满足如下方程更多还原

【Abstract】 Fractional Brownian motion is a continuous Gaussian process.Neither it is Markov process nor semimartingale.When Hurst index is greater than 1/2,it has long-term memory and the increments are positively correlated.While when Hurst index is small-er than 1/2,its increments are negatively correlated.The stochastic analysis theory related to fractional Brownian motion has been widely studied and applied to many fields,such as climatology,hydrology and financial Mathematics.This article mainly studies the option pricing problems described by the fractional Brownian motion and the optimal control problems of discrete stochastic control system driven by fractional noise.Since the famous Black-Scholes formula of European option pricing was put for-ward in the 1970s,there has been a qualitative leap in the study of financial market.However,although this theory is a milestone in option pricing,its pricing result is somewhat different from the actual situation.Therefore,constant volatility can not explain the existing price volatility.Then many scholars began to study the stochastic volatility model,among which Heston model,Hull-White model and Stein-Stein model are widely used.Compared with the standard normal distribution curve,the real yield distribution curve is inclined,peaked,long-term memory and incremental correlation.In order to describe the law of the price of the real financial market more appropriately,the standard Brownian motion was replaced by the fractional Brownian motion in the volatility equation.According to the above statement,two fractional stochastic volatility models are considered in this article.In order to better describe the actual market price trend,we adopt the volatility equations driven by both fractional Brownian motion and s-tandard Brownian motion.The introduction of the fractional Brownian motion also increases the difficulty of solving the problems.We have to apply Ito’s formula for fractional Brownian motion and Malliavin calculus to deal with the corresponding s-tochastic volatility problems.We first consider a general style of Hull-White stochastic volatility model.The volatility is not a mean reversion process and it increases expo-nentially.For increasing the flexibility of financial products and avoiding the risks to investors caused by excessive volatility,we add barriers to the volatility.Furthermore,we also consider another stochastic volatility model which satisfies fractional Ornstein-Uhlenbeck process.Because the volatility equation satisfy mean reversion,the price fluctuation is more stable and it also effectively reduces investors’risks.In these two models,we use the option replication strategy,Ito’s formula and Malliavin calculus to obtain the partial differential equation that the price of European call option meets.Then we solve the partial differential equation and obtain a explicit solution of the price of European call option.The numerical simulations for the results are also made for proving the rationality.Next,we give a brief introduction of the main results of the option pricing under the above two fractional stochastic volatility models.We consider option pricing under fractional Hull-White stochastic volatility model in chapter 3:where St is risky asset price process and vt is instantaneous volatility.Here,μis the drift rate of risk asset price process and β is the drift rate of the volatility process.γ1 and γ2 are volatilities of volatility.BtS and Btv are two standard Brownian motion on(Ω,F,P)and their correlation coefficient is ρ.BtH is fractional Brownian motion on(Ω,F,P)with Hurst parameter H>1/2 and independent with BtS and Btv.The explicit solution of the volatility is shown as follows:and it increases exponentially.So we add upper and lower barriers denoted by B and A.The partial differential equation of European call option price is obtained by using Malliavin calculus,Ito’s formula and option replication strategy.The equation is introduced as follows:where τ＝T-t and T is the due date of European call option.We solve the above partial differential equation by introducing the Green’s function and using the method of indeterminate coefficient.A closed-form solution is obtained asThe fractional Ornstein-Uhlenbeck process is considered to describe the stochastic volatility for pricing European option in chapter 4.The volatility is a mean reversion process and no longer increases exponentially.The stochastic volatility is written as where St is risky as,set price process and vt is instantaneous volatility.Here,μ is the drift rate of risk asset price process and β is the average recurrent rate of the volatility process.γ1 and γ2 are volatilities of volatility.BtsandBtv are two mutually independent standard Brownian motion on(Ω,F,P).BtH is fractional Brownian motion on(Ω,F,P)with Hurst parameter H>1/2 and independent with BtS and Btv.The partial differential equation of the price of European call option is obtained by using the option replicat,ion strategy.The equation is shown as follows:where τ=T-t,κ=β+λ,θ＝βα/β+λandWe solve the above partial differential equation and obtain an closed-form solution as where A(τ,k),B(τ,k)and C(τ,k)are derived as Here,As a necessary condition to solve the optimal control problems,the Pontryagin’s maximum principle was proposed by Pontryagin and his team in the 1950s.One al-ways obtains the discrete data in practical applications.Therefore,the discrete optimal control problems are more meaningful than the continuous case.However,the meth-ods to study optimal control problems in continuous case can not be directly applied to discrete cases.Some scholars considered to add some convex condition to obtain the maximum principle of discrete controlled system.The stochastic optimal control problems were studied extensively.A classical result was introduced by Peng[98]and he obtained a maximum principle of stochastic control system,where the control vari-ables were included in the diffusion term and the domain of the admissible control need not be convex.Compared to standard Brownian motion,the general fractional Brownian motion can better describe the actual phenomena and laws because of its long-term memory and self similarity.Some scholars gave results for discrete controlled system driven by white noise recently.According to the above statement,we consider maximum principle of the discrete stochastic optimal control problems driven by both fractional noise and white noise.Let{BtkH}k=0,1,2,…,N-1 be a sequence of d-dimunsion fractional Brownian motion that satisfies the following conditions:(ⅰ){BtkH}k=0,1,2,…,N is Fk-1-measurable.(ⅱ)The increments of(?)are stationary and need not be independent.(ⅲ)For every BtkH=(BtkH,1,BtkH,2,…,BtkH,d)T,BtkH,1,BtkH,2…,BtkH,dare indepen-dent R-valued Gaussian random variables.(ⅳ)(?)E[BtkH]=0,E[BtiH,lBtjH,l]=1/2(ti2H-|ti-tj|2H),wherel=1,2,…,d.Let gtk=Btk+1H-BtkHbe the fractional noise.Let D be a bounded domain and the space of admissible controls is defined asUad={Uk}(?)k=0N-1(?){vtk∈Uk∈Rn:Ω→D|vtkisFk-1-mcasurable,E[vtkTvtk]<+∞,k=0.1,…N-1｝.The discrete controlled stochastic system in chapter 5 is described as with the cost functional as Our stochastic optimal control problem is to minimize the cost functional J(v)over Uad.Namely,to find the optimal u*∈Uad satisfyingLet u*(·)be the optimal control and(a*(·),v*(·))be the optimal pair.A necessary condition which optimal control satisfies is derived by using classical variational method and Malliavin calculus and the main result is introduced as where DH(·)and D(·)are Malliavin derivatives with respect to fractional Brownian motion and standard Brownian motion.φtk satisfies the following equation:更多还原