分数Black-Scholes模型下美式期权定价的积分方程式

为得到分数Black-Scholes模型下美式期权价格的公式,文章以看涨期权为例,应用偏微分方程法,推导期权价格的积分方程式.由于美式期权的价格可分解为欧式期权的价格和由于提前实施需要增付的期权金,而提前实施期权金与最佳实施边界的位置有关,所以为导出最佳实施边界所满足的方程,文章首先研究分数Black-Scholes方程的基本解,然后建立美式看涨期权的分解公式,推导最佳实施边界适合的非线性积分方程,从而得到美式看涨期权价格的积分方程式.美式看跌期权价格的积分方程式类似得到.     

美式利率期权的最佳实施边界的分析

在Vasicek利率模型的假设下,应用变分不等式方法分析了美式利率期权自由边界的性质．首先我们得到美式利率期权自由边界的下界, 然后把自由边界问题化为变分不等式,通过引入惩罚函数证明了该变分不等式解的存在唯一性,最后证明了自由边界的单调性、 有界性和C∞光滑性．     

多尺度高维亚式期权定价的奇摄动解

讨论了一类含有快慢变换尺度的高维亚式期权定价随机波动率模型.根据Girsanov定理和Radon-Nikodym导数实现了期望回报率与无风险利率之间的转化;定义路径依赖型的新算术平均算法,借助Feynman-Kac公式,得到了风险资产期权价格所满足的相应的Black-Scholes方程,运用奇摄动渐近展开方法,得到了期权定价方程的渐近解,并得到其一致有效估计.     

跳扩散模型下美式期权的对称性

利用分析方法得到了跳扩散模型下美式看涨、看跌期权的价格和最佳实施边界间的对称性公式．美式看涨和看跌期权价格问的对称关系通常是利用概率理论得到,这里给出了这些结果在跳扩散模型下的另一种证明．此外,由本文所得结果和偏微分方程理论,可以得到跳扩散模型下美式看涨期权的最佳实施边界以及永久美式期权的若干性质．     

随机波动率与双指数跳扩散组合模型的美式期权定价

在股价满足Cox-Ingersoll-Ross(CIR)随机波动率与Kou的双指数跳扩散组合模型下,利用随机分析方法讨论了美式看跌期权函数及最佳实施边界的性质.应用一阶线性近似实施边界获得了期权价格的拟解析式和实施边界满足的非线性方程.进一步,应用梯形法离散处理方程式内积分表达式,建立了期权最佳实施边界和价格的数值算法.最后分别给出了常数波动率或CIR随机波动率的数值实例.     

混合高斯模型下带红利的永久美式期权定价

本文建立混合高斯模型下支付连续红利的永久美式期权定价模型.利用自融资策略和分数伊藤公式,得到永久美式期权价值所满足的偏微分方程.其次,由永久美式期权的实施条件与看涨-看跌期权的对称关系,获得看涨与看跌期权的定价公式与最佳实施边界.最后,利用平安银行的日收盘价对标的资产进行实证分析,结果表明:用混合高斯模型模拟出的股票价格与真实股票价格比较接近,能够反映股票的整体走势.     

一类渗流方程解在边界上的不稳定性和Blow-up

对一类具有非线性第二、第三边值条件的非线性渗流方程,证明了解的先验的界可以用初值和解在区域边界上的积分来估计和控制．这一先验估计是通过迭代技巧来建立的．根据这个估计,解可能在边界上爆破(Blow-up)从而解有渐近不稳定性．     

多尺度亚式期权定价模型的奇摄动解

讨论了一类多尺度亚式期权定价随机波动率模型问题,其中随机波动率采用了具有快慢变换的随机波动率模型.通过Feynman-Kac公式,得到了风险资产期权价格所满足的相应的Black-Scholes方程,运用奇摄动渐近展开方法,得到了期权定价方程的渐近解,并得到其一致有效估计.     

美式利率期权定价的抛物型变分不等式

应用PDE方法对美式利率期权定价问题进行理论分析.在CIR利率模型下美式利率期权定价问题可归结为一个退化的一维抛物型变分不等式.通过引入惩罚函数证明了该变分不等式的解的存在唯一性,然后研究了自由边界的一些性质,如单调性,光滑性和自由边界在终止期的位置.     

随机利率下期权定价

本文在随机利率的基础上,考虑股票价格过程和利率过程分别为扩散过程和Ito过程,并且在相关的假设下,运用鞅方法推导出欧式期权价值过程所满足的微分方程;以及利率满足一种特殊方程时,运用最优停止的鞅方法,得到了随机利率下美式期权的价格和最优停时.     

A NUMERICAL METHOD FOR DETERMINING THE OPTIMAL EXERCISE PRICE TO AMERICAN OPTIONS

American Options can be exercised prior to the date of expiration,the valuation of American options then constitutes a free boundary value problem.How to determine the free boundary,i.e. the optimal exercise price,is a key problem.In this paper,a nonlinear equation is given.The free boundary can be obtained by solving the nonlinear equation and the numerical results are better.     

The regularity of the free boundary in a multidimensional filtration problem

We prove the regularity of the free boundary for a filtration problem with capillarity in more than one space dimension. The free boundary is the interface between the saturated region (in which the governing equation is elliptic) and the unsaturated region (where a degenerate parabolic equation is to be solved).This work was partially supported by National Project Equazioni di Evoluzione e Applicazioni Fisico Matematiche (M.U.R.S.T.).     

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRODINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schr(o)dinger equation posed on finite interval α≤ x ≤β:{iyt yxx |y|2y = 0,y(α,t) = h1(t),y(β,t) = h2(t) for t ＞ 0 (S)is considered. It is shown that by choosing appropriate control inputs (hj), (j = 1,2) one can always guide the system (S) from a given initial state ψ∈ Hs(α,β),(s ∈ R) to a terminal state ψ∈ Hs(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schr(o)dinger equation posed on the whole line R. The discovered smoothing properties of Schr(o)dinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schr(o)dinger equation.     

A class of options with stochastic lives and an extension of the Black-Scholes formula

Certain options have a fixed date of maturity but may be cancelled prematurely. This can happen for a stock option in case of a merger or for an executive stock option in case the executive leaves his/her present job. The differential equation is given which governs the value of an option with a stochastic life. Solutions can be obtained through integration in certain cases. The main result is an extension of the Black-Scholes formula to options where the time to expiration is stochastic.     

半导体瞬态问题的修正迎风有限体积格式

半导体器件的瞬时状态由包含三个拟线性偏微分方程所组成的方程组的初边值问题来描述.其中电子位势方程足椭圆型的,电子和空穴浓度方程是对流扩散型的.对电子位势方程采用一次元有限体积法米逼近,对电子浓度和空穴浓度方程采用修正的迎风有限体积方法来逼近,并进行详细的理论分析,关于位势得到O(h Δt)阶的H1模误差估计结果,关于浓度得到O(h2 Δt)阶的L2模误差估计结果.最后,给出数值例子.     

Solutions of the equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $$\end{document}. Exact solutions of some partial differential equations in mechanics and physics

The solutions of the equation $ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $, for L? a linear operator are derived. Different forms for L? whether it is time independent or time dependent and self-commutative (or not) at different times are considered separately. By using the results obtained, exact solutions of some partial differential equations are found for the first time.     

On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory

The equation arising from Prandtl boundary layer theory $$\frac{\partial u}{\partial t} -\frac{\partial }{\partial x_i}\left( a(u,x,t)\frac{\partial u}{\partial x_i}\right)-f_i(x)D_iu+c(x,t)u=g(x,t)$$ is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since $a(\cdot,x,t)$ may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kružkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if $a_{x_i}(\cdot,x,t)\mid_{x\in \partial \Omega}=a(\cdot,x,t)\mid_{x\in \partial \Omega}=0$ and $f_i(x)\mid_{x\in \partial \Omega}=0$, the stability can be proved even without any boundary value condition.     

Novel numerical techniques based on Fokas transforms,for the solution of initial boundary value problems

The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10⁻¹⁵, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10⁻⁸ accuracy.     

非齐次边界条件泛定方程的代换选择

对具有非齐次边界条件的泛定方程齐次化过程中代换的选择进行研究和探讨.基于一些相关结论和齐次化的定义得出新的研究成果,即给出对三类非齐次边界条件齐次化都适用的代换W(x,t)=A(t)x<'3>+B(t).     

时标上动力方程在右局部边值条件下正解的存在性

主要研究时标上二阶动力学方程u~(△△)(t)+λ_p(t)f(t,u(σ(t)))=0在右局部边值条件u(0)=0=u~△(σ(1))下正解的存在性.应用格林函数和锥上Krasnoselskii不动点原理给出其正解存在的充分条件及正解存在的特征值区间.