## 金融代写|金融工程作业代写Financial Engineering代考|BE953

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Phase Diagrams for Linear Dynamical Systems

The following autonomous linear system is considered
$$\dot{x}=A x$$
The eigenvalues of matrix $A$ define the system dynamics. Some terminology associated with fixed points is as follows:

A fixed point for the system of Eq. (1.27) is called hyperbolic if none of the eigenvalues of matrix $A$ has zero real part. A hyperbolic fixed point is called a saddle if some of the eigenvalues of matrix $A$ have real parts greater than zero and the rest of the eigenvalues have real parts less than zero. If all of the eigenvalues have negative real parts then the hyperbolic fixed point is called a stable node or sink. If all of the eigenvalues have positive real parts then the hyperbolic fixed point is called an unstable node or source. If the eigenvalues are purely imaginary then one has an elliptic fixed point which is said to be a center.
Case 1: Both eigenvalues of matrix $A$ are real and unequal, that is $\lambda_1 \neq \lambda_1 \neq 0$. For $\lambda_1<0$ and $\lambda_2<0$ the phase diagram for $z_1$ and $z_2$ is shown in Fig. 1.4. In case that $\lambda_2$ is smaller than $\lambda_1$ the term $e^{\lambda_2 t}$ decays faster than $e^{\lambda_1 t}$. For $\lambda_1>0>\lambda_2$ the phase diagram of Fig. $1.5$ is obtained.

In the latter case there are stable trajectories along eigenvector $v_1$ and unstable trajectories along eigenvector $v_2$ of matrix $A$. The stability point $(0,0)$ is said to be a saddle point.
When $\lambda_1>\lambda_2>0$ then one has the phase diagrams of Fig. 1.6.
Case 2: Complex eigenvalues:
Typical phase diagrams in the case of stable complex eigenvalues are given in Fig. 1.7.
Typical phase diagrams in the case of unstable complex eigenvalues are given in Fig. 1.8.

Typical phase diagrams in the case of imaginary eigenvalues are given in Fig. 1.9.
Case 3: Matrix $A$ has nonzero eigenvalues which are equal to each other. The associated phase diagram is given in Fig. 1.10.

## 金融代写|金融工程作业代写Financial Engineering代考|Saddle-Node Bifurcations of Fixed Points

The considered dynamical system is given by $\dot{x}=\mu-x^2$. The fixed points of the system result from the condition $\dot{x}=0$ which for $\mu>0$ gives $x^*=\pm \sqrt{\mu}$. The first fixed point $x=\sqrt{\mu}$ is a stable one whereas the second fixed point $x=-\sqrt{\mu}$ is an unstable one. The phase diagram of the system is given in Fig. 1.14. Since there is one stable and one unstable fixed point the associated bifurcation (locus of the fixed points in the phase plane) will be a saddle-node one.

The bifurcations diagram is given next. The diagram shows how the fixed points of the dynamical system vary with respect to the values of parameter $\mu$. In the above case it represents a parabola in the $\mu-x$ plane as shown in Fig. 1.15.

For $\mu>0$ the dynamical system has two fixed points located at $\pm \sqrt{\mu}$. The one fixed point is stable and is associated with the upper branch of the parabola. The other fixed point is unstable and is associated with the lower branch of the parabola. The value $\mu=0$ is considered to be a bifurcation value and the point $(x, \mu)=(0,0)$ is a bifurcation point. This particular type of bifurcation where the one branch is associated with fixed points and the other branch is not associated to any fixed points is known as saddle-node bifurcation.

In pitchfork bifurcations the number of fixed points varies with respect to the values of the bifurcation parameter. The dynamical system $\dot{x}=x\left(\mu-x^2\right)$ is considered. The associated fixed points are found by the condition $\dot{x}=0$. For $\mu<0$ there is one fixed point at zero which is stable. For $\mu=0$ there is still one fixed point at zero which is still stable. For $\mu>0$ there are three fixed points, one at $x=0$, one at $x=+\sqrt{\mu}$ which is stable and one at $x=-\sqrt{\mu}$ which is also stable. The associated phase diagrams and fixed points are presented in Fig. 1.16.

The bifurcations diagram is given next. The diagram shows how the fixed points of the dynamical system vary with respect to the values of parameter $\mu$. In the above case it represents a parabola in the $\mu-x$ plane as shown in Fig. 1.17.

## 金融代写|金融工程作业代写Financial Engineering代考|Phase Diagrams for Linear Dynamical Systems

$$\dot{x}=A x$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|金融工程作业代写Financial Engineering代考|ICEFE2022

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Lyapunov Stability Approach

The Lyapunov method analyzes the stability of a dynamical system without the need to compute explicitly the trajectories of the state vector $x=\left[x_1, x_2, \cdots, x_n\right]^T$.
Theorem: The system described by the relation $\dot{x}=f(x)$ is asymptotically stable in the vicinity of the equilibrium $x_0=0$ if there is a function $V(x)$ such that
(i) $V(x)$ to be continuous and to have a continuous first order derivative at $x_0$
(ii) $V(x)>0$ if $x \neq 0$ and $V(0)=0$
(iii) $\dot{V}(x)<0, \forall x \neq 0$.
The Lyapunov function is usually chosen to be a quadratic (and thus positive) energy function of the system however there in no systematic method to define it.

Assume now, that $\dot{x}=f(x)$ and $x_0=0$ is the equilibrium. Then the system is globally asymptotically stable if for every $\varepsilon>0, \exists \delta(\varepsilon)>0$, such that if $|x(0)|<\delta$ then $|x(t)|<\varepsilon, \forall t \geq 0$.

This means that if the state vector of the system starts in a disc of radius $\delta$ then as time advances it will remain in the disc of radius $\varepsilon$, as shown in Fig. 1.3. Moreover, if $\lim _{t \rightarrow \infty}|x(t)|=x_0=0$ then the system is globally asymptotically stable.
Example 1: Consider the system
$$\begin{gathered} \dot{x}_1=x_2 \ \dot{x}_2=-x_1-x_3^2 \end{gathered}$$
The following Lyapunov function is defined
$$V(x)-x_1^2+x_2^2$$

The equilibrium point is $\left(x_1=0, x_2=0\right)$. It holds that $V(x)>0 \forall\left(x_1, x_2\right) \neq(0,0)$ and $V(x)=0$ for $\left(x_1, x_2\right)=(0,0)$. Moreover, it holds
$$\begin{gathered} \dot{V}(x)=2 x_1 \dot{x}1+2 x_2 \dot{x}_2=2 x_1 x_2+2 x_2\left(-x_1-x_2^3\right) \Rightarrow \ \dot{V}(x)=-2 x_2^4<0 \forall\left(x_1, x_2\right) \neq(0,0) \end{gathered}$$ Therefore, the system is asymptotically stable and $\lim {t \rightarrow \infty}\left(x_1, x_2\right)=(0,0)$.

## 金融代写|金融工程作业代写Financial Engineering代考|Local Stability Properties of a Nonlinear Model

Local stability of a nonlinear model can be studied round the associated equilibria. Local linearization can be performed round equilibria, using the set of differential equations that describe the nonlinear model $\dot{x}=h(x)$ and performing Taylor series expansion, that is $\dot{x}=h(x) \Rightarrow \dot{x}=\left.h\left(x_0\right)\right|_{x_0}+\nabla_x h\left(x-x_0\right)+\cdots$.
The nonlinear model is taken to have the generic form
$$\left(\begin{array}{l} \dot{x}_1 \ \dot{x}_2 \end{array}\right)=\left(\begin{array}{l} f\left(x_1, x_2\right) \ g\left(x_1, x_2\right) \end{array}\right)$$
where $f\left(x_1, x_2\right)=2 x_1+x_2^2$ and $g\left(x_1, x_2\right)=x_1^2+2 x_2$. The fixed points of this model are computed from the condition $\dot{x}_1=0$ and $\dot{x}_2=0$. Using these relations one finds the equilibria $\left(x_1^, x_2^\right)=(0,0)$ and $\left(x_1^, x_2^\right)=(0,0)$
The Jacobian matrix $\nabla_x h=M$ is given by
$$M=\left(\begin{array}{ll} \frac{\partial f}{\partial x_1} & \frac{\partial f}{\partial x_2} \ \frac{\partial g}{\partial x_1} & \frac{\partial g}{\partial x_2} \end{array}\right)$$
which results into the matrix
$$J=\left(\begin{array}{cc} 2 & 2 x_2 \ 2 x_1 & 2 \end{array}\right)$$
The eigenvalues of matrix $M$ define stability round fixed points (stable or unstable fixed point). To this end, one has to find the roots of the associated characteristic polynomial that is given by $\operatorname{det}(\lambda I-J)=0$ where $I$ is the identity matrix.

## 金融代写|金融工程作业代写Financial Engineering代考|Lyapunov Stability Approach

Lyapunov 方法分析动力系统的稳定性，无需显式计算状态向量的轨迹
$$x=\left[x_1, x_2, \cdots, x_n\right]^T \text {. }$$

(i) $V(x)$ 是连续的，并且有一个连续的一阶导数在 $x_0$
(二) $V(x)>0$ 如果 $x \neq 0$ 和 $V(0)=0$
(三) $\dot{V}(x)<0, \forall x \neq 0$. Lyapunov 函数通常被选择为系统的二次 (因此是正) 能量函数，但是没有系统的方法来定义 它。 现在假设， $\dot{x}=f(x)$ 和 $x_0=0$ 是平衡点。那么系统是全局渐近稳定的如果对于每个 $\varepsilon>0, \exists \delta(\varepsilon)>0$, 这样如果 $|x(0)|<\delta$ 然后 $|x(t)|<\varepsilon, \forall t \geq 0$. 这意味着如果系统的状态向量开始于半径为 $\delta$ 然后随着时间的推移它将保留在半径的圆盘中 $\varepsilon$ ， 如图1.3所示。此外，如果 $\lim _{t \rightarrow \infty}|x(t)|=x_0=0$ 那么系统是全局渐近稳定的。 示例 1：考虑系统 $$\dot{x}_1=x_2 \dot{x}_2=-x_1-x_3^2$$ 定义了以下 Lyapunov 函数 $$V(x)-x_1^2+x_2^2$$ 平衡点是 $\left(x_1=0, x_2=0\right)$. 它认为 $V(x)>0 \forall\left(x_1, x_2\right) \neq(0,0)$ 和 $V(x)=0$ 为了 $\left(x_1, x_2\right)=(0,0)$. 此外， 它持有
$$\dot{V}(x)=2 x_1 \dot{x} 1+2 x_2 \dot{x}_2=2 x_1 x_2+2 x_2\left(-x_1-x_2^3\right) \Rightarrow \dot{V}(x)=-2 x_2^4<0 \forall\left(x_1, x_2\right)$$

## 金融代写|金融工程作业代写Financial Engineering代考|Local Stability Properties of a Nonlinear Model

left $\left(x_{-} 1 \wedge, x_{-} _2 \wedge\right.$ right $)=(0,0)$

$$M=\left(\begin{array}{llll} \frac{\partial f}{\partial x_1} & \frac{\partial f}{\partial x_2} & \frac{\partial g}{\partial x_1} & \frac{\partial g}{\partial x_2} \end{array}\right)$$

$$J=\left(\begin{array}{lll} 2 & 2 x_2 2 x_1 & 2 \end{array}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|金融工程作业代写Financial Engineering代考|Best107

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Characteristics of the Dynamics of Nonlinear Systems

Main features characterizing the stability of nonlinear dynamical systems are defined as follows [121, 274]:

1. Finite escape time: It is the finite time within which the state-vector of the nonlinear system converges to infinity.
2. Multiple isolated equilibria: A linear system can have only one equilibrium to which converges the state vector of the system in steady-state. A nonlinear system can have more than one isolated equilibria (fixed points). Depending on the initial state of the system, in steady-state the state vector of the system can converge to one of these equilibria.
3. Limit cycles: For a linear system to exhibit oscillations it must have eigenvalues on the imaginary axis. The amplitude of the oscillations depends on initial conditions. In nonlinear systems one may have oscillations of constant amplitude and frequency, which do not depend on initial conditions. This type of oscillations is known as limit cycles.
4. Sub-harmonic, harmonic and almost periodic oscillations: A stable linear system under periodic input produces an output of the same frequency. A nonlinear system,

under periodic excitation can generate oscillations with frequencies which are several times smaller (subharmonic) or multiples of the frequency of the input (harmonic). It may also generate almost periodic oscillations with frequencies which are not necessarily multiples of a basis frequency (almost periodic oscillations).

1. Chaos: A nonlinear system in steady-state can exhibit a behavior which is not characterized as equilibrium, periodic oscillation or almost periodic oscillation. This behavior is characterized as chaos. As time advances the behavior of the system changes in a random-like manner, and this depends on the initial conditions. Although the dynamic system is deterministic it exhibits randomness in the way it evolves in time.
2. Multiple modes of behavior: It is possible the same dynamical system to exhibit simultaneously more than one of the aforementioned characteristics (1)-(5). Thus, a system without external excitation may exhibit simultaneously more than one limit cycles. A system receiving a periodic external input may exhibit harmonic or subharmonic oscillations, or an even more complex behavior in steady state which depends on the amplitude and frequency of the excitation.

## 金融代写|金融工程作业代写Financial Engineering代考|Computation of Isoclines

An autonomous second order system is described by two differential equations of the form
\begin{aligned} & \dot{x}_1=f_1\left(x_1, x_2\right) \ & \dot{x}_2=f_2\left(x_1, x_2\right) \end{aligned}
The method of the isoclines consists of computing the slope (ratio) between $f_2$ and $f_1$ for every point of the trajectory of the state vector $\left(x_1, x_2\right)$.
$$s(x)=\frac{f_2\left(x_1, x_2\right)}{f_1\left(x_1, x_2\right)}$$
The case $s(x)=c$ describes a curve in the $x_1-x_2$ plane along which the trajectories $\dot{x}_1=f_1\left(x_1, x_2\right)$ and $\dot{x}_2=f_2\left(x_1, x_2\right)$ have a constant slope.

The curve $s(x)=c$ is drawn in the $x_1-x_2$ plane and along this curve one also draws small linear segments of length $c$. The curve $s(x)=c$ is known as isocline. The direction of these small linear segments is according to the sign of the ratio $f_2\left(x_1, x_2\right) / f_1\left(x_1, x_2\right)$.
Example 1:
The following simplified nonlinear dynamical system is considered
$$\begin{gathered} \dot{x}_1=x_2 \ \dot{x}_2=-\sin \left(x_1\right) \end{gathered}$$

The slope $s(x)$ is given by the relation
$$s(x)=\frac{f_2\left(x_1, x_2\right)}{f_1\left(x_1, x_2\right)} \Rightarrow s(x)=-\frac{s i n\left(x_2\right)}{x_2}$$
Setting $s(x)=c$ it holds that the isoclines are given by the relation
$$x_2=-\frac{1}{c} \sin \left(x_1\right)$$
For different values of $c$ one has the following isoclines diagram depicted in Fig. 1.1. Example 2:

The following oscillator model is considered, being free of friction and with statespace equations
$$\begin{gathered} \dot{x}_1=x_2 \ \dot{x}_2=-0.5 x_2-\sin \left(x_1\right) \end{gathered}$$

## 金融代写|金融工程作业代写Financial Engineering代考|Characteristics of the Dynamics of Nonlinear Systems

1. 有限逃逸时间：是非线性系统的状态向量收敛到无穷大的有限时间。
2. 多重孤立平衡：一个线性系统只能有一个平衡点，该平衡点收敛于稳态系统的状态向量。一个非线性系统可以有多个孤立的平衡点（不动点）。根据系统的初始状态，在稳态下，系统的状态向量可以收敛到这些平衡点之一。
3. 极限环：对于表现出振荡的线性系统，它必须在虚轴上具有特征值。振荡的幅度取决于初始条件。在非线性系统中，可能存在幅度和频率恒定的振荡，不依赖于初始条件。这种类型的振荡被称为极限环。
4. 次谐波、谐波和几乎周期性振荡：在周期性输入下稳定的线性系统会产生相同频率的输出。一个非线性系统，

1. 混沌：处于稳态的非线性系统可能表现出一种不具有平衡、周期性振荡或几乎周期性振荡特征的行为。这种行为的特点是混乱。随着时间的推移，系统的行为会以类似随机的方式发生变化，这取决于初始条件。尽管动态系统是确定性的，但它随时间演化的方式表现出随机性。
2. 多种行为模式：同一动力系统可能同时表现出上述特征 (1)-(5) 中的一个以上。因此，没有外部激励的系统可能同时表现出多个极限循环。接收周期性外部输入的系统可能会表现出谐波或次谐波振荡，或者在稳态下表现出更复杂的行为，这取决于激励的幅度和频率。

## 金融代写|金融工程作业代写Financial Engineering代考|Computation of Isoclines

$$\dot{x}_1=f_1\left(x_1, x_2\right) \quad \dot{x}_2=f_2\left(x_1, x_2\right)$$

$$s(x)=\frac{f_2\left(x_1, x_2\right)}{f_1\left(x_1, x_2\right)}$$

$$\dot{x}_1=x_2 \dot{x}_2=-\sin \left(x_1\right)$$

$$s(x)=\frac{f_2\left(x_1, x_2\right)}{f_1\left(x_1, x_2\right)} \Rightarrow s(x)=-\frac{\sin \left(x_2\right)}{x_2}$$

$$x_2=-\frac{1}{c} \sin \left(x_1\right)$$

$$\dot{x}_1=x_2 \dot{x}_2=-0.5 x_2-\sin \left(x_1\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|金融工程作业代写Financial Engineering代考|BE953

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Local Stability Properties of a Nonlinear Model

Local stability of a nonlinear model can be studied round the associated equilibria. Local linearization can be performed round equilibria, using the set of differential equations that describe the nonlinear model $\dot{x}=h(x)$ and performing Taylor series expansion, that is $\dot{x}=h(x) \Rightarrow \dot{x}=\left.h\left(x_{0}\right)\right|{x{0}}+\nabla_{x} h\left(x-x_{0}\right)+\cdots$.
The nonlinear model is taken to have the generic form
$$\left(\begin{array}{l} \dot{x}{1} \ \dot{x}{2} \end{array}\right)=\left(\begin{array}{l} f\left(x_{1}, x_{2}\right) \ g\left(x_{1}, x_{2}\right) \end{array}\right)$$
where $f\left(x_{1}, x_{2}\right)=2 x_{1}+x_{2}^{2}$ and $g\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{2}$. The fixed points of this model are computed from the condition $\dot{x}{1}=0$ and $\dot{x}{2}=0$. Using these relations one finds the equilibria $\left(x_{1}^{}, x_{2}^{}\right)=(0,0)$ and $\left(x_{1}^{}, x_{2}^{}\right)=(0,0)$
The Jacobian matrix $\nabla_{x} h=M$ is given by
$$M=\left(\begin{array}{ll} \frac{\partial f}{\partial x_{1}} & \frac{\partial f}{\partial x_{2}} \ \frac{\partial g}{\partial x_{1}} & \frac{\partial g}{\partial x_{2}} \end{array}\right)$$
which results into the matrix
$$J=\left(\begin{array}{cc} 2 & 2 x_{2} \ 2 x_{1} & 2 \end{array}\right)$$

## 金融代写|金融工程作业代写Financial Engineering代考|Phase Diagrams for Linear Dynamical Systems

The following autonomous linear system is considered
$$\dot{x}=A x$$
The eigenvalues of matrix $A$ define the system dynamics. Some terminology associated with fixed points is as follows:

A fixed point for the system of Eq. (1.27) is called hyperbolic if none of the eigenvalues of matrix $A$ has zero real part. A hyperbolic fixed point is called a saddle if some of the eigenvalues of matrix $A$ have real parts greater than zero and the rest of the eigenvalues have real parts less than zero. If all of the eigenvalues have negative real parts then the hyperbolic fixed point is called a stable node or sink. If all of the eigenvalues have positive real parts then the hyperbolic fixed point is called an unstable node or source. If the eigenvalues are purely imaginary then one has an elliptic fixed point which is said to be a center.
Case 1: Both eigenvalues of matrix $A$ are real and unequal, that is $\lambda_{1} \neq \lambda_{1} \neq 0$. For $\lambda_{1}<0$ and $\lambda_{2}<0$ the phase diagram for $z_{1}$ and $z_{2}$ is shown in Fig. 1.4. In case that $\lambda_{2}$ is smaller than $\lambda_{1}$ the term $e^{\lambda_{2} t}$ decays faster than $e^{\lambda_{1} t}$. For $\lambda_{1}>0>\lambda_{2}$ the phase diagram of Fig. $1.5$ is obtained.

## 金融代写|金融工程作业代写Financial Engineering代考|Local Stability Properties of a Nonlinear Model

$$(\dot{x} 1 \dot{x} 2)=\left(f\left(x_{1}, x_{2}\right) g\left(x_{1}, x_{2}\right)\right)$$

$$M=\left(\begin{array}{lll} \frac{\partial f}{\partial x_{1}} \quad \frac{\partial f}{\partial x_{2}} & \frac{\partial g}{\partial x_{1}} & \frac{\partial g}{\partial x_{2}} \end{array}\right)$$

## 金融代写|金融工程作业代写Financial Engineering代考|Phase Diagrams for Linear Dynamical Systems

$$\dot{x}=A x$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|金融工程作业代写Financial Engineering代考|ICEFE 2022

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|The Phase Diagram

One can consider the following nonlinear model with the two state variables, $V$ and $\eta$. The dynamics of this model can be written as
\begin{aligned} \frac{d V}{d t} &=f(V, t) \ \frac{d \eta}{d t} &=g(V, \eta) \end{aligned}
The phase diagram consists of the points on the trajectories of the solution of the associated differential equation, i.e. $\left(V\left(t_{k}\right), \eta\left(t_{k}\right)\right)$.

At a fixed point or equilibrium it holds $f\left(V_{R}, \eta_{R}\right)=0$ and $g\left(V_{R}, \eta_{R}\right)=0$. The closed trajectories are associated with periodic solutions. If there are closed trajectories then $\exists T>0$ such that $\left(V\left(t_{k}\right), \eta\left(t_{k}\right)\right)=\left(V\left(t_{k}+T\right), \eta\left(t_{k}+T\right)\right)$.

Another useful parameter is the nullclines. The $V$-nullcline is characterized by the relation $\dot{V}=f(V, \eta)=0$. The $\eta$-nullcline is characterized by the relation $\dot{\eta}=g(V, \eta)=0$. The fixed points (or equilibria) are found on the intersection of nullclines.

## 金融代写|金融工程作业代写Financial Engineering代考|Lyapunov Stability Approach

The Lyapunov method analyzes the stability of a dynamical system without the need to compute explicitly the trajectories of the state vector $x=\left[x_{1}, x_{2}, \cdots, x_{n}\right]^{T}$.
Theorem: The system described by the relation $\dot{x}=f(x)$ is asymptotically stable in the vicinity of the equilibrium $x_{0}=0$ if there is a function $V(x)$ such that
(i) $V(x)$ to be continuous and to have a continuous first order derivative at $x_{0}$
(ii) $V(x)>0$ if $x \neq 0$ and $V(0)=0$
(iii) $\dot{V}(x)<0, \forall x \neq 0$.
The Lyapunov function is usually chosen to be a quadratic (and thus positive) energy function of the system however there in no systematic method to define it.

Assume now, that $\dot{x}=f(x)$ and $x_{0}=0$ is the equilibrium. Then the system is globally asymptotically stable if for every $\varepsilon>0, \exists \delta(\varepsilon)>0$, such that if $|x(0)|<\delta$ then $|x(t)|<\varepsilon, \forall t \geq 0$.

This means that if the state vector of the system starts in a disc of radius $\delta$ then as time advances it will remain in the disc of radius $\varepsilon$, as shown in Fig. 1.3. Moreover, if $\lim {t \rightarrow \infty}|x(t)|=x{0}=0$ then the system is globally asymptotically stable.
Example 1: Consider the system
$$\begin{gathered} \dot{x}{1}=x{2} \ \dot{x}{2}=-x{1}-x_{3}^{2} \end{gathered}$$
The following Lyapunov function is defined
$$V(x)=x_{1}^{2}+x_{2}^{2}$$

## 金融代写|金融工程作业代写Financial Engineering代考|The Phase Diagram

$$\frac{d V}{d t}=f(V, t) \frac{d \eta}{d t} \quad=g(V, \eta)$$

## 金融代写|金融工程作业代写Financial Engineering代考|Lyapunov Stability Approach

Lyapunov 方法无需明确计算状态向量的轨迹即可分析动力系统的稳定性 $x=\left[x_{1}, x_{2}, \cdots, x_{n}\right]^{T}$. 定理: 由关系描述的系统 $\dot{x}=f(x)$ 在平衡点附近是渐近稳定的 $x_{0}=0$ 如果有功能 $V(x)$ 这样 (i) $V(x)$ 是连续的并且在处具有连续的一阶导数 $x_{0}$
(二) $V(x)>0$ 如果 $x \neq 0$ 和 $V(0)=0$
$\Leftrightarrow \dot{V}(x)<0, \forall x \neq 0$. Lyapunov 函数通常被选择为系统的二次 (因此是正) 能量函数，但是没有系统的方法来定义它。 现在假设，那 $\dot{x}=f(x)$ 和 $x_{0}=0$ 是均衡。那么系统是全局渐近稳定的，如果对于每个 $\varepsilon>0, \exists \delta(\varepsilon)>0$, 这样 如果 $|x(0)|<\delta$ 然后 $|x(t)|<\varepsilon, \forall t \geq 0$.

$$\dot{x} 1=x 2 \dot{x} 2=-x 1-x_{3}^{2}$$

$$V(x)=x_{1}^{2}+x_{2}^{2}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|金融工程作业代写Financial Engineering代考|Best 107

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Characteristics of the Dynamics of Nonlinear Systems

Main features characterizing the stability of nonlinear dynamical systems are defined as follows [121, 274]:

1. Finite escape time: It is the finite time within which the state-vector of the nonlinear system converges to infinity.
2. Multiple isolated equilibria: A linear system can have only one equilibrium to which converges the state vector of the system in steady-state. A nonlinear system can have more than one isolated equilibria (fixed points). Depending on the initial state of the system, in steady-state the state vector of the system can converge to one of these equilibria.
3. Limit cycles: For a linear system to exhibit oscillations it must have eigenvalues on the imaginary axis. The amplitude of the oscillations depends on initial conditions. In nonlinear systems one may have oscillations of constant amplitude and frequency, which do not depend on initial conditions. This type of oscillations is known as limit cycles.
4. Sub-harmonic, harmonic and almost periodic oscillations: A stable linear system under periodic input produces an output of the same frequency. A nonlinear system,

under periodic excitation can generate oscillations with frequencies which are several times smaller (subharmonic) or multiples of the frequency of the input (harmonic). It may also generate almost periodic oscillations with frequencies which are not necessarily multiples of a basis frequency (almost periodic oscillations).

1. Chaos: A nonlinear system in steady-state can exhibit a behavior which is not characterized as equilibrium, periodic oscillation or almost periodic oscillation. This behavior is characterized as chaos. As time advances the behavior of the system changes in a random-like manner, and this depends on the initial conditions. Although the dynamic system is deterministic it exhibits randomness in the way it evolves in time.
2. Multiple modes of behavior: It is possible the same dynamical system to exhibit simultaneously more than one of the aforementioned characteristics (1)-(5). Thus, a system without external excitation may exhibit simultaneously more than one limit cycles. A system receiving a periodic external input may exhibit harmonic or subharmonic oscillations, or an even more complex behavior in steady state which depends on the amplitude and frequency of the excitation.

## 金融代写|金融工程作业代写Financial Engineering代考|Computation of Isoclines

An autonomous second order system is described by two differential equations of the form
\begin{aligned} &\dot{x}{1}=f{1}\left(x_{1}, x_{2}\right) \ &\dot{x}{2}=f{2}\left(x_{1}, x_{2}\right) \end{aligned}
The method of the isoclines consists of computing the slope (ratio) between $f_{2}$ and $f_{1}$ for every point of the trajectory of the state vector $\left(x_{1}, x_{2}\right)$.
$$s(x)=\frac{f_{2}\left(x_{1}, x_{2}\right)}{f_{1}\left(x_{1}, x_{2}\right)}$$
The case $s(x)=c$ describes a curve in the $x_{1}-x_{2}$ plane along which the trajectories $\dot{x}{1}=f{1}\left(x_{1}, x_{2}\right)$ and $\dot{x}{2}=f{2}\left(x_{1}, x_{2}\right)$ have a constant slope.

The curve $s(x)=c$ is drawn in the $x_{1}-x_{2}$ plane and along this curve one also draws small linear segments of length $c$. The curve $s(x)=c$ is known as isocline. The direction of these small linear segments is according to the sign of the ratio $f_{2}\left(x_{1}, x_{2}\right) / f_{1}\left(x_{1}, x_{2}\right)$
Example 1:
The following simplified nonlinear dynamical system is considered
$$\begin{gathered} \dot{x}{1}=x{2} \ \dot{x}{2}=-\sin \left(x{1}\right) \end{gathered}$$

## 金融代写|金融工程作业代写Financial Engineering代考|Characteristics of the Dynamics of Nonlinear Systems

1. 有限逃逸时间：这是非线性系统的状态向量收敛到无穷大的有限时间。
2. 多重孤立平衡：一个线性系统只能有一个平衡点，系统在稳态时的状态向量会收敛到该平衡点。一个非线性系统可以有多个孤立的平衡点（固定点）。根据系统的初始状态，在稳态下，系统的状态向量可以收敛到这些平衡之一。
3. 极限环：对于表现出振荡的线性系统，它必须在虚轴上具有特征值。振荡幅度取决于初始条件。在非线性系统中，可能会有恒定幅度和频率的振荡，它不依赖于初始条件。这种类型的振荡称为极限环。
4. 次谐波、谐波和几乎周期性的振荡：周期性输入下的稳定线性系统会产生相同频率的输出。非线性系统，

1. 混沌：稳态的非线性系统可以表现出不具有平衡、周期性振荡或几乎周期性振荡的特性。这种行为的特点是混乱。随着时间的推移，系统的行为会以类似随机的方式发生变化，这取决于初始条件。尽管动态系统是确定性的，但它在时间演化的方式上表现出随机性。
2. 多种行为模式：同一动力系统可能同时表现出上述特征 (1)-(5) 中的一个以上。因此，没有外部激励的系统可能同时表现出多个极限环。接收周期性外部输入的系统可能会表现出谐波或次谐波振荡，或者在稳态下表现出更复杂的行为，这取决于激励的幅度和频率。

## 金融代写|金融工程作业代写Financial Engineering代考|Computation of Isoclines

$$\dot{x} 1=f 1\left(x_{1}, x_{2}\right) \quad \dot{x} 2=f 2\left(x_{1}, x_{2}\right)$$

$$s(x)=\frac{f_{2}\left(x_{1}, x_{2}\right)}{f_{1}\left(x_{1}, x_{2}\right)}$$

$$\dot{x} 1=x 2 \dot{x} 2=-\sin (x 1)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|金融工程作业代写Financial Engineering代考|Multivariate Black-Scholes Model

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Black-Scholes Model for Several Assets

In many cases, an option’s payoff may depend on several assets, e.g., swap options, quantos, basket options, etc. Therefore one has to model the dynamical behavior of several assets.

Definition 2.1.1 $W=\left(W_{1}, \ldots, W_{d}\right)$ is a d-dimensional Brownian motion with correlation matrix $R$ if it is a continuous random vector starting at 0 , with independent increments, and such that $W(t)-W(s) \sim N_{d}(0, R(t-s))$, whenever $0 \leq s \leq t$.

Note that it follows that the components $W_{1}, \ldots, W_{d}$ are correlated Brownian motions with
$$\operatorname{Cov}\left{W_{j}(s), W_{k}(t)\right}=R_{j k} \min (s, t), \quad s, t, \geq 0, \quad j, k \in{1, \ldots, d}$$
One can now state the extension of the Black-Scholes model to several assets.

Definition 2.1.2 The values $S_{1}, \ldots, S_{d}$ of assets are modeled by geometric Brownian motions if
$$S_{j}(t)=S_{j}(0) e^{\left(\mu_{j}-\frac{\sigma_{j}^{2}}{2}\right) t+\sigma_{j} W_{j}(t)}, \quad t \geq 0, \quad j \in{1, \ldots, d},$$
where the Brownian motions $W_{i}$ are correlated, with correlation matrix $R$. One can also say that $S=\left(S_{1}, \ldots, S_{d}\right)$ are correlated geometric Brownian

motions with parameters $\mu$ and $\Sigma$, where the covariance matrix $\Sigma$ is defined $b y$
$$\Sigma_{j k}=\sigma_{j} \sigma_{k} R_{j k}, \quad j, k \in{1, \ldots, d}$$
Remark 2.1.1 As in the one-dimensional case, the geometric Brownian motions satisfy the following system of stochastic differential equations:
$$d S_{j}(t)=\mu_{j} S_{j}(t) d t+\sigma_{j} S_{j}(t) d W_{j}(t), \quad j \in{1, \ldots, d} .$$

## 金融代写|金融工程作业代写Financial Engineering代考|Representation of a Multivariate Brownian Motion

Using the properties of Gaussian random vectors (Section A.6.18), a $d-$ dimensional Brownian motion $W$ with correlation matrix $R$ can be constructed as a linear combination of $d$ independent univariate Brownian motions $Z=$ $\left(Z_{1}, \ldots, Z_{d}\right)^{\top}$ by setting $W(t)=b^{\top} Z(t)$, where $b^{\top} b=R$. We can then rewrite (2.1) as
$$\frac{S_{j}(t)}{S_{j}(0)}=e^{\left(\mu_{j}-\frac{\sigma_{1}^{2}}{2}\right) t+\sigma_{j} \sum_{k=1}^{d} b_{k j} Z_{k}(t)}, \quad t \geq 0, \quad j \in{1, \ldots, d}$$
An interesting decomposition of $R=b^{\top} b$ is when $b$ is an upper triangular matrix, as in Cholesky decomposition. In this case,
$$\frac{S_{j}(t)}{S_{j}(0)}=e^{\left(\mu_{j}-\frac{\Sigma_{j j}}{2}\right) t+\sum_{k=1}^{j} a_{k j} Z_{k}(t)}, \quad t \geq 0, \quad j \in{1, \ldots, d},$$
where the Brownian motions $Z_{1}, \ldots, Z_{d}$ are independent, and $a_{j k}=\sigma_{k} b_{j k}$. The matrix $a$ is upper triangular, contains all the information on the dependence, and is uniquely determined by the Cholesky decomposition $a^{\top} a=\Sigma$, i.e..
$$\Sigma_{j k}=\left(a^{\top} a\right){j k}=\sum{l=1}^{\min (j, k)} a_{l j} a l k=\sigma_{j} \sigma_{k} R_{j k}, \quad j, k \in{1, \ldots, d}$$
under the constraints $a_{j j}>0, j \in{1, \ldots, d}$. In the bivariate case, one can check that $a=\left(\begin{array}{cc}\sigma_{1} & R_{12} \sigma_{2} \ 0 & \sigma_{2} \sqrt{1-R_{12}^{2}}\end{array}\right)$.

## 金融代写|金融工程作业代写Financial Engineering代考|Numerical Method

In general, we cannot compute explicitly the maximum likelihood estimates of a model, so we have to rely on numerical methods for optimization. One of the main problems encountered with numerical methods for optimization is the problem of constraints on the parameters.

For example, for model (2.1), the matrix $R$ must be positive definite and symmetric. In two dimensions, this condition is simple since the correlation matrix $R$ is determined by the unique number $\rho=R_{12}$. In this case, one needs to assume that the correlation coefficient $\rho$ is in the interval $(-1,1)$. To get rid of this constraint, we can set $\rho=\tanh (\alpha)$. Its inverse, called the Fisher transformation, is $\alpha=\frac{1}{2} \ln \left(\frac{1+\rho}{1-\rho}\right)$. See Figure $2.1$.

For higher dimensions, it is impossibly difficult to find explicit constraints on a matrix coefficients to ensure that it is positive definite. This is where representation (2.4) becomes interesting. The only constraint on the upper triangular matrix $a$ is that its diagonal is positive. As parameters are expressed by vectors, it is therefore natural to work with the volatility vector $v$.

In addition, as we will see later, it is relatively easy to compute the estimator error on option prices in terms of the error on $v$.

## 金融代写|金融工程作业代写Financial Engineering代考|Black-Scholes Model for Several Assets

\operatorname{Cov}\left{W_{j}(s), W_{k}(t)\right}=R_{j k} \min (s, t), \quad s, t, \geq 0, \四边形 j, k \in{1, \ldots, d}\operatorname{Cov}\left{W_{j}(s), W_{k}(t)\right}=R_{j k} \min (s, t), \quad s, t, \geq 0, \四边形 j, k \in{1, \ldots, d}

Σjķ=σjσķRjķ,j,ķ∈1,…,d

d小号j(吨)=μj小号j(吨)d吨+σj小号j(吨)d在j(吨),j∈1,…,d.

## 金融代写|金融工程作业代写Financial Engineering代考|Representation of a Multivariate Brownian Motion

Σjķ=(一种⊤一种)jķ=∑l=1分钟(j,ķ)一种lj一种lķ=σjσķRjķ,j,ķ∈1,…,d

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|金融工程作业代写Financial Engineering代考|Estimation of Greeks using the Broadie-Glasserman

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Methodologies

While it is generally impossible to find explicit expressions for the option value, we can however fairly easily estimate them with a Monte Carlo approximation of the expected value in (1.8). On a similar note, expressions for the greeks are often not available. An easy way to circumvent this problem, which is often used in practice, is to estimate them with a finite difference approximation. For example, the delta could be approximated as
$$\Delta \approx \frac{C(t, s+\epsilon)-C(t, s)}{\epsilon}$$
where $\epsilon$ is a small positive scalar. However, such procedures are plagued by an inevitable tradeoff; a large $\epsilon$ will produce biased estimations of the greeks, while small $\epsilon$ values will results in high estimation variance.

Fortunately, Broadie and Glasserman [1996] proposed methods to estimate an option’s value, together with unbiased estimations of the greeks. They considered several models, including the Black-Scholes model.

According to formula (1.8), the value of a European option with payoff $\Phi$ at maturity is
\begin{aligned} C(t, s) &=e^{-r \tau} E\left[\Phi\left{s e^{\left(r-\frac{\alpha^{2}}{2}\right) \tau+\sigma \sqrt{\tau} Z}\right}\right] \ &=e^{-r \tau} \int_{-\infty}^{+\infty} \Phi\left{s e^{\left(r-\frac{\sigma^{2}}{2}\right) \tau+\sigma \sqrt{\tau} z}\right} \frac{e^{-z^{2} / 2}}{\sqrt{2 \pi}} d z \ &=e^{-r \tau} \int_{0}^{+\infty} \Phi(x) \frac{e^{-\frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{2}}{2}\right) \tau\right}^{2}}}{x \sigma \sqrt{2 \pi \tau}} d x \end{aligned}
where $Z \sim N(0,1)$ and $\tau=T-t$ is the time to maturity.
Suppose that one generates $Z_{1}, \ldots, Z_{N} \sim N(0,1)$, with $N$ large enough. Further set $\bar{S}{i}=s e^{\left(r-\frac{a^{2}}{2}\right) \tau+\sigma \sqrt{\tau} Z{i}}, i \in{1 \ldots, N}$.
Then, an unbiased and consistent estimation of $C(t, s)$ is given by
$$\hat{C}=\frac{1}{N} \sum_{i=1}^{N} e^{-r \tau} \Phi\left(\bar{S}_{i}\right)$$
This Monte Carlo approach was proposed a long time ago by Boyle [1977]. However, no unbiased estimation of the greeks was proposed until Broadie and Glasserman [1996]. In their article, the authors proposed in fact two methodologies to estimate greeks, based respectively on representations (1.19) and $(1.20)$. These methodologies have the advantage of being computed in parallel with the option price, not sequentially.

## 金融代写|金融工程作业代写Financial Engineering代考|Pathwise Method

The first methodology, called pathwise method, is based on representation (1.19). To be applicable, one has to assume that the payoff function $\Phi$ is differentiable “almost everywhere,” i.e., everywhere but possibly at a countable set of points ${ }^{11}$. However, note that the partial derivatives of any order for $\tilde{S}_{i}$ exist for any possible parameter $\theta \in{s, r, t, \sigma}$.

Proposition 1.7.1 Suppose that $\Phi$ is differentiable almost everywhere. Then simultaneous unbiased estimations of the option value and its first order derivatives are given respectively by
$$\hat{C}=\frac{1}{N} \sum_{i=1}^{N} e^{-r \tau} \Phi\left(\tilde{S}_{i}\right)$$

Remark 1.7.1 Since these estimations are averages of independent and identically distributed random vectors $X_{1}, \ldots, X_{N}$ with mean $\mathcal{G} \in \mathbb{R}^{p}$, one can determine the asymptotic behavior of the estimation errors. In fact, the central limit theorem (Theorem B.4.1) applies to yield
$$\sqrt{N}(\bar{X}-\mathcal{G}) \leftrightarrow N_{p}(0, V),$$
where $V$ is estimated by
$$\frac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)\left(X_{i}-\bar{X}\right)^{\top}$$
Example 1.7.1 For a European call option, $\Phi(s)=\max (s-K, 0)$, so $\Phi^{\prime}(s)=$ $\mathbb{I}(s>K)$ almost everywhere. As a result,
$$\hat{\Delta}=e^{-r \tau} \frac{1}{N} \sum_{i=1}^{N} \frac{\tilde{S}{i}}{s} \mathbb{I}\left(\bar{S}{i}>K\right)$$
and
$$\hat{\mathcal{V}}=e^{-r \tau} \frac{1}{N} \sum_{i=1}^{N}\left(Z_{i} \sqrt{\tau}-\sigma \tau\right) \tilde{S}{i} \mathbb{I}\left(\tilde{S}{i}>K\right) .$$
Therefore $\mathcal{G}=(C, \Delta, \mathcal{V})^{\top}$ can be estimated as the mean of the 3dimensional random vectors
$$X_{i}=e^{-r \tau} \mathbb{I}\left(\bar{S}{i}>K\right)\left(\bar{S}{i}-K, \frac{\bar{S}{i}}{s}, \bar{S}{i}\left(Z_{i} \sqrt{\tau}-\sigma \tau\right)\right)^{\top}$$
$i \in{1, \ldots, N} .$

## 金融代写|金融工程作业代写Financial Engineering代考|Likelihood Ratio Method

The second method proposed by Broadie and Glasserman $[1996]$ is based on representation $(1.20)$. For $x>0$, set
$$f(x)=\frac{e^{-\frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{2}}{2}\right) \tau\right}^{2}}}{x \sigma \sqrt{2 \pi \tau}} .$$
Then $f$ is the density of $\tilde{S}(T)$ given $\tilde{S}(t)=s$. Note that $f$ is differentiable with respect to every parameter $\theta \in{s, r, \sigma, t}$.

Proposition 1.7.2 Simultaneous unbiased estimations of the value of the option and derivatives of order 1 are given by
$$\hat{C}=\frac{1}{N} \sum_{i=1}^{N} e^{-r \tau} \Phi\left(\tilde{S}{i}\right)$$ and $$\widetilde{\partial{\theta} C}=-\hat{C} \partial_{\theta}(r \tau)+\left.e^{-r \tau} \frac{1}{N} \sum_{i=1}^{N} \Phi\left(\tilde{S}{i}\right) \partial{\theta}[\ln {f(x)}]\right|{x=\tilde{S}{i}}$$
In particular
$$\hat{\Delta}=e^{-r \tau} \frac{1}{N} \sum_{i=1}^{N} \frac{Z_{i}}{s \sigma \sqrt{\tau}} \Phi\left(\tilde{S}{i}\right)$$ and $$\hat{\mathcal{V}}=e^{-r \tau} \frac{1}{N} \sum{i=1}^{N} \frac{\left(Z_{i}^{2}-1-Z_{i} \sigma \sqrt{\tau}\right)}{\sigma} \Phi\left(\tilde{S}{i}\right) .$$ Moreover, an unbiased estimation of the gamma is given by $$\hat{\Gamma}=e^{-r \tau} \frac{1}{N} \sum{i=1}^{N} \frac{\left(Z_{i}^{2}-1-Z_{i} \sigma \sqrt{\tau}\right)}{s^{2} \sigma^{2} \tau} \Phi\left(\bar{S}_{i}\right)=\frac{\hat{\mathcal{V}}}{s^{2} \sigma \tau} .$$

## 金融代写|金融工程作业代写Financial Engineering代考|Methodologies

Δ≈C(吨,s+ε)−C(吨,s)ε

\begin{aligned} C(t, s) &=e^{-r \tau} E\left[\Phi\left{s e^{\left(r-\frac{\alpha^{2}}{2 }\right) \tau+\sigma \sqrt{\tau} Z}\right}\right] \ &=e^{-r \tau} \int_{-\infty}^{+\infty} \Phi\left {s e^{\left(r-\frac{\sigma^{2}}{2}\right) \tau+\sigma \sqrt{\tau} z}\right} \frac{e^{-z^{ 2} / 2}}{\sqrt{2 \pi}} d z \ &=e^{-r \tau} \int_{0}^{+\infty} \Phi(x) \frac{e^{- \frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{2}}{2}\right) \tau\对}^{2}}}{x \sigma \sqrt{2 \pi \tau}} d x \end{对齐}\begin{aligned} C(t, s) &=e^{-r \tau} E\left[\Phi\left{s e^{\left(r-\frac{\alpha^{2}}{2 }\right) \tau+\sigma \sqrt{\tau} Z}\right}\right] \ &=e^{-r \tau} \int_{-\infty}^{+\infty} \Phi\left {s e^{\left(r-\frac{\sigma^{2}}{2}\right) \tau+\sigma \sqrt{\tau} z}\right} \frac{e^{-z^{ 2} / 2}}{\sqrt{2 \pi}} d z \ &=e^{-r \tau} \int_{0}^{+\infty} \Phi(x) \frac{e^{- \frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{2}}{2}\right) \tau\对}^{2}}}{x \sigma \sqrt{2 \pi \tau}} d x \end{对齐}

C^=1ñ∑一世=1ñ和−rτ披(小号¯一世)

## 金融代写|金融工程作业代写Financial Engineering代考|Pathwise Method

C^=1ñ∑一世=1ñ和−rτ披(小号~一世)

ñ(X¯−G)↔ñp(0,在),

1ñ−1∑一世=1ñ(X一世−X¯)(X一世−X¯)⊤

Δ^=和−rτ1ñ∑一世=1ñ小号~一世s一世(小号¯一世>ķ)

X一世=和−rτ一世(小号¯一世>ķ)(小号¯一世−ķ,小号¯一世s,小号¯一世(从一世τ−στ))⊤

## 金融代写|金融工程作业代写Financial Engineering代考|Likelihood Ratio Method

Broadie 和 Glasserman 提出的第二种方法[1996]是基于表示(1.20). 为了X>0， 放
f(x)=\frac{e^{-\frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{ 2}}{2}\right) \tau\right}^{2}}}{x \sigma \sqrt{2 \pi \tau}} 。f(x)=\frac{e^{-\frac{1}{2 \sigma^{2} \tau}\left{\ln (x / s)-\left(r-\frac{a^{ 2}}{2}\right) \tau\right}^{2}}}{x \sigma \sqrt{2 \pi \tau}} 。

C^=1ñ∑一世=1ñ和−rτ披(小号~一世)和∂θC~=−C^∂θ(rτ)+和−rτ1ñ∑一世=1ñ披(小号~一世)∂θ[ln⁡F(X)]|X=小号~一世

Δ^=和−rτ1ñ∑一世=1ñ从一世sστ披(小号~一世)和在^=和−rτ1ñ∑一世=1ñ(从一世2−1−从一世στ)σ披(小号~一世).此外，伽马的无偏估计由下式给出Γ^=和−rτ1ñ∑一世=1ñ(从一世2−1−从一世στ)s2σ2τ披(小号¯一世)=在^s2στ.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|金融工程作业代写Financial Engineering代考|Greeks

statistics-lab™ 为您的留学生涯保驾护航 在代写金融工程Financial Engineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融工程Financial Engineering代写方面经验极为丰富，各种代写金融工程Financial Engineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Greeks

It is often important to measure the sensitivity of the option value with respect to the variables $t, s, r$, and $\sigma$. The so-called greeks are measures of sensitivity based on partial derivatives with respect to those parameters. Explicit formulas for greeks are known only in few cases, in particular the European call option [Wilmott, 2006]. In general, since there is no explicit expression for the option value, the greeks must be approximated. This will be done in Section 1.7. Here are the main definitions and interpretations for these useful parameters.

• The sensitivity of the option value with respect to the underlying asset price, called delta, is defined by
$$\Delta=\frac{\partial C}{\partial s}$$
The delta of an option is quite useful in hedging since it corresponds to the number of shares needed to create a risk-free portfolio replicating the value of the option at maturity; see Appendix 1.A.
• The sensitivity of the option value with respect to time, called theta, is defined by
$$\Theta=\frac{\partial C}{\partial t} .$$
Note that $-\Theta$, evaluated at $\tau=T-t$, yields the sensitivity with respect to the time to maturity $\tau$.
• The sensitivity of the option value with respect to the interest rate $r$, called $r o$, is defined by
$$\rho=\frac{\partial C}{\partial r}$$

Black-Scholes Model
15

• The sensitivity of the option value with respect to the volatility, called vega, is defined by
$$\mathcal{V}=\frac{\partial C}{\partial \sigma}$$
As shown next in Section 1.6.3, the vega is also important in determining the error on the option price due to the estimation of the volatility.
• A measure of convexity, the second order derivative of the option value with respect to the underlying asset prices, called gamma, is defined by
$$\Gamma=\frac{\partial^{2} C}{\partial s^{2}}$$
$\Gamma$ is useful in some approximations.

## 金融代写|金融工程作业代写Financial Engineering代考|Greeks for a European Call Option

Using the Black-Scholes formula (1.4), it is easy to check that

• $\Delta=\frac{\partial C}{\partial s}=\mathcal{N}\left(d_{1}\right)>0 .$
• $\Theta=\frac{\partial C}{\partial t}=-\frac{\sigma s}{2 \sqrt{T-t}} \frac{e^{-d_{1}^{2} / 2}}{\sqrt{2 \pi}}-K r e^{-r(T-t)} \mathcal{N}\left(d_{2}\right)<0 .$ $\rho=\frac{\partial C}{\partial r}=K(T-t) e^{-r(T-t)} \mathcal{N}\left(d_{2}\right)>0 .$
• $\mathcal{V}=\frac{\partial C}{\partial \sigma}=s \sqrt{T-t} \frac{e^{-d_{1}^{2} / 2}}{\sqrt{2 \pi}}>0 .$
Since the vega is positive, it means that the value of the option is an increasing function of the volatility. This property is essential in determining the so-called implied volatility.
• $\Gamma=\frac{\partial^{2} C}{\partial s^{2}}=\frac{1}{s \sigma \sqrt{T-t}} \frac{e^{-d_{1}^{2} / 2}}{\sqrt{2 \pi}}>0 .$
Since the gamma is positive, it means that the value of the option is a convex function of the underlying asset value.

Remark 1.6.1 For continuously paid dividends at rate $\delta$, using formula $(1.10)$, it is easy to check that $\Delta_{\delta}(t, s)=e^{-\delta \tau} \Delta_{0}\left(t, s e^{-\delta \tau}\right) .$ Also $\Gamma_{\delta}(t, s)=$ $e^{-2 \delta \tau} \Gamma_{0}\left(t, s e^{-\delta \tau}\right)$. Next, $\Theta_{\delta}(t, s)=\Theta_{0}\left(t, s e^{-\delta \tau}\right)+s \Delta_{\delta}(t, s)$. Finally, $\rho_{\delta}(t, s)=\rho_{0}\left(t, s e^{-\delta \tau}\right)$ and $\mathcal{V}{\delta}(t, s)=\mathcal{V}{0}\left(t, s e^{-\delta \tau}\right)$.

## 金融代写|金融工程作业代写Financial Engineering代考|Implied Distribution

One might ask why there is no sensitivity parameter corresponding to the partial derivative with respect to the strike price. In fact, there is one and it is related to the implied distribution [Breeden and Litzenberger, 1978]. Assuming that the value of a European call option is given by the expectation formula (1.8), and using the properties of expectations, namely (A.2), we obtain
$$C(t, s)=E_{Q}[\max {\tilde{S}(T)-K, 0} \mid \tilde{S}(t)=s]=\int_{K}^{\infty} Q{\tilde{S}(T)>y} d y$$
where $Q$ denotes the equivalent martingale measure. As a result,
$$\frac{\partial C}{\partial K}=-Q{\bar{S}(T)>K}=\tilde{F}(K)-1$$
where $\tilde{F}$ is the distribution function of $\bar{S}(T)$ given $\bar{S}(t)=s$, under the equivalent martingale measure $Q$. As a result $\frac{\partial C}{\partial K}$ is non-decreasing and it follows that $\frac{\partial^{2} C}{\partial K^{2}}=\tilde{f}(K) \geq 0$, where $\tilde{f}$ is the associated density, provided it exists. It also shows that the value of a call option is always a convex function of the strike. Note that in the case of the Black-Scholes model, the implied distribution is the log-normal, since $\ln {\tilde{S}(T)}$ has a Gaussian distribution with mean $\ln (s)+\left(r-\frac{\sigma^{2}}{2}\right) \tau$ and variance $\sigma^{2} \tau$, under the equivalent martingale measure. Since (1.18) is assumed to be always valid, not only for the BlackScholes model, the implied distribution function can be approximated from the market prices of the calls if there are enough strike prices available. See, e.g., Ait-Sahalia and Lo [1998].

## 金融代写|金融工程作业代写Financial Engineering代考|Option Value as an Expectation

C(吨,s)=和−r(吨−吨)和[披小号~(吨)∣小号~(吨)=s]

d小号¯(在)=r小号¯(在)d在+σ小号~(在)d在~(在),吨≤在≤吨

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。