### 金融代写|金融数学代写Financial Mathematics代考|ACTL20001

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• 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|金融数学代写Financial Mathematics代考|The Cox, Ross and Rubinstein model

We will now illustrate the different concepts introduced above using a specific case of financial market. This is a discretized version of the Black and Scholes model.
The market is considered to be made up of a risk-free asset $S_{n}^{0}=(1+r)^{n}$ and a single risky asset $S^{1}$ with the dynamic
$$S_{0}^{1}=1, \quad S_{n+1}^{1}=S_{n}^{1} T_{n+1}, n \geq 0,$$
where $\left(T_{n}\right){1 \leq n \leq N}$ is a sequence of random variable taking only two values $1+d$ and $1+u$ with -1{0}={\emptyset, \Omega}, \mathcal{F}{n}=\sigma\left(T{1}, \ldots, T_{n}\right), 1 \leq n \leq N . $$In particular, \mathcal{F}{N}=\sigma\left(T{1}, \ldots, T_{N}\right)=\mathcal{P}(\Omega) is the set of subsets of \Omega. We will now characterize viable markets in this model. In order to do this, we start by studying risk-neutral probabilities. PROPOSITION 5.2.-The discounted prices \left(\widetilde{S}{n}^{1}\right){0 \leq n \leq N} are a martingale under a probability \mathbb{P}^{} if and only if, for any 0 \leq n \leq N-1, we have$$ \mathbb{E}^{}\left[T_{n+1} \mid \mathcal{F}{n}\right]=1+r, $$with \mathbb{E}^{} denoting the expectation for the probability \mathbb{P}^{}. PROOF.- Let us proceed through double implication. Let us assume that the realized price is a martingale. For any 0 \leq n \leq N-1, we then have$$ \begin{aligned} & \mathbb{E}^{}\left[\widetilde{S}{n+1}^{1} \mid \mathcal{F}{n}\right]=\widetilde{S}{n}^{1} \ \Longrightarrow & \mathbb{E}^{}\left[\frac{S_{n+1}^{1}}{(1+r)^{n+1}} \mid \mathcal{F}{n}\right]=\frac{S{n}^{1}}{(1+r)^{n}} \ \Longrightarrow & \mathbb{E}^{*}\left[S_{n}^{1} T_{n+1} \mid \mathcal{F}{n}\right]=S{n}^{1}(1+r) \end{aligned}  \begin{aligned} &\Longrightarrow S_{n}^{1} \mathbb{E}^{}\left[T_{n+1} \mid \mathcal{F}{n}\right]=S{n}^{1}(1+r) \ &\Longrightarrow \mathbb{E}^{}\left[T_{n+1} \mid \mathcal{F}{n}\right]=(1+r) \end{aligned} $$since S{n}^{1} is the \mathcal{F}{n}-measurable. Conversely, if for any 0 \leq n \leq N-1, we have \mathbb{E}^{}\left[T{n+1} \mid \mathcal{F}{n}\right]=1+r; therefore$$ \begin{aligned} & \mathbb{E}^{}\left[T{n+1} \mid \mathcal{F}{n}\right]=(1+r) \ \Longrightarrow & S{n}^{1} \mathbb{E}^{}\left[T_{n+1} \mid \mathcal{F}{n}\right]=S{n}^{1}(1+r) \ \Longrightarrow & \mathbb{E}^{}\left[\frac{S_{n+1}^{1}}{(1+r)^{n+1}} \mid \mathcal{F}{n}\right]=\frac{S{n}^{1}}{(1+r)^{n}} \ \Longrightarrow & \mathbb{E}^{*}\left[\widetilde{S}{n+1}^{1} \mid \mathcal{F}{n}\right]=\widetilde{S}{n}^{1} \end{aligned} $$therefore \left(\widetilde{S}{n}^{1}\right) is indeed a martingale, as the measurability and integrability conditions are satisfied. ## 金融代写|金融数学代写Financial Mathematics代考|Portfolio optimization We now study a portfolio optimization problem in the Cox, Ross and Rubinstein model. Let V_{0} be the wealth of an investor at the time 0 . The investor can invest their money either in a risky asset or in a risk-free asset, following an admissible strategy. We use \phi_{n}^{0} and \phi_{n}^{1} to denote the number of shares in the risk-free asset and the number of shares in the risky asset, respectively, held between the time n-1 and n. Let \pi_{n} be the proportion of the wealth invested in the risky asset between the instants n-1 and n, that is,$$ \pi_{n}=\frac{\phi_{n}^{1} S_{n-1}^{1}}{V_{n-1}} $$1) Express \phi_{n}^{0} and \phi_{n}^{1} as the functions of \pi_{n}, S_{n-1}^{0}, S_{n-1}^{1} and V_{n-1} for any n. 2) Derive from this that for any n, the wealth at the time n, after the evolution of the prices and before the redistribution of the portfolio has the value:$$ V_{n}=\left(\pi_{n} T_{n}+\left(1-\pi_{n}\right)(1+r)\right) V_{n-1} $$3) On the same graph and for the same random sampled trajectory, represent the evolution of the risk-free asset and the evolution of the wealth for the following two strategies: a) The proportion of the wealth invested in the risky asset is fixed over time, at 1 / 4. b) The proportion of the wealth invested in the risky asset only takes the values 0 and 1 . It takes the value 1 when the price of a risky asset strictly exceeds that of the risk-free asset, and takes the value 0 when the risky asset is small than or equal to the risk-free asset, while remaining predictable. We will take the following parameters: initial wealth V_{0}=1, the risky asset evolves as per the Cox, Ross and Rubinstein model with parameters d=-2 \%, u=10 \% and q=0.52, interest rate r=4 \% and duration of investment: N=100 periods. Now consider that the investor’s utility function is logarithmic and we wish to find the strategy \pi^{}=\left(\pi_{n}^{}\right){1 \leq n \leq N} which maximizes the expectation of the utility of the wealth at maturity N :$$ \sup {\pi \text { admissible }} \mathbb{E}\left[\log V_{N}(\pi)\right] $$We will accept that the optimal strategy is constant over time and maximize the expression$$ q \log \left(\pi^{}(u-r)+1+r\right)+(1-q) \log \left(\pi^{}(d-r)+1+r\right) $$We wish to compare the performances of strategies 1 and 2 , given above, and that of the optimal strategy. We will use the same parameters as for the previous question. 4) Write a function optimal (u, d, r, q), which calculates the value of \pi^{*} the optimal proportion to invest in the risky asset. We can restrict ourselves to five decimals. What do we find for our parameters? 5) On the same graph, trace a trajectory of the logarithm of the wealth at each instant for the optimal strategy, and for strategies 1 and 2 . Does the optimal strategy always give the same result? Why? 6) On the same graph, trace the expectation of the logarithm of the wealth at each instant, for each of the three strategies. We will calculate the expectation using the Monte Carlo method. Which is the best strategy? Discuss. ## 金融代写|金融数学代写Financial Mathematics代考|Portfolio optimization with withdrawal In this section, the investor is allowed to withdraw a proportion c_{n} of their wealth at the instant n, after updating the asset prices, but before the redistribution of their portfolio for the next investment period. Therefore, they only reinvest the non-withdrawn part. The corresponding investment-withdrawal strategy \left(\pi_{n}, c_{n}\right) is no longer self-financed, but it must remain predictable and the wealth after the withdrawal must be positive or zero at each instant. Therefore, it can thus be shown that the new wealth at the time n after the evolution of the prices and when the value of the withdrawal is$$ V_{n}(\pi, c)=\prod_{i=1}^{n}\left(1-c_{i}\right)\left(\pi_{i} T_{i}+\left(1-\pi_{i}\right)(1+r)\right), $$such that the value of the wealth withdrawn at the instant n is R_{n}(\pi, c) with$$ R_{n}(\pi, c)= \begin{cases}\frac{c_{n} V_{n}(\pi, c)}{1-c_{n}} & \text { if } c_{n} \neq 1, \ V_{n-1}(\pi, c)\left(\pi_{n} T_{n}+\left(1-\pi_{n}\right)(1+r)\right) & \text { if not. }\end{cases} $$1) Graphically represent the evolution of the wealth for the investment strategy with the following withdrawal policy: a) The proportion of the wealth invested in the risky asset is fixed over time, at 1 / 4, b) The proportion of the wealth withdrawn at each instant is equal to 1 \% over the interval [1,80], 5 \% over the interval ] 80,90], 10 \% over the interval ] 90,95], 25 \% over the interval ] 95,100[ and upon maturity, all the remaining wealth is withdrawn. We will take the following parameters: the initial wealth V_{0}=1, the risky asset evolves as per the Cox, Ross and Rubinstein model with parameters d=-2 \%, u=10 \% and q=0.52, interest rate r=4 \%, duration of investment: N=100 periods. We now consider, once again, that the investor’s utility function is logarithmic and we want to find the investment strategy with the withdrawal \left(\pi_{n}, c_{n}\right) that maximizes the expectation of the cumulative sum of the withdrawal utility up to the date of maturity N :$$ \max {\left(\pi{n}, c_{n}\right)} \mathbb{E}\left[\sum_{n=1}^{N} \log \left(R_{n}(\pi, c)\right)\right] .$We will admit that the investment strategy with optimal withdrawal is given by$-\pi_{n}=\pi^{}$for any$1 \leq n \leq N$,$-c_{n}=\frac{1}{N+1-n}$for any$1 \leq n \leq N$, with the same$\pi^{}$as in the earlier practical exercise. ## 金融数学代考 ## 金融代写|金融数学代写Financial Mathematics代考|The Cox, Ross and Rubinstein model 我们现在将使用一个特定的金融市场案例来说明上面介绍的不同概念。这是 Black 和 Scholes 模型的离散版本。 市场被认为是由无风险资产组成的小号n0=(1+r)n和单一的风险资产小号1与动态 小号01=1,小号n+11=小号n1吨n+1,n≥0, 在哪里(吨n)1≤n≤ñ是一个随机变量序列，只取两个值1+d和1+在和−10=∅,Ω,Fn=σ(吨1,…,吨n),1≤n≤ñ.我np一个r吨一世C在l一个r,\mathcal{F}{N}=\sigma\left(T{1}, \ldots, T_{N}\right)=\mathcal{P}(\Omega)一世s吨H和s和吨○Fs在bs和吨s○F\欧米茄$。

⟹小号n1和[吨n+1∣Fn]=小号n1(1+r) ⟹和[吨n+1∣Fn]=(1+r)自从小号n1是个Fn- 可测量的。相反，如果对于任何0≤n≤ñ−1， 我们有和[吨n+1∣Fn]=1+r; 所以

## 金融代写|金融数学代写Financial Mathematics代考|Portfolio optimization

1) 快递φn0和φn1作为函数圆周率n,小号n−10,小号n−11和在n−1对于任何n.
2）由此得出对于任何n，当时的财富n，在价格演变之后和投资组合重新分配之前具有以下值：

3) 在同一张图上，对于同一个随机抽样轨迹，分别代表以下两种策略的无风险资产的演变和财富的演变：

a) 投资于风险资产的财富比例随着时间的推移是固定的，在1/4.
b) 投资于风险资产的财富比例仅取值 0 和 1 。当风险资产的价格严格超过无风险资产的价格时取值为 1，当风险资产小于或等于无风险资产的价格时取值为 0，同时保持可预测性。

q日志⁡(圆周率(在−r)+1+r)+(1−q)日志⁡(圆周率(d−r)+1+r)

4）写一个函数优化(在,d,r,q)，它计算的值圆周率∗投资于风险资产的最佳比例。我们可以将自己限制在小数点后五位。我们发现我们的参数是什么？
5) 在同一张图上，为最优策略以及策略 1 和 2 在每个时刻追踪财富对数的轨迹。最优策略总是给出相同的结果吗？为什么？
6) 在同一张图上，针对三种策略中的每一种，追踪每个时刻财富对数的期望值。我们将使用蒙特卡罗方法计算期望值。哪个是最好的策略？讨论。

## 金融代写|金融数学代写Financial Mathematics代考|Portfolio optimization with withdrawal

Rn(圆周率,C)={Cn在n(圆周率,C)1−Cn 如果 Cn≠1, 在n−1(圆周率,C)(圆周率n吨n+(1−圆周率n)(1+r)) 如果不。
1) 以图形方式表示具有以下退出政策的投资策略的财富演变：
a) 投资于风险资产的财富比例随着时间的推移是固定的，在1/4,
b) 每一刻提取的财富比例等于1%在区间内[1,80],5%在区间内]80,90],10%在区间内]90,95],25%在区间内]95,100[到期后，所有剩余的财富都将被提取。

−圆周率n=圆周率对于任何1≤n≤ñ, −Cn=1ñ+1−n对于任何1≤n≤ñ, 同圆周率和前面的实际练习一样。

## 有限元方法代写

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## 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。