统计代写|金融统计代写financial statistics代考| The Stock Price as a Stochastic Process

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  • Statistical Computing 统计计算
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统计代写|金融统计代写financial statistics代考| The Stock Price as a Stochastic Process

统计代写|金融统计代写financial statistics代考|The Stock Price as a Stochastic Process

Stock prices are stochastic processes in discrete time which only take discrete values due to the limited measurement scale. Nevertheless, stochastic processes in continuous time are used as models since they are analytically easier to handle than discrete models, e.g. the binomial or trinomial process. However, the latter is more intuitive and proves to be very useful in simulations.

Two features of the general Wiener process $d X_{t}=\mu d t+\sigma d W_{t}$ make it an unsuitable model for stock prices. First, it allows for negative stock prices, and second the local variability is higher for high stock prices. Hence, stock prices $S_{I}$ are modeled by means of the more general Itô-process:
d S_{t}=\mu\left(S_{l}, t\right) d t+\sigma\left(S_{l}, t\right) d W_{l}
This model, however, does depend on the unknown functions $\mu(X, t)$ and $\sigma(X, t)$. A useful and simpler variant utilizing only two unknown real model parameters $\mu$ and $\sigma$ can be justified by the following reflection: The percentage return on the invested capital should on average not depend on the stock price at which the investment is made, and of course, should not depend on the currency unit (EUR, USD,$\ldots$ ) in which the stock price is quoted. Furthermore, the average return should be proportional to the investment horizon, as it is the case for other investment

instruments. Putting things together, we request:
\frac{\mathrm{E}\left[d S_{l}\right]}{S_{l}}=\frac{\mathrm{E}\left[S_{l+d t}-S_{l}\right]}{S_{l}}=\mu \cdot d t
Since $\mathrm{E}\left[d W_{l}\right]=0$ this condition is satisfied if
\mu\left(S_{l}, t\right)=\mu \cdot S_{l}
for given $S_{l}$. Additionally,
\sigma\left(S_{l}, t\right)=\sigma \cdot S_{l}
takes into consideration that the absolute size of the stock price fluctuation is proportional to the currency unit in which the stock price is quoted. In summary, we model the stock price $S_{l}$ as a solution of the stochastic differential equation
d S_{t}=\mu \cdot S_{t} d t+\sigma \cdot S_{t} \cdot d W_{t}

统计代写|金融统计代写financial statistics代考|Itô’s Lemma

A crucial tool in dealing with stochastic differential equations is Itô’s lemma. If $\left{X_{I}, t \geq 0\right}$ is an Itô-process:
d X_{t}=\mu\left(X_{t}, t\right) d t+\sigma\left(X_{t}, t\right) d W_{l},

one is often interested in the dynamics of stochastic processes which are functions of $X_{l}: Y_{t}=g\left(X_{l}\right)$. Then $\left{Y_{l} ; t \geq 0\right}$ can also be described by a solution of a stochastic differential equation from which interesting properties of $Y_{I}$ can be derived, as for example the average growth in time $t$.

For a heuristic derivation of the equation for $\left{Y_{l} ; t \geq 0\right}$ we assume that $g$ is differentiable as many times as necessary. From a Taylor expansion it follows that:
Y_{t+d t}-Y_{l} &=g\left(X_{t+d t}\right)-g\left(X_{t}\right) \
&=g\left(X_{t}+d X_{t}\right)-g\left(X_{t}\right) \
&=\frac{d g}{d X}\left(X_{t}\right) \cdot d X_{t}+\frac{1}{2} \frac{d^{2} g}{d X^{2}}\left(X_{t}\right) \cdot\left(d X_{t}\right)^{2}+\ldots
where the dots indicate the terms which can be neglected (for $d t \rightarrow 0$ ). Due to Eq. (5.10) the drift term $\mu\left(X_{t}, t\right) d t$ and the volatility term $\sigma\left(X_{t}, t\right) d W_{t}$ are the dominant terms since for $d t \rightarrow 0$ they are of size $d t$ and $\sqrt{d t}$, respectively.

In doing this, we use the fact that $\mathrm{E}\left[\left(d W_{t}\right)^{2}\right]=d t$ and $d W_{t}=W_{t+d t}-W_{t}$ is of the size of its standard deviation, $\sqrt{d t}$. We neglect terms which are of a smaller size than $d t$. Thus, we can express $\left(d X_{l}\right)^{2}$ by a simpler term:
\left(d X_{t}\right)^{2} &=\left(\mu\left(X_{t}, t\right) d t+\sigma\left(X_{t}, t\right) d W_{t}\right)^{2} \
&=\mu^{2}\left(X_{t}, t\right)(d t)^{2}+2 \mu\left(X_{t}, t\right) \sigma\left(X_{t}, t\right) d t d W_{t}+\sigma^{2}\left(X_{t}, t\right)\left(d W_{t}\right)^{2}
We see that the first and the second terms are of size $(d t)^{2}$ and $d t \cdot \sqrt{d t}$, respectively. Therefore, both can be neglected. However, the third term is of size $d t$. More precisely, it can be shown that $d t \rightarrow 0$ :
\left(d W_{l}\right)^{2}=d t
Thanks to this identity, calculus rules for stochastic integrals can be derived from the rules for deterministic functions (as Taylor expansions for example). Neglecting terms which are of smaller size than dt we obtain the following version of Itôs lemma from (5.11):

统计代写|金融统计代写financial statistics代考|Recommended Literature

Exercise 5.1 Let $W_{I}$ be a standard Wiener process. Show that the following processes are also standard Wiener processes:
(a) $X_{t}=c^{-\frac{1}{2}} W_{c l}$ for $c>0$
(b) $Y_{l}=W_{T+l}-W_{T}$ for $T>0$
(c) $V_{t}= \begin{cases}W_{t} & \text { if } t \leq T \ 2 W_{T}-W_{t} & \text { if } t>T\end{cases}$
(d) $Z_{l}=t W_{\frac{1}{l}}$ for $t>0$ and $Z_{l}=0$.
Exercise $5.2$ Calculate $\operatorname{Cov}\left(W_{l}, 3 W_{s}-4 W_{t}\right)$ and $\operatorname{Cov}\left(W_{s}, 3 W_{s}-4 W_{l}\right)$ for $0 \leq s \leq t$.

Exercise $5.3$ Let $W_{t}$ be a standard Wiener process. The process $U_{t}=W_{t}-t W_{1}$ for $t \in[0,1]$ is called Brownian bridge. Calculate its covariance function. What is the distribution of $U_{1}$ ?
Exercise $5.4$ Calculate E $\left(\int_{0}^{2 \pi} W_{s} d W_{s}\right)$
Exercise 5.5 Consider the process $d S_{t}=\mu d t+\sigma d W_{t}$. Find the dynamics of the process $Y_{t}=g\left(S_{l}\right)$, where $g\left(S_{l}, t\right)=2+t+e^{S_{t}}$.

统计代写|金融统计代写financial statistics代考| The Stock Price as a Stochastic Process


统计代写|金融统计代写financial statistics代考|The Stock Price as a Stochastic Process


然而,这个模型确实依赖于未知函数μ(X,吨)和σ(X,吨). 一个有用且更简单的变体,仅使用两个未知的真实模型参数μ和σ可以通过以下反思来证明:投资资本的平均回报率不应取决于进行投资的股票价格,当然也不应取决于货币单位(欧元、美元、…) 报价的股票价格。此外,平均回报应与投资期限成正比,就像其他投资一样

给定的小号l. 此外,

统计代写|金融统计代写financial statistics代考|Itô’s Lemma

处理随机微分方程的一个关键工具是伊藤引理。如果\left{X_{I}, t \geq 0\right}\left{X_{I}, t \geq 0\right}是一个伊藤过程:

人们通常对随机过程的动力学感兴趣,这些动力学是Xl:是吨=G(Xl). 然后\left{Y_{l} ; t \geq 0\右}\left{Y_{l} ; t \geq 0\右}也可以用一个随机微分方程的解来描述是一世可以推导出来,例如平均时间增长吨.

对于方程的启发式推导\left{Y_{l} ; t \geq 0\右}\left{Y_{l} ; t \geq 0\右}我们假设G可以根据需要多次微分。从泰勒展开可以得出:
是吨+d吨−是l=G(X吨+d吨)−G(X吨) =G(X吨+dX吨)−G(X吨) =dGdX(X吨)⋅dX吨+12d2GdX2(X吨)⋅(dX吨)2+…
其中点表示可以忽略的项(对于d吨→0)。由于方程。(5.10) 漂移项μ(X吨,吨)d吨和波动率项σ(X吨,吨)d在吨是主要术语,因为对于d吨→0它们的大小d吨和d吨, 分别。

在这样做时,我们使用了以下事实和[(d在吨)2]=d吨和d在吨=在吨+d吨−在吨是其标准偏差的大小,d吨. 我们忽略尺寸小于d吨. 因此,我们可以表达(dXl)2用一个更简单的术语:
(dX吨)2=(μ(X吨,吨)d吨+σ(X吨,吨)d在吨)2 =μ2(X吨,吨)(d吨)2+2μ(X吨,吨)σ(X吨,吨)d吨d在吨+σ2(X吨,吨)(d在吨)2
我们看到第一项和第二项是大小(d吨)2和d吨⋅d吨, 分别。因此,两者都可以忽略。但是,第三项是大小d吨. 更准确地说,可以证明d吨→0 :
由于这个恒等式,随机积分的微积分规则可以从确定性函数的规则中推导出来(例如泰勒展开式)。忽略比 dt 更小的项,我们从 (5.11) 得到以下版本的 Itôs lemma:

统计代写|金融统计代写financial statistics代考|Recommended Literature

练习 5.1 让在一世是一个标准的维纳过程。证明以下过程也是标准维纳过程:
(C)在吨={在吨 如果 吨≤吨 2在吨−在吨 如果 吨>吨

锻炼5.4计算 E(∫02圆周率在sd在s)
练习 5.5 考虑过程d小号吨=μd吨+σd在吨. 发现过程的动态是吨=G(小号l), 在哪里G(小号l,吨)=2+吨+和小号吨.

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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