### 统计代写|金融统计代写financial statistics代考| Geometric Random Walks

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## 统计代写|金融统计代写financial statistics代考|Geometric Random Walks

The essential idea underlying the random walk for real processes is the assumption of mutually independent increments of the order of magnitude for each point in time. However, economic time series in particular do not satisfy the latter assumption. Seasonal fluctuations of monthly sales figures for example are in absolute terms significantly greater if the yearly average sales figure is high. By contrast, the relative or percentage changes are stable over time and do not depend on the

current level of $X_{t}$. Analogously to the random walk with i.i.d. absolute increments $Z_{t}=X_{t}-X_{t-1}$, a geometric random walk $\left{X_{t} ; t \geq 0\right}$ is assumed to have i.i.d. relative increments
$$R_{t}=\frac{X_{t}}{X_{l-1}}, \quad t=1,2, \ldots$$
For example, a geometric binomial random walk is given by
$$X_{t}=R_{l} \cdot X_{t-1}=X_{0} \cdot \Pi_{k=1}^{t} R_{k}$$
where $X_{0}, R_{1}, R_{2}, \ldots$ are mutually independent and for $u>1, d<1$ : $$\mathrm{P}\left(R_{k}=u\right)=p, \mathrm{P}\left(R_{k}=d\right)=1-p .$$ Given the independence assumption and $\mathrm{E}\left[R_{k}\right]=(u-d) p+d$ it follows from Eq. (4.5) that $\mathrm{E}\left[X_{t}\right]$ increases or decreases exponentially as the case may be $\mathrm{E}\left[R_{k}\right]>1$ or $\mathrm{E}\left[R_{k}\right]<1$ :
$$\mathrm{E}\left[X_{t}\right]=\mathrm{E}\left[X_{0}\right] \cdot\left(\mathrm{E}\left[R_{1}\right]\right)^{t}=\mathrm{E}\left[X_{0}\right] \cdot{(u-d) p+d}^{t} .$$
If $E\left[R_{k}\right]=1$ the process is on average stable, which is the case for
$$p=\frac{1-d}{u-d} .$$
For a recombining process, i.e. $d=\frac{1}{u}$, this relationship simplifies to
$$p=\frac{1}{u+1} .$$
Taking logarithms in Eq. (4.5) yields:
$$\log X_{t}=\log X_{0}+\sum_{k=1}^{t} \log R_{k}$$

## 统计代写|金融统计代写financial statistics代考|Binomial Models with State Dependent Increments

Binomial processes and more general random walks model the stock price at best locally. They proceed from the assumption that the distribution of the increments $Z_{l}=X_{l}-X_{t-1}$ is the same for each value $X_{l}$, regardless of whether the stock price is substantially greater or smaller than $X_{0}$. Absolute increments $X_{t}-X_{t-1}=$ $\left(R_{t}-1\right) X_{l-1}$ of a geometric random walk depend on the level of $X_{l-1}$. Thus, geometric random walks are processes which do not have independent absolute increments. Unfortunately, when modelling the stock price dynamics globally, the latter processes are too simple to explain the impact of the current price level on the future stock price evolution. A class of processes which take this effect into account are binomial processes with state dependent (and possibly time dependent) increments:
$$\begin{gathered} X_{t}=X_{t-1}+Z_{t}, \quad t=1,2, \ldots \ \mathrm{P}\left(Z_{t}=u\right)=p\left(X_{t-1}, t\right), \quad \mathrm{P}\left(Z_{t}=-d\right)=1-p\left(X_{t-1}, t\right) \end{gathered}$$
Since the distribution of $Z_{t}$ depends on the state $X_{t-1}$ and possibly on time $t$, increments are neither independent nor identically distributed. The deterministic functions $p(x, t)$ associate a probability to each possible value of the process at time $t$ and to each $t$. Stochastic processes $\left{X_{t} ; t \geq 0\right}$ which are constructed as in (4.6) are still markovian but without having independent increments.

Accordingly, geometric binomial processes with state dependent relative increments can be defined (for $u>1, d<1$ ):
$$\begin{gathered} X_{t}=R_{t} \cdot X_{t-1} \ \mathrm{P}\left(R_{t}=u\right)=p\left(X_{t-1}, t\right), \mathrm{P}\left(R_{t}=d\right)=1-p\left(X_{t-1}, t\right) \end{gathered}$$
Processes as defined in (4.6) and (4.7) are mainly of theoretic interest, since without further assumptions it is rather difficult to estimate the probabilities $p(x, t)$ from observed stock prices. But generalized binomial models (as well as the trinomial models) can be used to solve differential equations numerically, as the BlackScholes equation for American options for example.

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

Exercise $4.1$ Consider an ordinary random walk $X_{I}=\sum_{k=1}^{t} Z_{k}$ for $t=1,2, \ldots$, $X_{0}=0$, where $Z_{1}, Z_{2}, \ldots$ are i.i.d. with $\mathrm{P}\left(Z_{i}=1\right)=p$ and $\mathrm{P}\left(Z_{i}=-1\right)=1-p$. Calculate
(a) $\mathrm{P}\left(X_{I}>0\right)$
(b) $\mathrm{P}\left(X_{I}>0\right)$
(c) $\mathrm{P}\left(Z_{2}=1 \mid X_{3}=1\right)$
Exercise 4.2 Consider an ordinary random walk $X_{l}=\sum_{k=1}^{t} Z_{k}$ for $t=1,2, \ldots$, $X_{0}=0$, where $Z_{1}, Z_{2}, \ldots$ are i.i.d. with $\mathrm{P}\left(Z_{i}=1\right)=p$ and $\mathrm{P}\left(Z_{i}=-1\right)=1-p$. Let $\tau=\min \left{t:\left|X_{l}\right|>1\right}$ be a random variable denoting the first time $t$ when $\left|X_{t}\right|>1$. Calculate $\mathrm{E}[\tau]$.

Exercise 4.3 Consider an ordinary random walk $X_{L}=\sum_{k=1}^{l} Z_{k}$ for $t=1,2, \ldots$, $X_{0}=0$, where $Z_{1}, Z_{2}, \ldots$ are i.i.d. with $\mathrm{P}\left(Z_{i}=1\right)=p$ and $\mathrm{P}\left(Z_{i}=-1\right)=$ $1-p$. Consider also a process $M_{t}=\max {s \leq I} X{s}$. Calculate $\mathrm{P}\left(X_{3}=M_{3}\right)$ and $\mathrm{P}\left(M_{4}>M_{3}\right)$.

Exercise 4.4 Let $X_{t}=\sum_{k=1}^{t} Z_{k}$ be a general random walk for $t=1,2, \ldots$, $X_{0}=0$, and $Z_{1}, Z_{2}, \ldots$ are i.i.d. with $\operatorname{Var}\left(Z_{i}=1\right)$. Calculate $\operatorname{Cor}\left(X_{s}, X_{t}\right)$.

Exercise 4.5 Let $X_{t}=\sum_{k=1}^{t} Z_{k}$ be a general random walk for $t=1,2, \ldots$, $X_{0}=0$, and $Z_{1}, Z_{2}, \ldots$ are i.i.d. and symmetric random variables. Show that
$$\mathrm{P}\left(\max {i \leq t}\left|X{i}\right|>a\right) \leq 2 \mathrm{P}\left(\left|X_{t}\right|>a\right)$$
Exercise 4.6 It is a well known fact that the kernel density estimate $\hat{f_{h}}(x)=$ $n^{-1} \sum_{i=1}^{n} K_{h}\left(x-X_{i}\right)$ is biased. Therefore the comparison of a kernel density estimate with the analytical form of the true e.g. normal, density can be misleading. One has rather to compare the hypothetical density with the expected value $\mathrm{E}{\hat{f}{h}}(x)$ density given as $g(x)=\int_{-\infty}^{+\infty} K_{h}(x-u) f(u) d u$ where $f(u)$ is the true density. Illustrate this fact with a standard normal distribution. Plot the true density $f, a$ kernel density estimate and bias corrected density $g$.

Exercise 4.7 Consider a binomial process $X_{l}=\sum_{k=1}^{t} Z_{k}$ for $t=1,2, \ldots$, $X_{0}=0$, with state dependent increments. Let $\mathrm{P}\left(Z_{l}=1\right)=p\left(X_{l-1}\right)=$ $1 /\left(2^{\left|X_{t-1}\right|+1}\right)$ if $X_{l-1} \geq=0$ and $\mathrm{P}\left(Z_{l}=1\right)=p\left(X_{t-1}\right)=1-1 /\left(2^{\left|X_{t-1}\right|+1}\right)$ otherwise. To complete the setting $\mathrm{P}\left(Z_{l}=-1\right)=1-p\left(X_{l-1}\right)$. Calculate the distribution of $X_{t}$ for the first five steps.

## 统计代写|金融统计代写financial statistics代考|Geometric Random Walks

R吨=X吨Xl−1,吨=1,2,…

X吨=Rl⋅X吨−1=X0⋅圆周率ķ=1吨Rķ

p=1−d在−d.

p=1在+1.

## 统计代写|金融统计代写financial statistics代考|Binomial Models with State Dependent Increments

X吨=X吨−1+从吨,吨=1,2,… 磷(从吨=在)=p(X吨−1,吨),磷(从吨=−d)=1−p(X吨−1,吨)

X吨=R吨⋅X吨−1 磷(R吨=在)=p(X吨−1,吨),磷(R吨=d)=1−p(X吨−1,吨)
(4.6) 和 (4.7) 中定义的过程主要具有理论意义，因为没有进一步的假设，很难估计概率p(X,吨)从观察到的股票价格。但广义二项式模型（以及三项式模型）可用于数值求解微分方程，例如美式期权的 BlackScholes 方程。

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

（一）磷(X一世>0)
(二)磷(X一世>0)
（C）磷(从2=1∣X3=1)

$$\mathrm{P}\left(\max {i \leq t}\left|X {i}\right|>a\right) \leq 2 \mathrm{P}\left(\left|X_ {t}\right|>a\right)$$

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