### 金融代写|金融计量经济学代写Financial Econometrics代考|Random Walk Hypothesis and Martingales

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## 金融代写|金融计量经济学Financial Econometrics代考|Random Walk Hypothesis and Martingales

Martingale is often used likewise Random Walk Hypothesis in testing the Efficient Market Hypothesis. Let’s try to understand: What is martingale and how it is different from a Random walk?

Security price changes with the arrival of relevant new information and the arrival of relevant new information is a random process. Hence by the principle of EMH, security price will follow a random walk on arrival of the relevant new information associated with the security. The below Eq. (2.2) shows the degree of random walk followed by the security on arrival of the relevant new information associated with the security.
$$\text { Price }{t+1}-\text { Price }{t}=\varepsilon_{t+1}$$

In practice individual security’s historic prices data is considered in testing the efficient market hypothesis. As a result more generic version of EMH considers “Security prices follow a martingale” as shown below in Eq. ( $2.3)$
$$E\left(\text { Price }{t+1}-\text { Price }{t} \mid \Phi_{t}\right)=0$$
where $\Phi_{t}=$ Price $_{t}$, Price $_{t-1}, \ldots$
Hence, martingales are the random variables and on the basis of martingales’ present state information it is impossible to predict the future variations.

## 金融代写|金融计量经济学Financial Econometrics代考|Background

Robert Brown in the year 1827 first observed that the pollen grains suspended in the water follow a zigzag random motion. The zigzag random motion of these tiny particles suspended in water is known as the Brownian motion. Subsequently Louis Bachelier in his doctoral thesis in the year 1990 established a mathematical model of the stochastic process or Brownian motion or wiener process for valuing the stock options. Louis Bachelier work notably underlined the two fundamental features of Brownian motion namely Markov process and reflection principle as shown below in Eq. 3.1.
$$P\left{\max {0 \leq b \leq t} W(b) \leq \lambda\right}=\frac{1}{\sqrt{2 \pi t}} \int{0}^{\lambda} e^{-x^{2} / 2 t} d x$$
$W$ (b) represents position of Brownian motion at time b whereas the righthand side of the above cquation represents the simple random distribution or probability density function.

Norbert Wiener in the year 1923 formally formulated the mathematical foundation of the Brownian motion. Standardized Brownian motion is often referred to as the Wiener process. Louis Bachelicr is often attributed as the first person to introduce advanced mathematics into the field of finance labelled as the random walk model.

Standardized Brownian motion or Wiener process has these following propertics:

1. $W(0)=0$ represents that the Wiener process starts at the origin at time zero.
2. At any given time $t>0$ the position of Wiener process follows a normal distribution with mean $(\mu)=0$ and variance $\left(\sigma^{2}\right)=t$.
3. The random function or Wiener process $W()$ is a continuous function.
4. The displacement from $W(b)$ to $W(t)$ is time homogencous, independent and non-overlapping random progression.

However, Brownian Motion is not appropriate for modelling stock prices as Brownian Motion can take negative values. A Geometric Brownian Motion is represented by the following Eq. 3.2.
$$d b(t)=\mu \mathrm{b}(\mathrm{t}) d t+\sigma \mathrm{b}(\mathrm{t}) d W(t)$$
where
$b(t)$ is a random or stochastic process.
$\mu$ represents the drift term.
$\sigma$ volatility term.
$W(t)$ represents the Brownian motion or Wiener process.

## 金融代写|金融计量经济学Financial Econometrics代考|Multiple Choice Questions

1. Which of the following is not appropriate for modelling stock prices?
(a) Geometric Brownian Motion
(b) Brownian Motion
2. (c) Both
3. (d) None of the above
1. Which of the following is not belongs to the Greeks’ measures of an option?
(a) Delta
(b) Sigma
(c) Theta
(d) Rho
2. Wiener process is also known as the
(a) Simple Brownian motion
(b) Standardized Brownian motion
(c) Structured Brownian motion
(d) Second order Brownian motion
3. The “fOptions” $R$ package does not include which of the following binomial tree models for valuation of an option?
(a) CRR binomial tree model
(b) JR binomial tree model
(c) TIAN binomial tree model
(d) TRR binomial tree model
4. Which of the following Greeks’ value of an option measures the probable change in the option price for a percentage implied volatility change of the underlying asset?
(a) Delta
(b) Gamma
(c) Vega
(d) Theta
5. Which among the following measures the time decay value of an option?
(a) Delta
(b) Theta
(c) Vega
(d) Gamma

## 金融代写|金融计量经济学Financial Econometrics代考|Random Walk Hypothesis and Martingales

价格 吨+1− 价格 吨=e吨+1

## 金融代写|金融计量经济学Financial Econometrics代考|Background

P\left{\max {0 \leq b \leq t} W(b) \leq \lambda\right}=\frac{1}{\sqrt{2 \pi t}} \int{0}^{\ λ} e^{-x^{2} / 2 t} d xP\left{\max {0 \leq b \leq t} W(b) \leq \lambda\right}=\frac{1}{\sqrt{2 \pi t}} \int{0}^{\ λ} e^{-x^{2} / 2 t} d x

Norbert Wiener 在 1923 年正式制定了布朗运动的数学基础。标准化布朗运动通常被称为维纳过程。Louis Bachelicr 经常被认为是第一个将高等数学引入金融领域的人，被称为随机游走模型。

1. 在(0)=0表示维纳过程在零时刻从原点开始。
2. 在任何给定时间吨>0维纳过程的位置服从均值正态分布(μ)=0和方差(σ2)=吨.
3. 随机函数或维纳过程在()是一个连续函数。
4. 位移从在(b)至在(吨)是时间同质、独立且不重叠的随机级数。

db(吨)=μb(吨)d吨+σb(吨)d在(吨)

b(吨)是一个随机或随机的过程。
μ表示漂移项。
σ波动性术语。

## 金融代写|金融计量经济学Financial Econometrics代考|Multiple Choice Questions

1. 以下哪项不适合模拟股票价格？
(a) 几何布朗运动
(b) 布朗运动
2. (c) 两者
3. (d) 以上都不是
1. 以下哪一项不属于希腊人对期权的衡量？
(a) Delta
(b) Sigma
(c) Theta
(d) Rho
2. 维纳过程也称为
(a) 简单布朗运动
(b) 标准化布朗运动
(c) 结构化布朗运动
(d) 二阶布朗运动
3. “fOptions”R软件包不包括以下哪些用于期权估值的二叉树模型？
(a) CRR 二叉树模型
(b) JR 二叉树模型
(c) TIAN 二叉树模型
(d) TRR 二叉树模型
4. 以下哪项希腊人的期权价值衡量了期权价格在标的资产隐含波动率变化百分比下的可能变化？
(a) Delta
(b) Gamma
(c) Vega
(d) Theta
5. 以下哪一项衡量期权的时间衰减值？
(a) Delta
(b) Theta
(c) Vega
(d) Gamma

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