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## 金融代写|期权理论代写Mathematical Introduction to Options代考|STOCK PRICE MOVEMENTS

(i) Consider the evolution of a stock price: the prices observed at successive moments in time $\delta t$ apart are: $S_0, S_1, \ldots, S_N$. For simplicity, it is assumed that no dividend is paid in this period. The “price relative” in period $n$ is defined as the ratio $R_n=S_n / S_{n-1}$. This quantity is a random variable and we make the following apparently innocuous but far-reaching assumption: the price relatives are independently and identically distributed. If the price relatives are independently distributed, the next move does not depend on what happened at the last move. This is a statement of the so-called weak form market efficiency hypothesis, which maintains that the entire history of a stock is summed up in its present price; future movements depend only on new information and on changes in sentiment or the environment. People who believe in charts or concepts such as momentum for predicting stock prices clearly do not believe in this hypothesis; in consequence, they should not believe in the results of modern option theory.
If price relatives are identically distributed, then their expected means and variances are constant, i.e.
$$\mathrm{E}\left[\frac{S_n}{S_{n-1}}\right]=\mathrm{E}\left[\frac{S_{n+1}}{S_n}\right] ; \quad \operatorname{var}\left[\frac{S_n}{S_{n-1}}\right]=\operatorname{var}\left[\frac{S_{n+1}}{S_n}\right]$$
The first equation says that the expected growth of the stock remains constant. It will be seen below that the second equation is a statement that the volatility of the stock remains constant.
(ii) Consider the following simple identity:
\begin{aligned} \frac{S_N}{S_0} & \equiv \frac{S_1}{S_0} \frac{S_2}{S_1} \cdots \frac{S_N}{S_{N-1}} \ &=R_1 R_2 \cdots R_N \end{aligned}
Taking the logarithm of each side gives
$$x_N=r_1+r_2+\cdots+r_N \quad \text { where } \quad x_N=\ln \frac{S_N}{S_0} ; \quad r_n=\ln R_n=\ln \frac{S_n}{S_{n-1}}$$
Some very profound conclusions emerge from this trivial-looking equation: if the price relatives $R_n$ are identically distributed and independent, then so are their logarithms, $r_n$. They can be treated as independent random variables, drawn from the same infinite population. It follows that
$$\begin{gathered} \mathrm{E}\left[x_N\right]=\sum_{n=1}^N \mathrm{E}\left[r_n\right]=N \mathrm{E}\left[r_n\right]=N m_{\delta T} \ \operatorname{var}\left[x_N\right]=\sum_{n=1}^N \operatorname{var}\left[r_n\right]=N \operatorname{var}\left[r_n\right]=N \sigma_{\delta T}^2 \end{gathered}$$
where $m_{\delta T}$ and $\sigma_{\delta T}^2$ are the mean and variance of the logarithm of the price relatives.

## 金融代写|期权理论代写Mathematical Introduction to Options代考|PROPERTIES OF STOCK PRICE DISTRIBUTION

In the last section we defined $x_T=\ln S_T / S_0$ and showed that $x_T$ is normally distributed with mean $m T$ and variance $\sigma^2 T . S_T$ is then said to be lognormally distributed. Some of the more useful properties of the normal and lognormal distributions are explained in Appendix A.1.
(i) In terms of $x_t$, the stock price is given by $S_t=S_0 \mathrm{e}^{x_t}$ and the explicit probability distribution function for $x_t$ is
$$n\left(x_t\right)=\frac{1}{\sigma \sqrt{2 \pi t}} \exp \left{-\frac{1}{2}\left(\frac{x_t-m t}{\sigma \sqrt{t}}\right)^2\right}$$

The various moments for the distribution of $S_t$ may be written
\begin{aligned} \mathrm{E}\left[\left(\frac{S_t}{S_0}\right)^\lambda\right] &=\mathrm{E}\left[\mathrm{e}^{\lambda x_t}\right]=\int_{-\infty}^{+\infty} \mathrm{e}^{\lambda x_t} n\left(x_t\right) \mathrm{d} x_t \ &=\mathrm{e}^{\lambda m t+\frac{1}{2} \lambda^2 \sigma^2 t} \end{aligned}
This result is proved in Appendix A.1(v), item (D).
(ii) Mean and Variance of $S_T$ : The rate of return $\mu$ of a non-dividend-paying stock over a time interval $T$ is defined by $\mathrm{E}\left\langle S_T\right\rangle=S_0 \mathrm{e}^{\mu T}$. Using equation (3.3) with $\lambda=1$ gives
$$\mathrm{E}\left[S_T\right]=S_0 \mathrm{e}^{\mu T}=S_0 \mathrm{e}^{\left(m+\frac{1}{2} \sigma^2\right) T}$$
A relationship which simply falls out of the last equation and which is used repeatedly throughout this book is
$$\mu=m+\frac{1}{2} \sigma^2$$
Again using equation (3.3), this time with $\lambda=2$, gives
\begin{aligned} \operatorname{var}\left[\frac{S_T}{S_0}\right] &=\mathrm{E}\left[\frac{S_T}{S_0}\right]^2-\mathrm{E}^2\left[\frac{S_T}{S_0}\right] \ &=\mathrm{e}^{2 \mu T}\left(\mathrm{e}^{\sigma^2 T}-1\right) \end{aligned}
(iii) Variance and Volatility: $x_T$ is a stochastic variable with mean $m T$ and variance $\sigma^2 T . T$ is measured in units of a year and $\sigma$ is referred to as the (annual) volatility of the stock. The reader would be quite right to comment that $\sigma$ should be called the volatility of the logarithm of the stock price; but the two are closely related and for most practical purposes, the same. To see this, consider a small time interval $\delta T$ (maybe a day or a week) and write $S_{\delta T}-S_0=\delta S$.

# 期权理论代写

## 金融代写|期权理论代写期权数学介绍代考|股票价格变动

.股票价格变动

(i)考虑股票价格的演变:在$\delta t$之外的连续时刻观察到的价格为:$S_0, S_1, \ldots, S_N$。为了简单起见，假设在这一时期没有支付红利。在$n$期间的“相对价格”被定义为比率$R_n=S_n / S_{n-1}$。这个量是一个随机变量，我们做了以下看似无伤大雅但意义深远的假设:价格相关性是独立和同分布的。如果价格关系是独立分布的，那么下一步就不取决于上一步发生了什么。这是对所谓弱形式市场效率假说的一种陈述，该假说认为，一只股票的整个历史可以用它当前的价格来总结;未来的走势只取决于新的信息和情绪或环境的变化。那些相信用图表或动量等概念来预测股价的人显然不相信这个假设;因此，他们不应该相信现代期权理论的结果。

$$\mathrm{E}\left[\frac{S_n}{S_{n-1}}\right]=\mathrm{E}\left[\frac{S_{n+1}}{S_n}\right] ; \quad \operatorname{var}\left[\frac{S_n}{S_{n-1}}\right]=\operatorname{var}\left[\frac{S_{n+1}}{S_n}\right]$$

\begin{aligned} \frac{S_N}{S_0} & \equiv \frac{S_1}{S_0} \frac{S_2}{S_1} \cdots \frac{S_N}{S_{N-1}} \ &=R_1 R_2 \cdots R_N \end{aligned}

$$x_N=r_1+r_2+\cdots+r_N \quad \text { where } \quad x_N=\ln \frac{S_N}{S_0} ; \quad r_n=\ln R_n=\ln \frac{S_n}{S_{n-1}}$$

$$\begin{gathered} \mathrm{E}\left[x_N\right]=\sum_{n=1}^N \mathrm{E}\left[r_n\right]=N \mathrm{E}\left[r_n\right]=N m_{\delta T} \ \operatorname{var}\left[x_N\right]=\sum_{n=1}^N \operatorname{var}\left[r_n\right]=N \operatorname{var}\left[r_n\right]=N \sigma_{\delta T}^2 \end{gathered}$$
，其中$m_{\delta T}$和$\sigma_{\delta T}^2$是价格相关对数的均值和方差

## 金融代写|期权理论代写期权数学介绍代考|PROPERTIES OF STOCK – PRICE – DISTRIBUTION

(i)对于$x_t$，股票价格由$S_t=S_0 \mathrm{e}^{x_t}$给出，$x_t$的显式概率分布函数为
$$n\left(x_t\right)=\frac{1}{\sigma \sqrt{2 \pi t}} \exp \left{-\frac{1}{2}\left(\frac{x_t-m t}{\sigma \sqrt{t}}\right)^2\right}$$

\begin{aligned} \mathrm{E}\left[\left(\frac{S_t}{S_0}\right)^\lambda\right] &=\mathrm{E}\left[\mathrm{e}^{\lambda x_t}\right]=\int_{-\infty}^{+\infty} \mathrm{e}^{\lambda x_t} n\left(x_t\right) \mathrm{d} x_t \ &=\mathrm{e}^{\lambda m t+\frac{1}{2} \lambda^2 \sigma^2 t} \end{aligned}该结果在附录A.1(v)，项目(D)中得到证明。
(ii)的均值和方差 $S_T$ :收益率 $\mu$ 在一段时间内不分红的股票 $T$ 定义为 $\mathrm{E}\left\langle S_T\right\rangle=S_0 \mathrm{e}^{\mu T}$。用式(3.3)与 $\lambda=1$ 给出
$$\mathrm{E}\left[S_T\right]=S_0 \mathrm{e}^{\mu T}=S_0 \mathrm{e}^{\left(m+\frac{1}{2} \sigma^2\right) T}$$

$$\mu=m+\frac{1}{2} \sigma^2$$

\begin{aligned} \operatorname{var}\left[\frac{S_T}{S_0}\right] &=\mathrm{E}\left[\frac{S_T}{S_0}\right]^2-\mathrm{E}^2\left[\frac{S_T}{S_0}\right] \ &=\mathrm{e}^{2 \mu T}\left(\mathrm{e}^{\sigma^2 T}-1\right) \end{aligned}
(iii)方差和波动性: $x_T$ 随机变量是有均值的吗 $m T$ 方差 $\sigma^2 T . T$ 是以年为单位的 $\sigma$ 被称为股票的(年度)波动率。读者对此的评论是完全正确的 $\sigma$ 应该称之为股价波动率的对数;但这两者是密切相关的，在大多数实际目的上是相同的。为了理解这一点，考虑一个小的时间间隔 $\delta T$ (可能是一天或一周)并写下来 $S_{\delta T}-S_0=\delta S$.

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