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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|期权理论代写Mathematical Introduction to Options代考|INFINITESIMAL PRICE MOVEMENTS

(i) Let us return to equation (3.7) for the stock price evolution, and consider only small time intervals $\delta t$. We may write that equation as
$$S_t+\delta S_t=S_t \mathrm{e}^{m \delta t+\sigma \delta W_t} \quad \text { where } \quad \delta W_t=\sqrt{\delta t} z_{\delta t}$$
We are dealing with small time periods and small price changes so that we may use the standard expansion $\mathrm{e}^a=1+a+\frac{1}{2 !} a^2+\cdots$ in the last equation, giving
$$S_t+\delta S_t=S_t\left{1+\left(m \delta t+\sigma \delta W_t\right)+\frac{1}{2}\left(m \delta t+\sigma \delta W_t\right)^2+\cdots\right}$$
Normally, one might expect to drop the squared and higher terms in this equation; but recall the definition $\delta W_t=\sqrt{\delta t} z_{\delta t}$. The term $z_{\delta t}$ is a standard normal variate, taking the values $-1$ to $+1$ for about $67 \%$ of the time; values $-2$ to $+2$ for about $95 \%$ of the time; values $-3$ to $+3$ for about $99.5 \%$ of the time, etc. $\delta W_t$ is therefore not of the same order as $\delta t$ (written $\mathrm{O}[\delta t]$ ); it is $\mathrm{O}[\sqrt{\delta t}]$. To be consistent in the last equation then, we need to retain terms up to $\delta t$ together with terms up to $\delta W_t^2$. This gives us
$$\frac{\delta S_t}{S_t}=m \delta t+\sigma \delta W_t+\frac{1}{2} \sigma^2 \delta W_t^2$$
(ii) An appreciation of the significance of the last term in this equation is obtained by analyzing the following expectations and variances of powers of $\delta W_t$. First, recall from Appendix A.1(ii) that the moment generating function for a standard normal distribution is $\mathrm{M}[\Theta]=\mathrm{e}^{\Theta^2 / 2}$; the
$$\begin{gathered} \mathrm{E}\left[\delta W_t\right]=\sqrt{\delta t} \mathrm{E}\left[z_{\delta t}\right]=0 \ \operatorname{var}\left[\delta W_t\right]=\mathrm{E}\left[\delta W_t^2\right]=\delta t \mathrm{E}\left[z_{\delta t}^2\right]=\delta t \ \operatorname{var}\left[\delta W_t^2\right]=\mathrm{E}\left[\delta W_t^4\right]-\mathrm{E}^2\left[\delta W_t^2\right]=\delta t^2 \mathrm{E}\left[z_{\delta t}^4\right]-\delta t^2=2 \delta t^2 \end{gathered}$$
The quantity $\delta W_t^2$ has expected value $\delta t$ and variance proportional to $\delta t^2$. Thus as $\delta t \rightarrow 0$ the variance of $\delta W_t^2$ approaches zero much faster than $\delta t$ itself. But as the variance of $\delta W_t^2$ approaches zero, $\delta W_t^2$ approaches its expected value with greater and greater certainty, i.e. it ceases to behave like a random variable at all. This permits us to make the substitution
$$\lim _{\delta t \rightarrow 0} \delta W_t^2 \rightarrow \mathrm{E}\left[\delta W_t^2\right]=\delta t$$

金融代写|期权理论代写Mathematical Introduction to Options代考|ITO’S LEMMA

In the last section it was seen that an infinitesimal stock price movement $\delta S_t$ in an infinitesimal time interval $\delta t$ could be described by the Wiener process $\delta S_t=S_t(\mu-q) \delta t+S_t \sigma \delta W_t$. A more generalized Wiener process (also known as an Ito process) can be written
$$\delta S_t=a_{S_t t} \delta t+b_{S_t t} \delta W_t$$
where $a_{S_t t}$ and $b_{S_t t}$ are now functions of both $S_t$ and $t$. Consider any function $f_{S_t t}$ of $S_t$ and $t$, which is reasonably well behaved (i.e. adequately differentiable with respect to $S_t$ and $t$ ). Taylor’s theorem states that
$$\delta f_t=\frac{\partial f_t}{\partial S_t} \delta S_t+\frac{\partial f_t}{\partial t} \delta t+\frac{1}{2}\left{\frac{\partial^2 f_t}{\partial S_t^2} \delta S_t^2+\frac{\partial^2 f_t}{\partial S_t \partial t} \delta S_t \delta t+\frac{\partial^2 f_t}{\partial t^2} \delta t^2\right}+\cdots$$
where the subscript notation has been lightened a little for the sake of legibility. Substitute for $S_t$ from the generalized Wiener process and retain only terms of order $\delta t$ or lower, remembering that $\delta W_t \sim \mathrm{O}[\sqrt{\delta t}]$ :
$$\delta f_t=\frac{\partial f_t}{\partial S_t} \delta S_t+\frac{\partial f_t}{\partial t} \delta t+\frac{1}{2} \frac{\partial^2 f_t}{\partial S_t^2} b_t^2 \delta W_t^2$$
Put $\delta W_t^2 \rightarrow \delta t$ as explained in Section 3.3, to give
$$\delta f_t=\left(\frac{\partial f_t}{\partial t}+a_t \frac{\partial f_t}{\partial S_t}+\frac{1}{2} b_t^2 \frac{\partial^2 f_t}{\partial S_t^2}\right) \delta t+b_t \frac{\partial f_t}{\partial S_t} \delta W_t$$
This result is known as Ito’s lemma and is one of the cornerstones of option theory. It basically says that if $f_t$ is any well-behaved function of an Ito process and of time, then $f_t$ itself follows an Ito process. The function of particular interest in this book is the price of a derivative.
In the case of the simple Wiener process of equation (3.9), Ito’s lemma becomes
$$\delta f_t=\left(\frac{\partial f_t}{\partial t}+(\mu-q) S_t \frac{\partial f_t}{\partial S_t}+\frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 f_t}{\partial S_t^2}\right) \delta t+\sigma S_t \frac{\partial f_t}{\partial S_t} \delta W_t$$

期权理论代写

金融代写|期权理论代写期权数学介绍代考|INFINITESIMAL PRICE moves

.

$$S_t+\delta S_t=S_t \mathrm{e}^{m \delta t+\sigma \delta W_t} \quad \text { where } \quad \delta W_t=\sqrt{\delta t} z_{\delta t}$$我们正在处理小的时间段和小的价格变化，所以我们可以使用标准展开 $\mathrm{e}^a=1+a+\frac{1}{2 !} a^2+\cdots$ 在上一个方程中，给出
$$S_t+\delta S_t=S_t\left{1+\left(m \delta t+\sigma \delta W_t\right)+\frac{1}{2}\left(m \delta t+\sigma \delta W_t\right)^2+\cdots\right}$$通常，人们可能期望在这个方程中去掉平方项和更高的项;但是回想一下定义 $\delta W_t=\sqrt{\delta t} z_{\delta t}$。这个术语 $z_{\delta t}$ 标准正态变量，取这些值吗 $-1$ 到 $+1$ 大约 $67 \%$ 时间的;价值观 $-2$ 到 $+2$ 大约 $95 \%$ 时间的;价值观 $-3$ 到 $+3$ 大约 $99.5 \%$ 时间的，等等。 $\delta W_t$ 因此，与 $\delta t$ (书面) $\mathrm{O}[\delta t]$ );是的 $\mathrm{O}[\sqrt{\delta t}]$。为了在上一个方程中保持一致，我们需要保留到 $\delta t$ 连同条款至 $\delta W_t^2$。得到
$$\frac{\delta S_t}{S_t}=m \delta t+\sigma \delta W_t+\frac{1}{2} \sigma^2 \delta W_t^2$$(ii)通过分析下列期望和幂的方差，可以了解这个方程中最后一项的显著性 $\delta W_t$。首先，回顾附录a .1(ii)，标准正态分布的矩产生函数为 $\mathrm{M}[\Theta]=\mathrm{e}^{\Theta^2 / 2}$;
$$\begin{gathered} \mathrm{E}\left[\delta W_t\right]=\sqrt{\delta t} \mathrm{E}\left[z_{\delta t}\right]=0 \ \operatorname{var}\left[\delta W_t\right]=\mathrm{E}\left[\delta W_t^2\right]=\delta t \mathrm{E}\left[z_{\delta t}^2\right]=\delta t \ \operatorname{var}\left[\delta W_t^2\right]=\mathrm{E}\left[\delta W_t^4\right]-\mathrm{E}^2\left[\delta W_t^2\right]=\delta t^2 \mathrm{E}\left[z_{\delta t}^4\right]-\delta t^2=2 \delta t^2 \end{gathered}$$

$$\lim _{\delta t \rightarrow 0} \delta W_t^2 \rightarrow \mathrm{E}\left[\delta W_t^2\right]=\delta t$$

金融代写|期权理论代写Options数学介绍代考|ITO ‘S LEMMA

.

$$\delta S_t=a_{S_t t} \delta t+b_{S_t t} \delta W_t$$
，其中$a_{S_t t}$和$b_{S_t t}$现在是$S_t$和$t$的函数。考虑$S_t$和$t$的任何函数$f_{S_t t}$，它表现得相当好(即对$S_t$和$t$可充分微分)。泰勒定理指出
$$\delta f_t=\frac{\partial f_t}{\partial S_t} \delta S_t+\frac{\partial f_t}{\partial t} \delta t+\frac{1}{2}\left{\frac{\partial^2 f_t}{\partial S_t^2} \delta S_t^2+\frac{\partial^2 f_t}{\partial S_t \partial t} \delta S_t \delta t+\frac{\partial^2 f_t}{\partial t^2} \delta t^2\right}+\cdots$$
，为了便于阅读，下标符号略去了一些。将广义维纳过程中的$S_t$替换为$\delta t$或更低阶项，记住$\delta W_t \sim \mathrm{O}[\sqrt{\delta t}]$:
$$\delta f_t=\frac{\partial f_t}{\partial S_t} \delta S_t+\frac{\partial f_t}{\partial t} \delta t+\frac{1}{2} \frac{\partial^2 f_t}{\partial S_t^2} b_t^2 \delta W_t^2$$

$$\delta f_t=\left(\frac{\partial f_t}{\partial t}+a_t \frac{\partial f_t}{\partial S_t}+\frac{1}{2} b_t^2 \frac{\partial^2 f_t}{\partial S_t^2}\right) \delta t+b_t \frac{\partial f_t}{\partial S_t} \delta W_t$$

$$\delta f_t=\left(\frac{\partial f_t}{\partial t}+(\mu-q) S_t \frac{\partial f_t}{\partial S_t}+\frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 f_t}{\partial S_t^2}\right) \delta t+\sigma S_t \frac{\partial f_t}{\partial S_t} \delta W_t$$

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