### 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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## 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:
\begin{aligned} 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 \end{aligned}
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:
\begin{aligned} \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} \end{aligned}
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

d小号吨=μ(小号l,吨)d吨+σ(小号l,吨)d在l

μ(小号l,吨)=μ⋅小号l

σ(小号l,吨)=σ⋅小号l

d小号吨=μ⋅小号吨d吨+σ⋅小号吨⋅d在吨

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

dX吨=μ(X吨,吨)d吨+σ(X吨,吨)d在l,

(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

(d在l)2=d吨

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

(a)X吨=C−12在Cl为了C>0
(二)是l=在吨+l−在吨为了吨>0
（C）在吨={在吨 如果 吨≤吨 2在吨−在吨 如果 吨>吨
(d)从l=吨在1l为了吨>0和从l=0.

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