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数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|The Linear SDE

In this section we will study the linear SDE, which in the scalar case has the form
\left{\begin{aligned} d X_t &=a X_t d t+\sigma d W_t, \ X_0 &=x_0 . \end{aligned}\right.
This equation turns up in various physical applications, and we will also meet it below in connection with interest rate theory.

In order to get some feeling for how to solve this equation we recall that the linear ODE
$$\frac{d x_t}{d t}=a x_t+u_t,$$
where $u$ is a deterministic function of time, has the solution
$$x_t=e^{a t} x_0+\int_0^t e^{a(t-s)} u_s d s .$$
If we, for a moment, reason heuristically, then it is tempting to formally divide eqn (5.17) by $d t$. This would (formally) give us
$$\frac{d X_t}{d t}=a X_t+\sigma \frac{d W_t}{d t},$$
and, by analogy with the ODE above, one is led to conjecture the formal solution
$$X_t=e^{a t} X_0+\sigma \int_0^t e^{a(t-s)} \frac{d W_s}{d s} d s=e^{a t} X_0+\sigma \int_0^t e^{a(t-s)} d W_s .$$
Generally speaking, tricks like this will not work, since the solution of the ODE is based on ordinary calculus, whereas we have to use Itô calculus when dealing with SDEs. In this case, however, we have a linear structure, which means that the second order term in the Itô formula does not come into play. Thus the solution of the linear SDE is indeed given by the heuristically derived formula above. We formulate the result for a slightly more general situation, where we allow $X$ to be vector-valued.

数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|The Infinitesimal Operator

Consider, as in Section 5.1, the $n$-dimensional SDE
$$d X_t=\mu\left(t, X_t\right) d t+\sigma\left(t, X_t\right) d W_t .$$
Through the Itô formula, the process above is closely connected to a partial differential operator $\mathcal{A}$, defined below. The next two sections are devoted to investigating the connections between, on the one hand, the analytical properties of the operator $\mathcal{A}$, and on the other hand the probabilistic properties of the process $X$ above.

Definition $5.4$ Given the SDE in (5.21), the partial differential operator $\mathcal{A}$, referred to as the infinitesimal operator of $X$, is defined, for any function $h(x)$ with $h \in C^2\left(R^n\right)$, by
$$\mathcal{A} h(t, x)=\sum_{i=1}^n \mu_i(t, x) \frac{\partial h}{\partial x_i}(x)+\frac{1}{2} \sum_{i, j=1}^n C_{i j}(t, x) \frac{\partial^2 h}{\partial x_i \partial x_j}(x),$$
where as before
$$C(t, x)=\sigma(t, x) \sigma^{\star}(t, x) .$$

数学代写连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代 考|The Linear SDE

$\$ \$$left {$$
d X_t=a X_t d t+\sigma d W_t, X_0=x_0 .
$$\正确的。 \ \$$

$$\frac{d x_t}{d t}=a x_t+u_t,$$

$$x_t=e^{a t} x_0+\int_0^t e^{a(t-s)} u_s d s .$$

$$\frac{d X_t}{d t}=a X_t+\sigma \frac{d W_t}{d t},$$

$$X_t=e^{a t} X_0+\sigma \int_0^t e^{a(t-s)} \frac{d W_s}{d s} d s=e^{a t} X_0+\sigma \int_0^t e^{a(t-s)} d W_s .$$

数学代写连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代 考|The Infinitesimal Operator

$$d X_t=\mu\left(t, X_t\right) d t+\sigma\left(t, X_t\right) d W_t .$$

$$\mathcal{A} h(t, x)=\sum_{i=1}^n \mu_i(t, x) \frac{\partial h}{\partial x_i}(x)+\frac{1}{2} \sum_{i, j=1}^n C_{i j}(t, x) \frac{\partial^2 h}{\partial x_i \partial x_j}(x),$$

$$C(t, x)=\sigma(t, x) \sigma^{\star}(t, x)$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Arbitrage Pricing in Continuous Time, 数学代写, 连续时间的期权定价理论

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数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|STOCHASTIC INTEGRALS

The purpose of this book is to study asset pricing on financial markets in continuous time. We thus want to model asset prices as continuous time stochastic processes, and the most complete and elegant theory is obtained if we use diffusion processes and stochastic differential equations as our building blocks. What, then, is a diffusion?

Loosely speaking we say that a stochastic process $X$ is a diffusion if its local dynamics can be approximated by a stochastic difference equation of the following type.
$$X(t+\Delta t)-X(t)=\mu(t, X(t)) \Delta t+\sigma(t, X(t)) Z(t) .$$
Here $Z(t)$ is a normally distributed disturbance term which is independent of everything which has happened up to time $t$, while $\mu$ and $\sigma$ are given deterministic functions. The intuitive content of (4.1) is that, over the time interval $[t, t+\Delta t]$, the $X$-process is driven by two separate terms.

A locally deterministic velocity $\mu(t, X(t))$.

A Gaussian disturbance term, amplified by the factor $\sigma(t, X(t))$.
The function $\mu$ is called the (local) drift term of the process, whereas $\sigma$ is called the diffusion term. In order to model the Gaussian disturbance terms we need the concept of a Wiener process.

数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|Information

Let $X$ be any given stochastic process. In the sequel it will be important to define “the information generated by $X$ ” as time goes by. To do this in a rigorous fashion is outside the main scope of this book, but for most practical purposes the following heuristic definitions will do nicely. See the appendices for a precise treatment.

Definition 4.2 The symbol $\mathcal{F}t^X$ denotes “the information generated by $X$ on the interval $[0, t]$ “, or alternatively “what has happened to $X$ over the interval $[0, t]$ “. If, based upon observations of the trajectory ${X(s) ; 0 \leq s \leq t}$, it is possible to decide whether a given event $A$ has occurred or not, then we write this as $$A \in \mathcal{F}_t^X,$$ or say that “A is $\mathcal{F}_t^X$-measurable”. If the value of a given stochastic variable $Z$ can be completely determined given observations of the trajectory ${X(s) ; 0 \leq s \leq t}$, then we also write $$Z \in \mathcal{F}_t^X .$$ If $Y$ is a stochastic process such that we have $$Y(t) \in \mathcal{F}_t^X$$ for all $t \geq 0$ then we say that $Y$ is adapted to the filtration $\left{\mathcal{F}_t^X\right}{t \geq 0}$. For brevity of notation, we will sometimes write the filtration as $\left{\mathcal{F}t^X\right}{t \geq 0}=\mathbf{F}$.

数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代 孝|STOCHASTIC INTEGRALS

$$X(t+\Delta t)-X(t)=\mu(t, X(t)) \Delta t+\sigma(t, X(t)) Z(t) .$$

数学代写连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代 考|Information

$$A \in \mathcal{F}_t^X,$$

$$Z \in \mathcal{F}_t^X .$$

$$Y(t) \in \mathcal{F}_t^X$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Arbitrage Pricing in Continuous Time, 数学代写, 连续时间的期权定价理论

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数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|Portfolios and Arbitrage

We will study the behaviour of various portfolios on the $(B, S)$ market, and to this end we define a portfolio as a vector $h=(x, y)$. The interpretation is that $x$ is the number of bonds we hold in our portfolio, whereas $y$ is the number of units of the stock held by us. Note that it is quite acceptable for $x$ and $y$ to be positive as well as negative. If, for example, $x=3$, this means that we have bought three bonds at time $t=0$. If on the other hand $y=-2$, this means that we have sold two shares of the stock at time $t=0$. In financial jargon we have a long position in the bond and a short position in the stock. It is an important assumption of the model that short positions are allowed.
Assumption 2.1.1 We assume the following institutional facts.

• Short positions, as well as fractional holdings, are allowed. In mathematical terms this means that every $h \in R^2$ is an allowed portfolio.
• There is no bid-ask spread, i.e. the selling price is equal to the buying price of all assets.
• There are no transactions costs of trading.
• The market is completely liquid, i.e. it is always possible to buy and/or sell unlimited quantities on the market. In particular it is possible to borrow unlimited amounts from the bank (by selling bonds short).

数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|Contingent Claims

Let us now assume that the market in the preceding section is arbitrage free. We go on to study pricing problems for contingent claims.

Definition 2.7 A contingent claim (financial derivative) is any stochastic variable $X$ of the form $X=\Phi(Z)$, where $Z$ is the stochastic variable driving the stock price process above.

We interpret a given claim $X$ as a contract which pays $X$ SEK to the holder of the contract at time $t=1$. See Fig. 2.2, where the value of the claim at each node is given within the corresponding box. The function $\Phi$ is called the contract function. A typical example would be a European call option on the stock with strike price $K$. For this option to be interesting we assume that $s dK$ then we use the option, pay $K$ to get the stock and then sell the stock on the market for $s u$, thus making a net profit of $s u-K$. If $S_1<K$ then the option is obviously worthless. In this example we thus have
$$X= \begin{cases}s u-K, & \text { if } Z=u, \ 0, & \text { if } Z=d,\end{cases}$$
and the contract function is given by
\begin{aligned} &\Phi(u)=s u-K, \ &\Phi(d)=0 . \end{aligned}

数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代 考|Contingent Claims

$$X={s u-K, \quad \text { if } Z=u, 0, \quad \text { if } Z=d,$$

$$\Phi(u)=s u-K, \quad \Phi(d)=0 .$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。