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# 金融代写|金融工程代考FINANCIAL ENGINEERING代写|MGF657 Black–Scholes Formula

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## 金融代写|金融工程代考FINANCIAL ENGINEERING代写|Black–Scholes Formula

We shall present an outline of the main results for European call and put options in continuous time, culminating in the famous Black-Scholes formula. Our treatment of continuous time is a compromise lacking full mathematical rigour, which would require a systematic study of Stochastic Calculus, a topic treated in detail in more advanced texts. In place of this, we shall exploit an analogy with the discrete time case.

As a starting point we take the continuous time model of stock prices developed in Chapter 3 as a limit of suitably scaled binomial models with time steps going to zero. In the resulting continuous time model the stock price is given by
$$S(t)=S(0) \mathrm{e}^{m t+\sigma W(t)},$$
where $W(t)$ is the standard Wiener process (Brownian motion), see Section 3.3.2. This means, in particular, that $S(t)$ has the log normal distribution.
Consider a European option on the stock expiring at time $T$ with payoff $f(S(T))$. As in the discrete-time case, see Theorem $8.4$, the time 0 price $D(0)$ of the option ought to be equal to the expectation of the discounted payoff $\mathrm{e}^{-r T} f(S(T))$
$$D(0)=E_{}\left(\mathrm{e}^{-r T} f(S(T))\right),$$ under a risk-neutral probability $P_{}$ that turns the discounted stock price process $\mathrm{e}^{-r t} S(t)$ into a martingale. Here we shall accept this formula without proof, by analogy with the discrete time result. (The proof is based on an arbitrage argument combined with a bit of Stochastic Calculus, the latter beyond the scope of this book.)

## 金融代写|金融工程代考FINANCIAL ENGINEERING代写|Black–Scholes Formula

The time $t$ price of a European call with strike price $X$ and exercise time $T$, where $t<T$, is given by
$$C^{\mathrm{E}}(t)=S(t) N\left(d_{1}\right)-X \mathrm{e}^{-r(T-t)} N\left(d_{2}\right)$$

with
$$d_{1}=\frac{\ln \frac{S(0)}{X}+\left(r+\frac{1}{2} \sigma^{2}\right)(T-t)}{\sigma \sqrt{T-t}}, \quad d_{2}=\frac{\ln \frac{S(0)}{X}+\left(r-\frac{1}{2} \sigma^{2}\right)(T-t)}{\sigma \sqrt{T-t}} .$$

## 金融代写|金融工程代考FINANCIAL ENGINEERING代写|Black-Scholes Formula

$$S(t)=S(0) \mathrm{e}^{m t+\sigma W(t)}$$

$$D(0)=E\left(\mathrm{e}^{-T T} f(S(T))\right)$$

## 金融代写|金融工程代考FINANCIAL ENGINEERING代写|Black-Scholes Formula

$$C^{\mathrm{E}}(t)=S(t) N\left(d_{1}\right)-X \mathrm{e}^{-r(T-t)} N\left(d_{2}\right)$$

$$d_{1}=\frac{\ln \frac{S(0)}{X}+\left(r+\frac{1}{2} \sigma^{2}\right)(T-t)}{\sigma \sqrt{T-t}}, \quad d_{2}=\frac{\ln \frac{S(0)}{X}+\left(r-\frac{1}{2} \sigma^{2}\right)(T-t)}{\sigma \sqrt{T-t}}$$

## 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。