Posted on Category:数学代写

# 数学代写|VOLATILITY MODEL代考

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Introduction to the Mathematics of Finance. Take-Home Midterm.
Due March10, 2023, $11.59$ p.m.
Please write a pledge that you did not copy solutions from the work of other students. You can consult TAs if you have any difficulties. This midterm uses base matlab models that can be downloaded from Courseworks.

\begin{prob}

1.Matlab option model. Download from Courseworks matlab option model files BlackScholesStocks.m and BlackScholesGraph.m and put them in the same directory. BlackScholesStocks.m contains the function that calculates the Black Scholes price for options on nondividend paying stocks. BlackScholesGraph.m is a script that makes a graph of option price as a function of stock price. Type at Matlab prompt $>$ BlackScholesGraph and the script will be executed and the graph will appear. Now modify the file BlackScholesStocks.m so that the function now calculates the price of options on stocks paying continuous dividends at a rate q. Modify the file BlackScholesGraph.m so that it now plots graph of a call with the same parameters as before but with the dividend yield $\mathrm{q}=2.5 \%$ and the new strike price 12. Submit printouts of code and graph.

\end{prob}

\begin{prob}

2 Download matlab Brownian motion model from the courseworks. Modify it to Geometric Brownian motion with starting value $\mathrm{Xo}_o=80$, growth rate $\mu=0.05$, volatility $\sigma=0.22$ and 5000 trajectories. Check that the code works. Try out 50,000 trajectories. Try out 100,000 trajectories. Submit the code printout and the graph printout for 5000 trajectories.

\end{prob}

\begin{prob}

1. Using arbitrage arguments explain why the price of an American call option on a stock paying no dividends should be the same as the price of a corresponding European call. Why American calls on a nondividend paying stock should not be exercised early.

\end{prob}

\begin{prob}

1. Why when the stock pays dividends the argument of the problem No.3 can not be used. Give a numerical example (choosing $\mathrm{x}, \mathrm{k}, \mathrm{r}, \mathrm{T}-\mathrm{t}, \sigma$ ) in which it is obvious (without any formulas) that American put price on a nondividend paying stock is larger then the corresponding European put price.

\end{prob}

\begin{prob}

1. (a) The stock price is 60 the volatility of the stock is $25 \%$. Assuming that the time to expiration is 3 months and the interest rate is $2 \%$ per annum calculate the price $\mathrm{P}$ of the European call option with strike 60 .
(b) Calculate $\Delta, \Gamma, \rho$, Vega using formulas for these parameters. Calculate the same parameters approximately using the options calculator.
(c) Check that following relationship holds
$$\Theta+r x \Delta+\frac{1}{2} \sigma^2 x^2 \Gamma=r P$$

\end{prob}

\begin{prob}

1. What are the parameters affecting prices European and American calls and puts. How do the prices change when one of the parameters changes with all the others remaining the same?

\end{prob}

\begin{prob}

1. Suppose that we have three European calls with strikes 30,35 , and 40 and the same maturity 1 month. Their prices are $10.00,8.00,5.00$. Is it possible to do an arbitrage?

\end{prob}

\begin{prob}

1. Suppose that current stock price is $100 \$$. Its annualized volatility is$20 \%$and annualized return$12 \%$i.e. we assume that the stock price follows$\mathrm{dXt}=0.12 \mathrm{Xt} \mathrm{dt}+0.2 \mathrm{Xt} \mathrm{dW} \mathrm{t}\$. Write the probability density function for the stock in 1 year. What is the mean and standard deviation of the terminal stock price? (standard deviation of price, not of return)

\end{prob}

## 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。