数学代写|金融衍生品代写Financial derivatives代考|ECON6042

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数学代写|金融衍生品代写Financial derivatives代考|Interest Rate Swap

An interest rate swap (IRS) is a contract that entails periodic payments between two counterparties for the term of the trade. The swap market is very large and liquid. Swaps are used to hedge long term interest rate risks. The market standard IRS (or plain vanilla IRS) is the “Fixed-Floating” trade with two legs:

Fixed leg: one party will pay the cash-flows calculated with a fixed interest rate as predetermined in the contract;
Floating leg (or Variable leg): the other party will pay the cash-flows calculated with a reference (e.g. USD 3M Libor) + spread. For each period, the Libor rate based floating rate will be fixed at the beginning of the period and the cashflow will be paid at the $e n d$ of the period.

In the market vocabulary, the fixed rate payer (receiver) is referred to as the “swap payer” (“swap receiver”). Other combinations (“Fixed-Fixed”, “Floating-Floating”) can be composed by “Fixed-Floating” trades.
An illustration example of a swap is as follows:

The valuation of an interest rate swap in general is based on the cost of replication. The time-0 PV of a receiver swap is simply the PV of the cash-flows of fixed leg minus the PV of the cash-flows of floating leg:
Swap $P V=P V^f-P V^v=\sum_{j=1}^m P\left(t_j\right) s \delta_j-\sum_{i=1}^n P\left(t_i\right) F\left(t_{i-1}, t_i\right) \delta_i$,
where
$m, n$ : respective number of remaining cash-flows for the fixed leg and the floating leg;
$s:$ rate of the fixed leg;
$F\left(t_{i-1}, t_i\right)$ : expected forward rate ${ }^8$ for the $i$-th period $\left(t_{i-1}, t_i\right)$;
$\delta_i$ : the day count fraction for the $i$-th period $\left(t_{i-1}, t_i\right)$.

数学代写|金融衍生品代写Financial derivatives代考|Standard Swap

For standard (Vanilla) IRS, the valuation can be further simplified. From Proposition $2.1$, the market quote for the floating rate is $F\left(t_{i-1}, t_i\right)=\frac{1}{\delta_i}\left(\frac{P\left(t_{i-1}\right)}{P\left(t_i\right)}-1\right)$. Replacing $F\left(t_{i-1}, t_i\right)$ in the PV of the floating leg:
\begin{aligned} P V^v=& \sum_{i=1}^n P\left(t_i\right) F\left(t_{i-1}, t_i\right) \delta_i \ =& {\left[P\left(t_0\right)-P\left(t_1\right)\right]+\left[P\left(t_1\right)-P\left(t_2\right)\right] \cdots+\left[P\left(t_{n-2}\right)-P\left(t_{n-1}\right)\right] } \ &+\left[P\left(t_{n-1}\right)-P\left(t_n\right)\right] \ =& P\left(t_0\right)-P\left(t_n\right) \end{aligned}
At the inception of the swap, $P\left(t_0\right)=1$. Hence, we obtain $P V^v=1-P\left(t_n\right)$. This can be understood by the fact that the PV of an investment receiving regular market floating rate based interests and the initial investment back at the end should just be the value of the initial investment. A market swap is quoted with the fixed rate level which makes the swap valuation at zero at inception, i.e. $P V^f=P V^v$. From the above, we have
$$\sum_{j=1}^m P\left(t_j\right) s \delta_j=1-P\left(t_n\right)$$
Hence the rate is given as
$$s=\frac{1-P\left(t_n\right)}{A\left(t_m\right)}$$
where $A\left(t_m\right)=\sum_{j=1}^m P\left(t_j\right) \delta_j$ is called the annuity factor.

金融衍生品代写

数学代写|金融衍生品代写金融衍生品代考|利率掉期

$m, n$:固定支腿和浮动支腿的剩余现金流的各自数量;
$s:$固定支腿的比率;

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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。

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